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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 ;
B is being_line implies B is being_line
a <> c
T c= S
D c= B
c in dom ( f | ( c | ( c | n ) ) ;
b <> c ;
X is being_line ;
b in D ;
x = e * e ;
let m be Nat ;
h is one-to-one implies h is one-to-one
N in K ;
let i be element ;
j = 1 - 1 ;
x = u . x ;
let n be Nat ;
let k be Nat ;
y in A ;
let x be element ;
let x be element ;
m m c= y ;
F is onto ;
let q be element ;
m = 1 |^ m ;
1 < k ;
G is finite implies G is finite
b in A ;
d divides a
i < n ;
s . b <= b . b ;
b in B ;
let r be Real ;
B is finite implies B is finite
R is total
x = 2 * x ;
d in D ;
let c be element ;
let c be element ;
b = Y or b = Y ;
0 < k ;
let b be Real ;
let n be Nat ;
r <= b ;
x in X ;
i >= 8 ;
let n be Nat ;
let n be Nat ;
y in f .: ( X \/ Y ) ;
let n be Nat ;
1 < j ;
a in L ;
C is being_line ;
a in A ;
1 < x ;
S is finite ;
u in I ;
z *' *' *' *' *' *' *' *' *' *' *' *' *' *' *' *' *' *' *' *'
x in V ;
r < t ;
let t be element ;
x c= y ;
a <= b ;
m in NAT ;
assume f is one-to-one ;
x in Y ;
z = + + + + + + + - - \infty ;
let k be Nat ;
K is being_line ;
assume n >= N ;
assume n >= N ;
assume X is positive-implicative ;
assume x in I ;
q is yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding yielding
assume c in { c } ;
1-p > 0 ;
assume x in Z ;
assume x in Z ;
1 <= kkkkkkkkkkkkkkkkkkk
assume m <= i ;
assume G is finite ;
assume a divides b ;
assume that P [ n ] ;
cn > 0 ;
assume q in A ;
W is bounded ;
f is onto implies f is one-to-one
assume A is being_line ;
g is one-to-one implies g is one-to-one
assume i > j ;
assume t in X ;
assume n <= m ;
assume x in W ;
assume r in X ;
assume x in A ;
assume b is odd ;
assume i in I ;
assume 1 <= k ;
X is non empty implies X is non empty
assume x in X ;
assume n in M ;
assume b in X ;
assume x in A ;
assume T c= W ;
assume that s is universal and s is universal ;
b `2 <= c `2 ;
A meets W /\ W ;
i + j <= j ;
assume that H is universal and H is universal ;
assume x in X ;
let X be set ;
let T be DecoratedTree of D ;
let d be element ;
let t be element ;
let x be element ;
let x be element ;
let s be element ;
k <= 12 ;
let X be set ;
let X be set ;
let y be element ;
let x be element ;
P [ 0 ] ;
let E be set , f be Function ;
let C be Subset of T ;
let x be element ;
let k be Nat ;
let x be element ;
let x be element ;
let e be element ;
let x be element ;
P [ 0 ] ;
let c be element ;
let y be element ;
let x be element ;
let a be Real ;
let x be element ;
let X be set ;
P [ 0 ] ;
let x be element ;
let x be element ;
let y be element ;
r in { r } ;
let e be element ;
n is being_line implies n is being_line
Q [ n + 1 ] ;
x in [#] hhhgh ;
M < m + 1 ;
T is open ;
z in b *^ a *^ a *^ b *^ a *^ a *^ b *^ a *^ b *^ a *^ b *^ a *^ b
R is well-ordering
1 <= k + 1 ;
i > n + 1 ;
q is one-to-one implies q is one-to-one
let x be element ;
P is one-to-one implies P is one-to-one
n <= n + 1 ;
1 <= k + 1 ;
1 <= k + 1 ;
let e be Real ;
i < i + 1 ;
p in P ;
p in K ;
y in C ;
k + 1 <= n + 1 ;
let a be Real ;
X |- p => q ;
x in A ;
let n be Nat ;
let k be Nat ;
let k be Nat ;
let m be Nat ;
0 < k + 1 ;
f | ( x | ( x | ( y | ( x | ( x | ( y | ( x | ( x | (
let x be element ;
let E be Ordinal ;
o , b1 // o , b1 ;
O <> 0 ;
let r be Real ;
f be Function ;
let i be set ;
let n be Nat ;
A c= A ;
L c= bool bool bool X ;
A /\ M = M /\ ( M /\ ( M /\ A ) ) ;
let V be VectSp of K ;
s in Y ^ Z ;
rng f <= rng f ;
b "/\" c "/\" b = b "/\" c ;
m = m * n ;
t in D . ( ( h . ( D . ( ) ) ) ;
P [ 0 ] ;
assume z = x * y ;
S . ( n + 1 ) is bounded ;
let V be non empty VectSpStr over K ;
P [ 0 ] ;
P [ 0 ] ;
C is being_line ;
H = G . i ;
1 <= i + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
-a -a -a -a -a -a -a -a -a -a -a -a -a -a -a -a -a . x -a -a . x -a . x -a
R [ 0 ] ;
b in X .: X ;
assume q = p . ( n + 1 ) ;
x in \Omega V ;
f . ( u , v ) = 0 ;
assume that that e > 0 and 0 < e ;
let V be non empty VectSpStr Str ;
s is empty implies s is empty
dom c = Q ;
P [ 0 ] ;
f . ( n + 1 ) in T ;
N . j in S ;
let T be TopSpace ;
the carrier of F is map of F ;
sgn ( x , 1 ) = sgn ( x , 1 ) ;
k in support a ;
1 in Seg 1 ;
rng f = X ;
len T in T ;
v < bb ;
{ S : S is bounded } ;
assume p = q `1 ;
len f = n ;
assume x in P ;
i in dom q ;
let U be Subset of T ;
p = c . p ;
j in dom h ;
let k be Nat ;
f | Z is continuous ;
k in dom G ;
UBD C = UBD C ;
1 <= len M ;
p in \mathop { \rm Ball } ( x , r ) ;
1 <= j ;
set A = QC-alphabet WFF WFF ;
card X c= c / a ;
e in rng f ;
assume B (+) A is (+) B ;
H |= ( H => '&' '&' '&' '&' '&' '&' '&' 'not' 'not' 'not' H ) ;
assume that n <= m ;
T is ordinal implies T is limit_ordinal
e <> 0 ;
Z c= dom g ;
dom p = X ;
H is being_line implies H is being_line
i + 1 <= n + 1 ;
v <> 0. V ;
A c= conv ( A ) ;
S c= dom F ;
m in dom f ;
X be set ;
c = sup rng f ;
R is connected iff R is connected
assume x in { x } ;
Im f is bounded ;
x in Int Cl Cl Int Cl Cl Int Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl
dom F = M ;
a in W *^ W ;
assume e in A ;
C c= C ;
m <> 0 ;
let x be Element of Y ;
let f be be yielding Function ;
n in Seg 3 ;
assume X in X .: A ;
assume that p <= m and m <= n and n <= m ;
assume u in { v } ;
d is Element of A ;
A ^ B misses B ;
e in v `1 ;
- - y in I - I ;
let A be Subset of T ;
P1 [ 0 ] ;
assume r in F ( k ) ;
assume f is is_integrable_on M ;
let A be Subset of T ;
rng f c= dom f ;
assume that P [ k + 1 ] ;
f <> {} ;
o be Ordinal ;
assume x is is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_is_.|| .|| f ;
assume v in { v } ;
let I be set ;
assume 1 < j and j < l ;
v = - u - v ;
assume that s . b > 0 and s . b > 0 ;
d in { d } ;
assume t . ( 1 - 1 ) in A ;
Y is open implies Y is open
assume a in ]. s , t .[ ;
let S be non empty ManySortedSign ;
a , b // a , b ;
a * b = p * a ;
assume x in the carrier of X ;
assume x in \Omega ( ( \Omega ) ) | ( \Omega ( ( ( ( TOP-REAL 2 ) | ( ( \Omega ) ) | ( \Omega (
a in X ;
m <> {} ;
M + N c= M + N ;
assume M is cf which is cf yielding ;
assume f is not empty ;
let x , y be element ;
let T be non empty TopSpace ;
b , c // a , b ;
k in dom p ;
v be Element of V ;
T in T ;
assume len p = n ;
assume C in rng f ;
k = ( ( ( k + 1 ) ) * ( k + 1 ) ) * ( k + 1 ) ;
m + 1 < n + 1 ;
s in S \/ S ;
n + 1 >= i + 1 ;
assume Im ( y - z ) . 0 = 0 ;
k <= j ;
f | X is continuous ;
f . ( x / b ) / ( x / b ) / ( x / b ) < b / ( x / b ) ;
assume y in dom h ;
x * y in B * ( B * A ) ;
set X = Seg n ;
1 <= i + 1 ;
k + 1 <= k + 1 ;
p ^ q = p ^ q ;
j mod m divides y ;
set m = max ( m , n ) ;
[ x , y ] in R ;
assume x in succ 0 ;
a in sup phi phi phi phi phi phi phi phi phi phi phi phi phi phi phi phi phi phi phi phi phi phi phi phi phi phi
C is being_line ;
{ q } c= C
a < c / 2 ;
s . ( 2 * s ) is DataLoc of SCMPDS ;
IC s = IC s ;
s . ( s . ( s . ( ) ) ) = s . ( s . s ) ;
let V be non empty VectSpStr over K ;
let x , y be element ;
let x be Element of T ;
assume a in rng F ;
x in dom T ;
S , S be Element of L ;
y <> 0 ;
y <> 0 ;
0 = 0. V ;
x2 <> 0. V ;
R [ 0 ] ;
let a , b , c be Real ;
let a be Element of C ;
let x be Element of G ;
let o be OperSymbol of C , a be Element of C ;
r (#) P = P (#) P ;
let i , j be Nat ;
let s be State of A , f be Function of A , B ;
s . ( n + 1 ) = N . ( n + 1 ) ;
set y = ( x * y ) * ( x * y ) ;
reconsider i = i as Element of dom g ;
l . ( l + 1 ) = ( l . ( l + 1 ) ;
|. g . y .| <= |. g . y .| ;
f . ( x in C ;
V is open ;
let x be Element of X ;
0 <> f . ( ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g .
f2 /* ( f2 /* seq ) is convergent ;
f . ( i ) is measurable ;
assume \xi in N ;
reconsider i = i as Element of NAT ;
r * v / ( 0 - r ) = 0 ;
rng f c= dom f ;
set G = the InstructionsF of SCM+FSA ;
let A be Subset of X ;
assume that A is open and A is open and A is open ;
|. f . x .| <= |. f . x .| ;
let x be Element of R ;
let b be Element of L ;
assume x in { W } ;
P [ k + 1 ] ;
let X be Subset of L ;
let b be Element of B ;
let A , B , C be Matrix of K ;
set X = \mathop { \rm Vars } ( C ) ;
let o be OperSymbol of S ;
let R be connected RelStr ;
n + 1 = succ n ;
{ x } c= Z ;
dom f = the carrier of C ;
assume that a in X and b in X ;
Im ( s ) is convergent ;
assume that a = b * a ;
A = Cl ( Int Cl Cl Cl Cl Cl A ) ;
a <= b or a <= a ;
n + 1 in dom f ;
let F be set ;
assume that r > 0 and r > 0 ;
Y be non empty set ;
2 * x in W ;
m in dom ( g | ( m + 1 ) ) ;
n in dom ( g | n ) ;
k + 1 in dom f ;
not not bound bound bound bound variables ( A ) in still_not-bound_in s ;
assume that x <> 0. X and y <> 0. X ;
v in { V } ;
b `2 <> b ;
i + 1 = i + 1 ;
T T c= ( T ^ ( T ^ T ) ) ^ T ;
( l ) ^2 = ( ( ( l ) ^2 ) ^2 ;
let n be Nat ;
( t `2 ) ^2 = ( t `2 ) ^2 ;
A is integrable ;
set t = ( t . x ) . x ;
let A , B , C be set ;
k <= len G ;
C misses C ;
\prod s is being_line
e <= f . e ;
ex q being sequence of NAT st q is finite & q is continuous & q is continuous ;
assume c = b . c ;
assume h in [. q , p .] ;
1 + 1 <= len C ;
c in B ( B ) ;
cluster R .: R .: X -> empty ;
p . ( n + 1 ) = ( p . ( n + 1 ) ;
assume v is ||. v .|| ;
IC SCM+FSA = IC SCM+FSA ;
k in N or K in N or K in N ;
F \/ F \/ F c= F \/ F ;
Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int Int
( z ) . ( ( z ) ) ^2 = ( ( ( z ) ) ^2 ;
p <> 0. TOP-REAL 2 ;
assume z in { y } ;
MaxADSet ( MaxADSet ( a ) ) c= MaxADSet ( a ) ;
sup sup ( s , S ) in S ;
f . x <= f . ( x - y ) ;
let T be non empty TopSpace ;
q `2 >= 1 ;
a >= b & b >= a & a >= b ;
assume that a <> b and c <> b ;
F . ( c . c ) = g . c ;
G is onto implies G is one-to-one
A \/ B c= B \/ C ;
0 = 0 ;
I be Instruction of S ;
( f . ( 1 - 1 ) ) = ( f . ( 1 - 1 ) ) . ( 1 - 1 ) ;
assume z \ x \ 0. X = 0. X ;
C = card C ;
let B be Subset of \Sigma ;
assume that X = p ^ q ;
n + 1 in { l + 1 } ;
f .: ( P /\ Q ) is compact ;
assume that x in { 1 + 1 } and y in { 1 + 1 } ;
p = 0. K ;
M . ( M . ( M . ( 0 qua Nat ) ) = ( M . ( 0 qua Nat ) ) . 0 ;
( ( \varphi *^ \varphi ) *^ \varphi in \varphi ;
MB is opers_closed ;
assume that z <> 0. L and 0. L <> 0. L ;
n < N ;
0 <= s . ( s . ( n + 1 ) ) ;
- ( q - p ) - p = - p - q ;
v in B ;
set g = f | A ;
R is finite
set R = the carrier of R ;
{ p } c= P ;
x in ]. 0 , 1 .[ ;
f . y in dom F ;
T be topological space ;
inf the carrier of S in the carrier of S ;
types types a = types a "\/" b ;
P , Q , P , Q , P , Q , P , Q , P , Q , Q , Q , Q , P , Q , Q , P , Q , Q ,
assume x in { r } ;
2 |^ m < m |^ m ;
x + y = z + y ;
x \ ( x \ y ) = x \ ( x \ y ) ;
||. \mathopen .|| <= ||. y-y .|| ;
assume Y c= field Q ;
a , b // a , b ;
assume a in A ( i ) ;
k in dom ( q | ( k + 1 ) ) ;
p is not empty ;
i -' 1 -' 1 = i-1 -' 1 ;
f | A is one-to-one implies f | A is one-to-one
assume x in f .: X ;
i - 1 - 1 - 1 / 2 < i - 1 ;
j + 1 <= i + 1 ;
g " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " "
K <> {} ;
cluster mode multF -> associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative associative
|. q .| ^2 > 0 ;
|. p .| = |. p .| ;
s - 2 - 2 > 0 ;
assume x in { G } \ { G } ;
min ( W , C ) in C ;
assume x in { G } \ { G } ;
assume i + 1 = len G ;
assume i + 1 = len G ;
dom I = Seg n ;
assume that k in dom C and n <> 0 ;
1 + 1 <= j-1 + 1 ;
dom S = dom F ;
let s be Element of NAT ;
let R be Relation ;
let n be Nat ;
let S be non void ManySortedSign ;
let f be Function ;
let z be Element of X ;
u in { b } ;
2 * n < 2 * n ;
let x , y be element ;
BB c= B ;
assume I is_closed_on s , P ;
U = U \ ( U \ ( U ) ;
M /. 1 = z /. 1 .= z /. 1 ;
{ x } = { x } ;
i + 1 < n + 1 ;
x in { {} } ;
f . ( ( ( f . n ) ) * f . ( n + 1 ) ) <= f . ( n + 1 ) ;
l be Element of L ;
x in dom F ;
let i be Element of NAT ;
r is non empty ;
assume o <> o ;
s . ( ( x * y ) = 1 / ( 1 - y ) ;
card K in M ;
assume X in U ;
let D be Subset of TopSpaceMetr M ;
set r = q | k ;
y = ( W * ( x - y ) ) * ( x - y ) ;
assume dom g = dom f ;
let X , Y be non empty set ;
x + ( A + B ) is bounded ;
|. ( 0 qua number ) . 0 .| = 0 ;
cluster -> strict -> strict for Lattice ;
( a in B ) . ( b + 1 ) ;
V be VectSp of F ;
A * B <> {} ;
f = ( f | ( n + 1 ) ) | ( n + 1 ) .= f | n .= f | n .= f | n .= f | n .= f |
A , B , C , C , C , D , C be non empty set ;
z = ( P /. ( j + 1 ) ) . j ;
assume that f is one-to-one and g is one-to-one and g is one-to-one ;
reconsider j = i as Element of M ;
a , b , c , d , d be Element of L ;
assume q in A \/ B ;
dom ( F | ( A ) ) = ( ( F | ( A ) ) /\ ( F | ( A /\ ( A /\ ( B ) ) ) ;
set S = Z --> 0 ;
z in dom ( ( A * ( ( ( ( ( ( ( ( ( A ) ) ) * ( ( A * ( A * ( ) ) ) ) ) ) )
P [ y , z ] ;
not x in dom ( ( ( 0 in dom f ) ) \ ( dom f ) ;
B be non-empty over S ;
cos cos * \pi < \pi * \pi ;
reconsider z = 0 as Element of NAT ;
LIN a , c , a9 ;
y in { I } ;
( Q . ( 0 , 1 ) = 0 ;
set j = x -' m ;
assume a in { x } ;
j - j - j > 0 ;
I \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \pi \! \pi \! \pi \! \pi = \! \! \pi ;
y in F \ { d } ;
let f be Function ;
set A = B .: ( C , B ) ;
s . ( n + 1 ) < s . ( n + 1 ) ;
j -' 1 = 0 ;
set m = n + 1 ;
reconsider t = t . ( n + 1 ) as bag of n ;
I . ( m + 1 ) = ( m . m ) . ( m + 1 ) ;
i mod n mod n mod n mod n = 0 ;
set g = f | ( D , j ) ;
assume X is bounded and X is bounded ;
( p `1 ) ^2 = ( ( ( ( p `1 ) / ( 1 - 1 ) ) ^2 ;
a < ( p `1 / ( ( ( p `1 / ( 1 - ( ( ( ( ( p / ( ( ( 1 - ( ( ( ) ) ) ) ) ) ) ) )
L /\ L misses UBD C ;
x in Ball ( x , r ) ;
a in { c } ;
1 <= i + 1 ;
1 <= i + 1 ;
i + 1 <= len h ;
x = ( W /\ ( P ) ) ` ;
[: x , y :] in [: X , Z :] ;
assume y in [. 0 , 1 .] ;
assume p = <* 1 , 2 *> ;
len ( A * ( 1 , 1 ) ) = len ( A * ( 1 , 1 ) ;
set H = ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g
card ( b * a ) = a * b ;
Shift ( Shift ( w , v ) , v ) \models v ;
set h = h * ( h * ( 2 * h ) ) ;
assume x in X /\ Y ;
||. h . n - ||. seq . n .|| < d ;
x in the support ( f ) ;
f . ( y ) = f . ( y ) ;
for n holds X [ n ] ;
k + l -' l = l ;
p ^ q = p ^ q ;
let S be non empty ManySortedSign ;
P , Q , t , s , r , s , t be Real ;
Q /\ M c= union ( F /\ M ) ;
f = b * ( b * f ) ;
let a , b be Element of G ;
f .: X c= f .: X ;
let L be non empty doubleLoopStr ;
S is empty implies S is being_line & S is being_line
let r be Real ;
M \models v ;
v + w = 0 ;
P [ 0 ] ;
assume InsCode InsCode InsCode InsCode InsCode I = InsCode I ;
the carrier of M = M ;
cluster z (#) ( ( ( z * z ) (#) ( z (#) ( z * z ) ) (#) ( z (#) ( z (#) ( z * z ) (#) ( z (#) ( z * z )
let O be Subset of C ;
||. f | X .|| is bounded ;
x = g . ( j + 1 ) ;
for S being ++VERUM holds S `2 >= S-bound S
reconsider l = 1 / 2 as Element of NAT ;
v is vertex vertex sequence of G ;
T is SubSpace of T ;
Q /\ Q <> {} ;
let k be Nat ;
q " " is open ;
F ( t ) = t . t .= t . ( t . t ) ;
assume that n <> 1 and 1 <> n and n <> 1 ;
set e = EmptyBag ( n ) ;
let b be bag of n , L be bag of n ;
for i being Nat holds b . i = ( b . i ) * b . i ;
x is Element of product ( p ) ;
r in ]. p , q .[ ;
let R be Relation ;
S <> b ;
R <> {} ;
|. p - x .| >= |. p .| ;
1 / ( s / ( s * ( s * ( ) ) ) ) = s * ( s * ( s * ( s * ( s * ( s * ( s * ( s * )
let x be Element of X ;
f be Function of C , C ;
for a being Real holds a + 1 = a + 1
s . ( s . ( s . ( ) ) ) = s . ( s . ( s . ( ) ) ) ;
H + ( GI ) = GI - ( - 0. I ) .= 0. I ;
{ C } . x = ( C . x ) . x ;
f = f . 1 ;
Sum ( p ^ q ) = Sum ( p ^ q ) ;
assume v + u = v + u ;
a = ( a |^ n ) |^ n .= ( a |^ n ) |^ n .= ( a |^ n ) |^ n .= a |^ n .= a |^ n .= a |^ n .= a |^ n .= a |^
a , b // a , b ;
o , b1 // o , b1 ;
I is total implies I is total
I is being_line ;
sup rng ( ( f | X ) ) = sup X ;
x = a * b ;
|. p .| ^2 >= 1 ;
assume that j + 1 < len f and 1 < j ;
rng s c= dom ( s * ( n + 1 ) ) ;
assume not a support b misses support ( a ) ;
let L be non empty doubleLoopStr ;
s / 2 < s / 2 ;
p . ( c / ( 1 - c ) ) = ( 1 - c / ( 1 - c ) / ( 1 - c ) ;
R . n <= R . n ;
Directed I = Directed I ;
set f = ( f + g ) (#) ( f + g ) ;
cluster Ball ( x , r ) -> open ;
consider r such that r in A and r in A ;
ex q being Function st q = ( the InstructionsF of SCM+FSA ) * ( q , n ) ;
let X be non empty Subset of S ;
S be non empty full SubRelStr of L ;
assume that \lambda c= \lambda and \lambda c= \lambda and \lambda c= \lambda ;
1 / ( a / ( 1 - a ) / ( 1 - a ) = a / ( 1 - a ) ;
( q `2 ) ^2 = ( ( ( q `2 ) ^2 ^2 ;
i- i - 1 > 0 ;
assume 1 <= t `1 ;
card B = card B + 1 ;
x in union rng f ;
assume x in the carrier of R ;
d in dom d ;
f . ( ( 1 - 1 ) ) = F ( ( 1 - 1 ) ;
the Sorts of G = the Sorts of G ;
let G be _Graph , G be _Graph ;
let e , v be Element of V ;
c . i in rng c ;
f2 /* ( f2 /* seq ) is convergent ;
set z = - ( - 1 ) / 2 , z = - ( ( 1 - 2 ) / 2 ;
assume w is Element of S ;
set f = p \! q ;
c be Element of C ;
ex a being Element of A st P [ a , b ] ;
let x be Element of X ;
let I be Subset of X ;
reconsider p = p as Element of NAT ;
v , w // X , w ;
let s be State of SCM+FSA ;
p is being_line implies p is being_line
stop stop stop I c= stop stop I ;
set Bi = f /. i , f /. i = f /. i ;
w ^ ( w ^ ( w ^ <* s *> ) ^ <* w *> ) ^ <* w ^ s ^ <* w *> ;
W /\ ( W /\ ( W /\ ( W /\ ( W /\ ( W /\ ( W /\ ( W /\ ( W /\ ( W /\ ( W /\ ( W /\ ( W /\ ( W /\ (
f . ( j ) = ( f . j ) . j ;
let x , y be Element of T ;
ex d , c , d being Real st a , b // d , b ;
a <> 0 & b <> 0 ;
( ( n -' 1 ) mod x = ( ( ( n + 1 ) div ( n + 1 ) ) mod ( n + 1 ) ) ^2 ;
set g = lim ( s * g ) ;
2 * x / 2 >= 1 ;
assume ( a 'imp' c ) 'imp' c <> ( a 'imp' c ) 'imp' ( a 'imp' c ) ;
f \circ ( g , f ) in Hom ( c , d ) ;
Hom ( c , d ) <> {} ;
assume 2 * ( ( 2 * ( m * n ) ) - ( m * n ) ) > 0 ;
L . ( ( 1 - 1 ) / ( 1 - 1 ) = 0 ;
id X \/ Y = id X \/ Y ;
( the function of #Z n ) * ( ( ( - 1 ) * ( ( - 1 ) ) ) * ( ( - 1 ) ) ) ) <> 0 ;
( the function of #Z n ) / ( ( 1 - ( ( x - x ) ) ) ^2 > 0 ;
o in X /\ Y ;
let e , v be Element of V ;
r / 2 > 0 ;
x in P .: ( F .: ( F .: ( F .: ) ) ;
J be ideal of SCM+FSA ;
h . ( ( p ) = ( ( p ) * ( q + 1 ) ) * ( p + 1 ) ;
Index ( Index ( p , f ) , i1 + 1 ) + 1 <= len f ;
len M = n ;
the support support support support L c= A ;
dom f c= union rng F ;
k + 1 in Seg ( n + 1 ) ;
let X be set ;
\llangle x , y \rrangle in R \/ S ;
i = D or i = D ;
assume a mod b = 0 ;
h . ( ( x ) ) = ( g . ( x + 1 ) ;
F c= the carrier of X ;
reconsider w = ( w * ( 1 - w ) ) * ( 1 - w ) as Point of TOP-REAL 2 ;
1 / 2 < m / 2 ;
dom f = the carrier of TOP-REAL 2 ;
[#] TOP-REAL ( ( K ) = [#] ( ( K ) ) ;
cluster - ( x - y ) -> real ;
assume that d c= A and A c= B ;
cluster TOP-REAL n -> non empty for Real ;
w be Element of M ;
let x be Element of dyadic ( n ) ;
u in W & v in W ;
reconsider y = x as Element of L ;
N is full SubRelStr of T ;
sup rng ( x "/\" y ) = sup y ;
g . n = ( g . n ) * ( g . n ) * ( g . n ) ;
h . ( u , v ) = EqClass ( u , v ) ;
reconsider s = ( - s ) - ( - s ) as Point of X ;
dist ( ( x , y ) < r / 2 ;
reconsider m = n as Element of NAT ;
- - ( x - r ) < - r - 1 ;
reconsider P = P as Subset of N ;
set g = p ^ q ;
let n , m , k be Nat ;
assume 0 < f | A ;
( D . ( ( ( ( ( x ) ) \ { x } ) ) ) ) ) ) ) ) /\ ( ( ( ( x \ { x } ) ) ) /\ ( ( ( x \
cluster [#] T -> finite-ind for finite-ind ;
P be Subset of TOP-REAL 2 ;
LSeg ( G , 1 ) in LSeg ( \pi , 1 ) ;
let n be Element of NAT ;
reconsider S9 = S as Subset of T ;
dom ( i \dotlongmapsto 0 ) = dom ( i .--> 0 ) ;
X be set ;
X be set ;
op op ( op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op
reconsider m = m as Element of NAT ;
reconsider d = x as Element of X ;
let s be State of SCMPDS ;
let t be State of SCMPDS ;
b , b // a , b ;
assume i = n \/ { n } ;
f be PartFunc of X , Y ;
N >= 0 ;
reconsider t = T . t as Point of TOP-REAL 2 ;
set q = p ^ q ;
z in ( ( y | ( z | ( z | y ) ) | ( z | ( z | ( y | ( z | ( z | ( y | ( z | ( z | ( y | ( z | (
A ^ { 0 } = {} ;
len W = n + 1 ;
len h in dom h ;
i + 1 in Seg len ( s | ( i + 1 ) ) ;
z in dom ( ( f | X ) | X ) ;
assume that p = 0. TOP-REAL 2 and p = 0. TOP-REAL 2 ;
len G + 1 <= i ;
f1 (#) f2 is convergent & f1 (#) f2 is convergent ;
let s be Real ;
assume j in dom ( M * ( i + 1 ) ) ;
let A , B , C , D be Subset of X ;
let x , y be Point of X ;
b / ( c - a ) ^2 >= 0 ;
<* y *> ^ x = y ^ x ^ <* x *> ;
a , b // a , b ;
len p = len p ;
ex x being element st x in R & R [ x , y ] ;
len q = n ;
s = Initialize s ;
consider w such that q = w + q ;
x ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
k = 0 or k = 0 or k = 0 ;
assume that X is open and X is open and X is open ;
for x being element holds x in L implies x in L
||. f /. /. x0 .|| <= ||. f /. x0 .|| ;
c in ]. p , q .[ ;
reconsider V = V as Subset of TOP-REAL n ;
N , M , N , L , L , L , L , L , L , L ;
compactbelow z >= compactbelow ( x "/\" ( compactbelow ( y ) ) ) ;
M = f | ( M , f ) ;
( ( ( ( EmptyBag n ) '&' ( 1 / 2 ) ) '&' ( ( 1 / 2 ) '&' ( ( 1 / 2 ) ) '&' ( ( 1 / 2 ) '&' ( ( 1 / 2 ) ) ) ) ) )
dom g = X /\ dom f ;
A mode left_....................n....n....n....n.........
[: i , j :] in Indices M ;
reconsider s = x as Element of H ;
let f be Element of REAL ;
( F - G ) . ( 1 - 1 ) = ( F - G ) . ( 1 - 1 ) ;
cluster \measuredangle ( a , b ) -> \measuredangle ;
let a , b , c , d be Real ;
rng s c= dom ( f | Z ) ;
curry ( F , ( ( curry ( F ) ) | ( ( F | ( ( F | ( F | ( ( F | ( F | ( F | ( F | ( F | ( F | ( F | ( F
set k = card ( B \/ C ) ;
set G = the Sorts of A ;
reconsider a = [ x , s ] as Element of TS ;
let a , b be Element of M ;
reconsider s = ( s . 1 ) as Element of S ;
rng p c= the carrier of L ;
let d be Element of the carrier of A ;
( ( x | ( 0 qua Nat ) ) | ( 0 -tuples_on REAL ) ) | ( 0 -tuples_on REAL ) = 0 ;
I in dom stop I ;
g , f be Function ;
reconsider D = Y as Subset of X ;
reconsider i = i as Element of NAT ;
dom f = the carrier of S ;
rng h c= the carrier of Lin ( A ) ;
cluster \forall func All ( x , p ) -> valid for All of x , p ;
d * N / 2 > 0 ;
]. a , b .[ c= ]. a , b .[ ;
set g = f | ( ( D \/ ( ( ( ) ) ) | ( ( D \/ B ) ) ;
dom ( p | ( m + 1 ) ) = Seg m ;
3 + 1 <= k + 1 ;
the function tan is differentiable ;
x in rng ( p | ( ( n + 1 ) ) ;
f , g , f , f be Function of D , D ;
p in the carrier of S ;
rng f = dom f ;
( the target of G ) . e = ( the target of G ) . e ;
width G < width G ;
assume v in rng ( ( E | ( n + 1 ) ) ;
assume x is not zero & g is not zero ;
assume 0 in rng ( g | A ) ;
let q be Point of TOP-REAL 2 ;
let p be Point of TOP-REAL 2 ;
dist ( ( u , v ) <= ( u + v ) * ( u + v ) ;
assume dist ( ( x , y ) < dist ( x , y ) ;
<* S , T *> is Sororororororororororororor;
i <= len G ;
let p be Point of TOP-REAL 2 ;
x1 in the carrier of X ;
set p = f /. 1 ;
g in { r : g . g < g / 2 } ;
Q = { Q } .: ( Q .: Q ) ;
( 1 - ( ( - 1 ) ) * ( 1 - 1 ) ) ^2 > 0 ;
- p + ( - p ) c= - ( p - p ) ;
n < ( ( ( n + 1 ) / 2 ) / 2 ) / 2 ;
CurInstr ( P1 , Comput ( P1 , s1 , i ) ) = halt SCM+FSA ;
A /\ ( A /\ ( B \/ A ) ) <> {} ;
rng f c= ]. r , s .[ ;
g , f be Function ;
f | ( ( ( X , Y ) | ( X | Y ) ) is bounded ;
reconsider z = ( ( z ) . x as Element of CompactSublatt CompactSublatt CompactSublatt CompactSublatt L ;
f be Function of T , T ;
reconsider g = g . c as Function of c , d ;
[: s , I :] in [: A , I :] ;
len ( the connectives of C ) = n ;
let C , B , C , D be non empty set ;
reconsider V = { V } as Subset of X ;
assume that p is valid and p is valid ;
assume X c= dom f ;
H \ ( a \ b ) is open ;
let A be Element of E , E be Element of O ;
p , q , r is_collinear ;
consider x such that x in K and v in K ;
x in { 0. TOP-REAL 2 } ;
p in [#] [: [: [: [: { 0 } , { 1 } :] ;
0 < ( ( ( ( ( - M ) ) ) . ( ( - M ) ) . ( ( - ( M ) ) . ( ( - M ) ) . ( ( - M ) ) ) ) ;
op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op op
consider c such that c in G and a in G ;
( a in dom ( F | ( ( n + 1 ) ) ;
cluster lattice -> Boolean for Lattice ;
set i = the Element of NAT ;
let s be State of SCM+FSA ;
assume y in ( ( the carrier of A ) \ ( the carrier of A ) ;
f ( len f ) = ( f /. len f ) ^ ( f /. ( len f ) ;
x , y \bfparallel f . ( x , y ) ;
If X c= Y & Y c= Z implies X c= Y
let y be Element of REAL , x be Element of REAL ;
cluster x \ ( x \ y ) -> empty ;
set S = EmptyBag n , T = EmptyBag n ;
set T = [. 0 / 2 , 1 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 , 1 = [. - 1 , - 1 / 2 / 2 / 2 , 1 / 2
1 in dom mid ( f , 1 , 1 ) ;
sqrt 4 / 4 / 4 < 1 / 4 ;
{ x } in dom ( f | X ) ;
O c= dom ( I --> {} ) ;
( the carrier of G ) /\ ( the carrier of G ) = the carrier of G ;
Support ( f , T ) c= Support f ;
reconsider h = R . ( n + 1 ) as Element of NAT ;
ex b being element st b = x & b = y * b ;
let x , y be Element of G ;
reconsider h = f . ( i + 1 ) as Function ;
( p `1 ) ^2 = ( ( ( ( p `1 ) / ( 1 - sn ) ) ^2 ;
i + 1 <= len Cage ( C , n ) ;
len P = n ;
set N = the Tran of over over n ;
ggy - gy >= gy - x ;
a <> b or a <> b ;
reconsider r = r * ( v * r ) as Point of TOP-REAL 2 ;
consider d such that x = d * c ;
consider u such that u in W and v = v + u ;
len f = n -' n ;
set q = ( the TopStruct of TOP-REAL 2 ) | D ;
set S = S --> ( ( ( ( ( S , T ) ) --> x ) ) --> ( S , T ) ) --> ( S , T ) ) ;
MaxADSet ( b ) c= MaxADSet ( a ) ;
Ball ( G , ( m , n ) c= Ball ( G . m , r ) ;
f .: ( D /\ D ) meets ( ( D /\ C ) ;
reconsider D = E as Subset of L ;
H = ( H '&' ( H '&' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' H ) '&' ( H '&' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' H ) ;
assume t is Element of product ( F ) ;
rng f c= the carrier of S ;
consider y being element such that y in X and x = y ;
f . ( ( 1 - 1 ) / 2 = ( ( 1 - 1 ) / 2 ) / 2 ;
the carrier of G = the carrier of G ;
reconsider m = m as Element of NAT ;
set S = LSeg ( p , q ) ;
i in the carrier of M ;
assume P c= Seg m ;
for k being Nat st X [ k ] holds X [ k + 1 ]
consider p such that p in a and a in X ;
L . ( p , q ) = p . ( p , q ) ;
p . ( ( i + 1 ) ) = ( p . ( i + 1 ) ;
let P , Y be Subset of X ;
then 0 < r / 2 ;
rng ( ( ( ( the TopStruct of X ) | ( X , Y ) ) | ( X , Y ) ) ) = the TopStruct of X ;
reconsider x = x as Element of K ;
consider k such that z = ( k + 1 ) * ( k + 1 ) ;
consider x such that x in X and y in X ;
len ( s * ( ) = len s ;
reconsider x1 = x as Element of L ;
Q in the topology of T ;
dom ( f | ( n + 1 ) ) c= dom ( f | ( n + 1 ) ;
If m divides n & n divides m implies m |^ n
reconsider x = x as Point of TOP-REAL 2 ;
a in [#] T ;
{ y : not 0 in the bound } ;
Hom ( ( a , b ) <> {} ;
consider k such that p < k and k < n ;
consider c such that c in dom f and f . c = d ;
[: x , y :] in [: [: [: g , g :] , dom g :] ;
set S = the TopStruct of S ;
l = m * ( i + 1 ) ;
{ x } in { A } ;
reconsider p = x as Point of TOP-REAL 2 ;
reconsider I = { 0. } as Subset of Lin ( B ) ;
f . ( p , q ) <= f . ( p , q ) ;
( F . ( x ) ) `1 <= ( ( F . x ) `1 ;
( x `2 ) ^2 = ( ( ( ( ( ( ( x ) ) ^2 ) ^2 ) ^2 ;
for n being Nat holds P [ n + 1 ] ;
J , K , L , L , L , L , L , L , L , L , K , L , L , K , L , L , K , L , L , L , L , K , L , L , L , L , L ;
assume 1 <= i & i <= len a ;
0 |-> 0 = 0
X ( X \ ( i ) ) in ( A \ ( i + 1 ) ) \ ( A \ ( i + 1 ) ) ;
<* 0 , 1 *> in dom ( ( ( 0 .--> 1 ) --> 1 ) ;
assume that P [ a ] ;
reconsider s = intpos intpos intpos n as Element of SCMPDS ;
k-1 -' 1 <= j ;
\Omega [: S , T :] c= the topology of T ;
cluster W /\ V -> open ;
assume that k in dom mid ( f , i , k ) and mid ( f , i , k ) . k in dom mid ( f , i , k ) ;
let P be Subset of TOP-REAL 2 ;
let A , B , C be Matrix of K , n , K ;
- a * b - b * a = - b * a - b * b ;
let A be Subset of T ;
id ( ( ( o , b1 ) ) in the carrier of [: o , b1 :] ;
If \times X is non empty & X is open & X is open ;
let N , M be Matrix of G , G ;
j >= ( ( ( Rev g ) ^ ( g /^ 1 ) ) ^ ( g /^ 1 ) ;
b = Q . ( ( Q . ( i + 1 ) ) ;
f2 (#) ( f2 /* ( f1 + f2 ) ) is convergent ;
reconsider h = f | ( N , M ) as Function of N , M ;
assume that a <> 0 and b <> 0 ;
t in the carrier of A ;
v |-- ( v |-- |-- \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow T ) ) is \rightarrow \rightarrow \rightarrow T ;
{} = the carrier of L ;
Directed I " I c= card I ;
Initialized ( p , s ) = p +* q ;
reconsider N = ( the mapping of N ) . x as net of T ;
reconsider Y = ( the carrier of CompactSublatt CompactSublatt CompactSublatt CompactSublatt CompactSublatt CompactSublatt CompactSublatt CompactSublatt ( L ) ) as Subset of CompactSublatt L ;
"/\" ( X \ Y ) <> {} ;
consider j such that i = j + 1 ;
[: s , s :] in the carrier of S ;
m in B /\ ( C \/ D ) ;
n <= len P ;
( ( ( ( x ) * ( y - x ) ) * ( y - x ) ) ) * ( ( y - x ) ) ) = ( ( ( ( y - x ) * ( y - x ) * ( y - x ) ) * ( y - x ) )
InputVertices S = { x } ;
let x , y be Element of TOP-REAL n ;
p `1 = ( ( p `1 ) / ( 1 - 1 ) ) / ( 1 - ( ( ( p ) / ( 1 - 1 ) ) / ( 1 - 1 ) ) ^2 ;
g * h = g * h .= g * h * g .= g * h * h * h * g .= g * h * h * h * h * h * h .= g * h * h * h * h * h * h .= g * h * h * h
let p , q be Element of CQC-WFF ( V ) ;
{ x } in dom ( ( ( x - y ) (#) ( x - y ) ) (#) ( x - y ) ) ;
R * ( ( R ) ) = ( R * ( R * ( R * ( ( R * ( ( R * ( ( R * ( ( R * ) ) ) ) ) * ( R * ( R * ) ) ) * ( R * ( R * (
n in Seg ( len p ) ;
for s being Real holds s . s in REAL
rng s c= dom ( ( s * ( n + 1 ) ) ;
We We notation notation the notation notation notation notation notation notation the notation of X for X holds X is Cn ;
1 / ( ( K ) ) / ( 1 - ( 1 - K ) = ( - 1 / ( 1 - K ) / ( 1 - K ) / ( 1 - K ) / ( 1 - K ) ;
set S = Segm ( A , P ) ;
ex w being element st w in REAL & w in REAL & w in REAL & ( ex y being element st w in REAL st y in REAL & w in REAL & w in REAL & w in REAL ;
curry ( ( P , k ) ) is invertible ;
cluster T .: T -> open ;
len ( f | n ) = n - 1 ;
sqrt ( i / 2 ) < ( i / 2 ) / 2 ;
let x , y be Element of OSSub ;
b , c // b , c ;
consider p such that p in dom p and p . i = c ;
assume that f is onto and f is onto ;
assume that CurInstr ( F , s ) = halt ( SCMPDS , s ) ;
" not CurInstr ( Reloc ( J , card I ) ;
not halt SCM+FSA ( SCM+FSA ) ;
set m = LifeSpan ( p , s ) , m = p . m ;
IC Exec ( i , s ) in dom ( p +* q ) ;
dom t = the carrier of SCM ;
( S ^ <* f *> ) ^ ( ( ( S ^ ) ^ ) ^ ( ( S ^ ) ) ) ) ) ) ) ^ ( ( S ^ <* f *> ) ) ) ) ) ) ) = ( ( S ^ <* f *> ^ ( (
let a , b , c be Element of U0 ;
union F c= union F ;
the carrier of X is open ;
assume not LIN a , b ;
consider i such that i = d . i and d . i = M . i ;
assume Y = {} implies Y = {}
M \models v ;
consider m such that m in meet ( F . m ) ;
reconsider A = ( the Sorts of X ) . x as MSSubset of X ;
card A c= card A + 1 ;
assume that that a <> b and b <> c ;
cluster s -compound -> relational ;
( L | ( n + 1 ) | ( n + 1 ) = L | ( n + 1 ) ;
let P be Subset of TOP-REAL 2 ;
assume r in { p } ;
let A be Subset of T ;
assume that m in Indices M1 and n in Seg m ;
0 <= ( ( 1 - ( p - 1 ) ) / 2 * ( 1 - 1 ) ) / 2 ;
( F ( N ) \ ( N \/ ( N \/ M ) ) = ( N \/ ( N \/ M ) ) \ ( N \/ ( N \/ M ) ) ;
If X c= Y & Y c= Z implies X c= Y
( z * ( y - z ) * ( y - z ) <> 0. TOP-REAL 2 ;
1 + 1 <= card X ;
set g = z \circlearrowleft z ;
assume that k . 1 = ( ( p . k ) * ( p . ( k + 1 ) ) * ( p . ( k + 1 ) ) ;
cluster func C -total -> total for PartFunc of C ;
reconsider B = A as Subset of T ;
let a , b , c , d be Element of Y ;
L . ( i + 1 ) = ( ( i + 1 ) * ( i + 1 ) .= ( ( ( i + 1 ) * ( i + 1 ) ) * ( i + 1 ) ) * ( i + 1 ) .= ( ( ( i +
Plane ( ( Plane ( X , Y ) ) c= Plane ( X , Y ) ;
n <= indx ( D2 , D1 ) ;
( g * ( ( g - f ) ) * g ) . x = ( g * ( g - f ) * g ;
j + 1 <= len f ;
set W = the carrier of TOP-REAL 2 ;
S . ( a , b ) = a * b + b ;
1 in Seg ( len M ) ;
dom ( f * ( i + 1 ) ) = dom f ;
( ( ( ( ( ( x , y ) ) | ( a , a ) ) | ( a , a ) ) ) | ( a , a ) ) = ( a | ( a , q ) ) ;
set Q = \mathop { \rm EqRel } ( R , p ) ;
cluster mode ManySortedFunction of A -> non-empty for ManySortedSet of S ;
ex A being Subset of T st A is open & A is open ;
reconsider z = x as Element of product G ;
rng f c= rng f \/ ( rng f ) ;
consider x such that x in dom f and f . x = f . x ;
f = the carrier of A ;
E \models _ { E } => { x } ;
reconsider n = ( ( ( n + 1 ) * ( n + 1 ) ) * ( n + 1 ) as Point of TOP-REAL n ;
assume that P is being_line and P is being_line and P is being_line and P is being_line and P is being_line and P is being_line and P is being_line ;
card B c= card ( B \/ C ) + card C ;
card ( card ( x \ y ) ) = card ( card ( x \ y ) ) ;
g + s in R ;
set q = ( q , s ) *' ) *' ;
for x being element holds X [ x , y ] ;
h /. ( i + 1 ) = ( ( h /. ( i + 1 ) ) /. ( i + 1 ) .= ( ( ( h /. ( i + 1 ) ) /. ( i + 1 ) ) .= ( ( ( ( h /. ( i + 1
set w = \mathop { \rm max } ( B , n ) , N = max ( B , n ) , N = max ( B , n ) , N = max ( B , n ) ;
t in Seg n ;
reconsider X = the carrier of C as Element of Fin C ;
not IncAddr ( i , k ) = i ;
S is compact ;
If R is open , then then R is being_line ;
assume 0 <= a * b ;
u in ( ( c + d ) /\ ( d + c ) ;
u in ( ( c / ( d - c ) ) / ( d - c ) ) / ( d - c ) ;
len C + 1 >= 0 ;
x , y // x , y ;
a ^ b = a ^ b ;
\langle 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
set y = <* <* x , y *> , z = <* y , z *> ;
{ F _ 1 } in rng ( Line ( M , 1 ) ) ;
p . m . m <> r . m ;
( p `1 ) ^2 = ( ( ( ( ( ( p ) / ( 1 - sn ) ) ^2 ) ^2 ;
lower_bound X = lower_bound X ;
0 + ( ( ( ( p + r ) ) * p ) / 2 ) ^2 >= 0 ;
x in dom g /\ ( g | ( ( 0 , 1 ) ) ;
f /* ( s1 ^\ k ) is convergent ;
reconsider u = ( u | X ) | X as Point of X ;
p \! \! \! \! \! \! \! S2 \! S2 \! S2 = ( p \! \! \! S2 ) \! \! S2 ;
len x < i + 1 ;
assume I is not empty ;
set i = halt SCMPDS , i1 = halt SCM+FSA ;
x in { {} } implies x in { {} }
consider y being element such that y in B and x in B ;
len S = n ;
reconsider m = M . i as Element of I ;
A ( A ) = ( ( B \/ A ) \/ ( B \/ A ) ) \/ ( B \/ A ) ;
set N = the InstructionsF of SCM+FSA ;
rng F c= the carrier of Lin ( A ) ;
( ( Q , n ) \ ( Q , m ) is binary
f . ( k + 1 ) in rng f ;
h .: ( P /\ Q ) = [#] ( P /\ Q ) ;
g in dom ( ( f | ( n + 1 ) ) \ ( g | ( n + 1 ) ) ;
ggX /\ ( X /\ Y ) = {} ;
consider n such that n in Z and Z c= Y ;
set d = \rho ( x , y ) , \rho ( y , z ) ;
b / 2 < 1 / 2 ;
reconsider f = the carrier of X as continuous continuous Function of X , COMPLEX ;
If i <> 0 & i mod 2 = 0 ;
j in Seg ( len g ) ;
dom ( i .--> a ) = dom ( i .--> a ) ;
cluster sec | ( ( ( n + 1 ) ) | ( n + 1 ) ) is increasing ;
Ball ( Ball ( u , e ) Ball ( u , e ) = Ball ( u , e ) ;
reconsider x1 = ( ( ( ( ( ( ( id S ) | | | ( S ) ) | ( S | ( S | ( S | ( S | ( S | ( S | ( S | ( S | ( S | ( S | ( S | ( S | ( S | ( S | ( S
reconsider R = R as Relation ;
consider a , b being Element of A such that a = b "/\" ( a , b ) ;
( ( 1 - 1 ) * ( 1 - 1 ) ) ^ ( ( 1 - 1 ) ) ^ ( ( 1 - 1 ) ) ) ) ) ^ ( ( 1 - 1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) = ( ( 1 - 1 ) ^ ( (
S = { S } \/ { S } ;
( ( the function ) `| Z ) (#) ( ( ( the function ) (#) ( ( the function ) (#) ( ( the function ) ^ ) (#) ( ( the function ) ^ ) (#) ( ( ( the function ) ^ ) (#) ( ( ( the function ) ^ ) (#) ( ( ( the function ) ^ ) (#) ( ( the
cluster id ( C ) -> one-to-one ;
set C = 1GateCircStr ( f , x ) , z = f /. ( x , y ) ;
|. ( ( E ) | ( ( ( ) ) | ( ( ( ( ( ( ( ( E | ) | ( ( ) ) | ( ( ( ) ) | ( ( ( ( ) ) | ( ( ( ) | ( ( ( ) | ( ( ( ) | ( ( ) | ( ( ( ) ) ) |
( ( ( - ( ( - 1 ) ) * ( - 1 ) ) * ( ( - 1 ) ) ) * ( ( - 1 ) ) ) is_differentiable_on Z ;
sup A = sup A & sup A = inf A ;
F ( F , f ) = F ( f , f ) ;
reconsider p = q `1 as Point of TOP-REAL 2 ;
g . ( ( ( ( [#] X ) ) | ( [#] Y ) ) | ( [#] Y ) ) is open ;
let C be Subset of TOP-REAL 2 ;
LSeg ( f , i1 ) = LSeg ( f , i1 ) ;
rng s c= dom ( f | X ) ;
assume x in dom ( Rev ( h ) ) ;
reconsider n = m as Element of NAT ;
for y being element holds y in rng g implies y in rng g
for k being Nat holds P [ k + 1 ] ;
m = ( m + 1 ) + 1 ;
assume for n holds ( ( ( ( n + 1 ) (#) ( n + 1 ) ) (#) ( ( ( n + 1 ) (#) ( ( n + 1 ) (#) ( ( n + 1 ) (#) ( ( n + 1 ) (#) ( ( n + 1 ) (#) ( ( n + 1 )
set BB = the carrier of X ;
ex d being Element of L st d in D & d in D ;
assume R is open and R is being_line ;
t in ]. r , s .[ ;
z in { v } ;
{ x } \ { y } = { x } ;
If for x holds ( x - 1 ) ^2 > 0 ;
assume that p - q is not zero and p - q is not zero ;
set q = f ^ <* p *> ^ q ;
f be PartFunc of REAL , REAL ;
( n mod m ) mod n = ( n mod m ) mod m .= ( n mod m ) mod m .= ( n mod m ) mod m .= ( n mod m ) mod m ;
dom T = [: T , T :] ;
consider x such that x in { w where w is Element of c : w < c } ;
assume ( F ( v ) ) * F . v = ( F . v ) * ( F . ( v * ( ( F . v ) * ( F . v ) ) ;
assume the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of T
reconsider A = [. A , B .] as Subset of R^1 ;
consider y being element such that y in dom F and F . y = x ;
consider s such that s in dom ( o | a ) ;
set p = ( p `1 - q `2 - q `2 - q `2 / 2 ;
n + 1 <= len f ;
( q *^ ( O , a ) *^ ( O , a ) ) *^ ( O , a ) = ( ( O *^ ( a , b ) *^ ( O , a ) ) *^ ( O , a ) ) *^ ( O , a ) ) *^ ( O , a ) ) *^ ( O , a ) ) ) *^ (
set C = ( the InstructionsF of SCM+FSA ) . k ;
Sum ( L * ( p * p ) = Sum ( L * p ) .= Sum ( L * p ) .= Sum ( L * p ) .= Sum ( L * p ) .= Sum ( L * p ) .= Sum ( L * p ) .= Sum ( L * p ) .= Sum ( L *
consider i such that i in dom p and p . i = q . i ;
defpred Q [ set ] means ex Q being set st Q [ $1 , ( $1 + 1 ) , $1 + 1 ) ;
set t2 = Comput ( P1 , s1 , 1 ) , P1 = P1 +* I , P1 = P1 +* I , P1 = P1 +* I , P1 = P1 +* I , P1 = P1 +* I , P1 = P1 +* I , P1 = P1 +* I , P1 = P1 +* I1 , P1 = P1
let l be Nat , A be Matrix of K , K ;
reconsider U = union { G } as Subset of T ;
consider r such that r > 0 and p > 0 ;
( h | ( n + 1 ) | n = ( ( ( ( n + 1 ) | n ) | n ) | n ;
reconsider B = the carrier of X as Subset of X ;
p `2 = |[ p `1 , p `2 ]| ;
cluster f | ( ( n + 1 ) \ f | ( n + 1 ) -> finite ;
consider b such that b in dom F and F . b = F . b ;
{ x } < card { x } ;
If X c= B & B c= B & B c= C implies B c= C
func w / ( x , y ) -> Real equals ( w / ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) ;
\measuredangle ( x , y , z ) = 0 ;
assume 1 <= s & s = s . 1 ;
f . ( ( ( f . n ) ) ) c= f . ( n + 1 ) ;
the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: , the carrier of [: the carrier of [: the carrier of [: the carrier of [: , the carrier :] :] :] , the carrier of [: the carrier of [: the carrier
p - q in HP ;
- ( ( t - ( 1 - t ) ) ^2 < ( - 1 - t ) ^2 ;
U . ( ( 1 - 1 ) ) = U . ( 1 - 1 ) .= U . ( 1 - 1 ) .= U . ( 1 - 1 ) .= U . ( 1 - 1 ) .= U . ( 1 - 1 ) .= U . ( 1 - 1 ) .= U . ( 1 - 1 ) .=
f .: ( the carrier of A ) = the carrier of A ;
the carrier of [: the carrier of [: n , Seg n :] = [: the carrier of [: n , Seg n :] ;
for n being Nat holds P [ n + 1 ] ;
ex V being Subset of M st V in M & M in M & M = M . V ;
ex f being Function of A , B st f is Function of A , B & f is one-to-one & f is one-to-one ;
h . ( ( 0 , 1 ) in the InternalRel of G ;
s . ( s + 1 ) = ( Initialize ( s ) ) . ( ( Initialize s ) . ( ( Initialize s ) ;
[ w , v ] <> [ w , v ] ;
reconsider t = t as Element of REAL ;
C \/ ( [#] ( ( ( G ) ) \ ( G \/ ( ( G \/ A ) ) ) ) c= [#] ( G \/ ( G \/ A ) ) ;
f .: ( [#] X ) in [#] ( X ) ;
x in the carrier of A ;
g . ( x * ( ( x * ( 1 - x ) ) ) / ( 1 - x ) ) <= ( 1 - x ) / ( 1 - x ) * ( 1 - x ) ;
InputVertices S = { x } ;
for n being Nat holds P [ n + 1 ] ;
set R = Line ( M , i ) ;
assume that M is being_line and M is being_line and M is being_line ;
reconsider a = ( ( ( a * b ) * a ) * a as Element of K ;
len ( B ^ <* x *> ) = len ( ( B ^ A ) .= len ( ( B ^ A ) ^ ( ( B ^ A ) ^ ( B ^ A ) ) ;
len ( f | n ) = n ;
dom ( f - g ) = dom f /\ dom g ;
( the Sorts of U0 ) . x = ( ( the Sorts of U0 ) . x ;
dom ( p ^ q ) = dom q ;
M . ( 1 , 1 ) = ( 1 - 1 ) * ( 1 - 1 ) .= 1 * ( 1 - 1 ) ;
assume W is open ;
LSeg ( C , i ) = LSeg ( f /. i ) ;
{ C , p } |- p => q ;
for b being element st b in rng f holds b "/\" f . b >= sup rng f
- ( 1 - ( ( 1 - 1 ) ) / ( 1 - ( 1 - 1 ) ) ^2 = 1 - ( 1 - ( 1 - 1 ) ) ^2 ;
( ( ( L \/ R ) \/ ( L \/ R ) ) \/ ( L \/ R ) ) ) \/ ( L \/ R ) ) ) c= L ;
consider p such that p in \widetilde L~ f and not p in L~ f ;
the carrier of X = [: X , Y :] ;
consider s such that s => q => p ;
( ( ( ( ( ( ( 0. ) ) ) * ( 0. M ) ) * ( ( 0. M ) ) ) ) * ( ( 0. M ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) is 0. ;
cluster f . ( x , y ) -> Element of D ;
consider g such that g . t = t . ( g . t ) ;
p in LSeg ( f , n ) ;
set R = ]. - b , a / 2 / 2 / 2 / 2 / 2 ;
IncAddr ( I , S ) = I ;
( s . m ) . m <= ( ( s . m ) . ( ( s . m ) * ( s . m ) ;
a + b = ( a + b ) / ( a + b ) / ( a + b ) ;
id X /\ ( id X ) = id X ;
for x being element holds x in dom ( f | ( x \ y ) implies x in dom f | ( x \ y )
reconsider H = U \/ ( ( U \/ U ) ) as Subset of TOP-REAL 2 ;
u in ( ( ( u + v ) ) /\ ( ( ( ( u + v ) /\ ( ( u + v ) ) /\ ( ( ( u + v ) /\ ( ( u + v ) ) ) ) ) ) ;
consider y being element such that y in dom ( B | A ) and not y in A ;
consider A being Subset of R such that A = R .: A ;
p in rng p ;
len s > 1 ;
( ( ( ( ( TOP-REAL 2 ) | P ) ) | P ) | P ) = ( ( ( TOP-REAL 2 ) | P ) | P ;
Ball ( Ball ( p , r ) c= Ball ( p , r ) ;
f . ( ( a " ) " = ( f " ) " ;
( ( s ^\ k ) (#) ( s ^\ k ) (#) ( s ^\ k ) ) (#) ( s ^\ k ) is convergent ;
( g `| Z ) `| Z = g `| Z ;
the internal relation of S in the InternalRel of S ;
deffunc F ( set , set , set , set ) = ( ( ( the Tran of A ) * ( $1 , $2 ) ^ ( $1 , $2 ) ^ ( $1 , $1 ) ^ ( $1 , $1 ) ;
F . ( ( ( ( F . ( s . ( ) ) ) ) ) ) ) = ( F . ( s . ( A ) ) ;
x `2 = ( ( ( A * ( ( ( ( ( ( A ) ) * ( ( A * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
card ( f .: ( X \/ Y ) ) c= card ( f .: ( X /\ Y ) ) ;
the topology of product ( T ) c= the topology of T ;
attr o is constructor means : Def3 : o is constructor ;
assume that X is finite and card X <> card Y ;
the carrier of SCM+FSA is compact ;
LIN a , b , c ;
e . ( ( 2 * ( 1 - 1 ) ) / ( ( 1 - 1 ) ) = ( 1 - 1 ) / ( 1 - 1 ) ;
{ E } in S & not E in S ;
set JJ = J +* ( l , u ) ;
set A = [ a , b ] , B = [ a , b ] , C ] , C = [ a , b ] , B = [ a , b ] , C ] , B = [ a , b ] , C = [ a , b ] , B = [ a , b ] , C ] , B = [ a , b ] , C
set c = [ [ c , d ] , [ c , d ] , [ c , d ] , [ c , d ] , [ c , d ] , [ c , d ] , [ c , d ] , [ c , d ] , [ c , d ] , [ c , d ] , [ c , d ] , [ c , d ]
x * ( z * x ) in N * ( z * x ) ;
for x being element holds ( ( ( x in X ) \ Y ) \ Y ) implies x in Y
Int cell ( f , i1 , i2 ) misses Int L~ f ;
U /\ ( U /\ C ) is closed
set f = f .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .:
If for S being non empty ManySortedSign holds S is convergent & S is convergent & S is convergent implies S is convergent & S is convergent & S is convergent implies S is convergent
f . ( 0 qua Nat ) = 0 ;
cluster pcs-(# (# (# reflexive , symmetric #) -> symmetric symmetric symmetric symmetric symmetric symmetric symmetric symmetric symmetric symmetric symmetric symmetric symmetric symmetric symmetric symmetric symmetric symmetric symmetric symmetric symmetric symmetric ;
consider d such that d in dom f and f . d = g . d ;
b in dom Start-At ( card I + 2 ) ;
( z + y ) + z = z + ( z + y ) ;
len ( l | A ) = 0 ;
t ^ <* t *> ^ <* {} *> is {} ;
t = t . ( p ^ q ) ;
set p = ( W-min L~ f ) /. n ;
k -' 1 = ( ( ( k + 1 ) - 1 ) / 2 ) / 2 ;
consider u such that u in dom ( p | ( D \ ( B \/ C ) ) ;
len ( a |^ ( n -' 1 ) ) = width ( a |^ ( n -' 1 ) ;
( F . ( x ) ) in dom ( F . x ) ;
set H = the carrier of TOP-REAL 2 ;
set H = the carrier of X ;
Comput ( P1 , s , m ) . intpos ( m + 1 ) = s . intpos m ;
IC Comput ( P1 , s1 , k + 1 ) = IC Comput ( P1 , s1 , k + 1 ) ;
dom ( ( ( ( the function ) ) ) | ( ( n + 1 ) ) | ( n + 1 ) ) = the carrier of TOP-REAL 2 ;
reconsider l = l *^ ( 1 - 1 ) as Element of ( TOP-REAL 2 ) | ( ( 1 - 1 ) ) -tuples_on NAT ;
set b = [ b , p ] , p = [ b , p ] , p ] ;
Line ( M , i ) = Line ( M , i ) ;
n in dom ( ( ( the Sorts of A ) * ( ( the Sorts of A ) * ( ( the Sorts of A ) * ( ( the Sorts of A ) ) ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A )
cluster f + g -> Lipschitzian for PartFunc of REAL , REAL ;
consider y being Point of X such that y = r * y ;
set x = intpos ( n + 1 ) , y = intpos ( n + 1 ) , y = intpos ( intpos ( n + 1 ) , intpos ( n + 1 ) , intpos ( n + 1 ) , intpos ( n + 1 ) , y = intpos ( n + 1 ) , intpos ( n + 1 ) , intpos (
set p = stop I , s = stop I +* I ;
consider a such that a in D and b in D ;
A \/ B = { A } ;
let A , B , C , D be set ;
|. p - ( ( p `1 ) / 2 ) ^2 >= 0 ;
l - l - ( l - 1 ) * ( l - 1 ) = ( l - 1 ) * ( l - 1 ) ;
x = ( v + ( w - ( a - b ) ) * ( a - b ) ;
the topological topological structure of ( L ) = the topological structure of L ;
consider y being element such that y in dom ( ( ( ( ( x | y ) | ( x | y ) ) | ( x | ( y | ( y | ( y | ( y | ( y | ( y | ( x | ( y | ( y | ( y | ( x | ( x | ( y | ( x | ( y | ( y | (
{ f : { { v } } misses Free ( { v _ 1 } , { v _ 1 } ) ;
cluster Y -> finite for set ;
2 * 2 * N * N in N & N * ( N + 1 ) in N ;
let s be State of SCM+FSA ;
for x being element holds ( ( for x st x in Z holds ( ( id Z ) * ( ( id Z ) * ( ( id Z ) * ( x + 1 ) ) ) * ( ( - 1 ) ) * ( ( ( - 1 ) ) ) ) * ( ( - 1 ) ) ) ) ) * ( ( - 1 ) ) is_differentiable_on Z )
rng ( ( ( ( ( f | A ) ) | A ) | ( ( A | ( ( A | ( ( ) ) ) | ( ( ( A | ( ( ) ) ) | ( ( A | ( ( A | ( ( ( ) ) ) | ( ( ( ) ) ) | ( ( ( ) ) ) | ( ( A | ( (
j + 1 <= len f ;
reconsider R = R * ( n + 1 ) as Function of REAL ;
C . ( ( ( ( C . ( n + 1 ) ) ) ) ) `2 = ( ( C . n ) * ( C . ( n + 1 ) ) ) * ( C . ( n + 1 ) ) ;
1. K = 0. K .= 0. K .= 0. K .= 0. K ;
t = ( the array of S ) . I ;
support ( f | support f ) is support support support f ;
ex N being Nat st N = ( ( ( N + 1 ) (#) ( ( ( ( ( N + 1 ) (#) ( ( ( N + 1 ) (#) ( ( N + 1 ) (#) ( ( ( N + 1 ) ) (#) ( ( ( N + 1 ) (#) ( ( N + 1 ) ) (#) ( ( N + 1 ) ) ) ) )
for P being non empty doubleLoopStr holds P [ n ] ;
\ \ { x } is open ;
h = ( h . i ) * ( h . i ) ;
ex x being Element of A st x = N * ( x * N ) ;
set X = ( ( q , m ) --> ( q , m ) --> ( q , m ) , q ) , q = ( ( q , m ) --> q ) --> q ) , q = ( q , m ) , q = ( q , m ) --> q ) , q = ( q , m ) --> q ) , q = ( q , m ) , q = ( q , m ) ,
b . ( n + 1 ) in { g : g . ( n + 1 ) < g . ( n + 1 ) ;
f /* ( s ) is convergent ;
the lattice of lattice of lattice ( the lattice of L ) = the lattice of L ;
( a '&' b ) '&' ( a '&' b ) '&' ( a '&' ( b '&' c ) '&' ( b '&' ( b '&' c ) '&' ( b '&' c ) ) ) ) '&' ( b '&' ( b '&' c ) '&' ( b '&' ( c '&' ( b '&' ( c '&' ( b '&' ( a '&' ( ) '&' ( b '&' ( b '&' c )
reconsider B9 = ( ( the TopStruct of ( n + 1 ) ) ^ ( ( ( the TopStruct of ( n + 1 ) ^ ( ( n + 1 ) ) ^ ( ( n + 1 ) ) ^ ( ( n + 1 ) ) ) ;
( 1 - ( ( ( ( ( ( ( ( ( - ( ( - ( ( - ( - 1 ) ) ) ) - ( 1 - ( ( - ( ( ( ( ( ( ( ( - ( ( ( ( - ( ( ( ( ( ( ( - ( ( - - ) ) ) ) ) - ( ( - 1 ) ) ) ) ) ) ) )
set K = integral ( ( f , A ) , ( ( integral ( f , A ) ) * ( integral ( f , A ) ) , ( integral ( f , A ) , ( integral ( f , A ) ) ) , ( integral ( f , A ) ) = integral ( ( f , A ) ) ;
assume e in { w where w is Element of over n : w in { w where w is Element of of_w.r.t. w.r.t. w.r.t. w.r.t. w.r.t. F : w in F } ;
reconsider d = F . a as set ;
LIN f , f , f ;
assume X in T ( K ) \ T ;
assume Hom ( c , d ) <> {} ;
dom S = the carrier of S ;
assume x in { H } ;
* ( a * ( b - a ) ) = a * ( b - a ) .= a * ( a - a ) .= a * ( b - a ) .= a * ( a - a ) .= a * ( a - a ) .= a * ( b - a ) .= a - a ;
( D . ( ( ( ( ( ( ( ( ( ( r / 2 ) ) ) ) * ( 1 - r ) ) ) ) / ( 1 - r ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) / ( ( 1 - r ) ) ) ) ) / ( ( 1 - r ) ) ) ) ) / ( ( 1 -
ex p being Point of TOP-REAL 2 st p in P & p in P & q in P ;
for c being element holds f . c = g . c implies f . c = g . c
dom ( ( ( ( ( f | X ) ) | X ) | X ) | X ) ) ) = X /\ ( X /\ Y ) ;
1 = p * 1 .= p * p * p .= p * 1 .= p * p .= p * p * p .= p * p .= p * p .= p * 1 .= p * p .= p * p .= p * p .= p * p .= p * p .= p * p .= p * p .= p * p .= p * p .=
len g = len f + 1 ;
dom ( F | ( ( n + 1 ) ) = [: n , Seg n :] ;
dom ( f | ( t | ( q ) ) ) = dom ( f | ( q ) ) ;
assume a in ( ( ( ( ( a ) ) | D ) | ( ( D , D ) ) ) .: ( ( ( the carrier of S ) | ( D , D ) ) ) ) ) ;
assume that g is one-to-one and g is one-to-one and g is one-to-one ;
( ( ( ( x \ y ) ) \ ( x \ y ) ) \ ( x \ ( y \ z ) ) ) ) ` = ( ( ( x \ z ) \ ( x \ z ) ) \ ( x \ z ) ) \ ( x \ z ) ) ;
consider f such that f * f = id Z and f * f = id Z ;
( ( the function of TOP-REAL 2 ) | K1 ) | K1 is continuous ;
Index ( p , L ) -' 1 <= len p -' L ;
reconsider t = ( t , s = ( t , s ) . t as Element of S ;
Sup ( Frege Frege ( h ) . ( ( Frege h ) . ( ( Frege h ) ) ) ;
defpred P [ set ] means ex f being Function st f . $1 = f . $1 ;
Q [ x , y ] ;
consider x such that x in dom ( F . x ) and y in dom F . x ;
l . i < r . i ;
the Sorts of A = ( the Sorts of A ) +* ( the Sorts of A ) +* ( the Sorts of A ) ;
consider s such that rng s = rng s and rng s c= dom ( F * ( s * ( ) * ( F * ( F * ( F * ) ) ) ;
dist ( ( ( b , a ) ) - ( b + a ) < ( b - a ) - ( b - a ) ;
( \widetilde C ) /\ \widetilde L~ Cage ( C , n ) = \widetilde L~ Cage ( C , n ) ;
q `1 <= E-bound L~ f ;
LSeg ( f , i ) /\ LSeg ( f , i ) = {} ;
consider extended extended extended extended extended extended extended real-membered set such that [. a , b .] c= [. a , b .] ;
consider a , b such that a , b // a , b and a , b // a , b ;
set X = { b where b is Element of NAT : b <= n } ;
( ( ( ( x * y ) * z ) \ z ) \ z = ( ( x * z ) \ ( x * z ) ;
set { x } = [ x , y ] , z ] , z = [ z , y ] ;
l /. ( n + 1 ) = ( ( l /. n ) /. ( l + 1 ) .= ( ( ( l /. n ) /. ( l + 1 ) ) .= ( ( ( l /. n ) /. ( l /. ( ) ) ) .= ( ( ( ( ( l /. ( n + 1 ) ) /. ( n + 1 ) ) ) /. ( n + 1 ) ) .=
( ( ( ( - 1 ) - ( 1 - 1 ) ) / ( 1 - 1 ) ) ^2 = - 1 ;
( ( ( ( - 1 ) / ( 1 - 1 ) ) / ( 1 - ( ( 1 - 1 ) ) / ( 1 - 1 ) ) ^2 < ( - 1 ) / ( 1 - 1 ) ;
( S ) /\ ( ( S \/ Y ) ) = {} ;
( ( ( s - ( 1 - ( ( 1 - ( ( - ( 2 ) ) ) ) ) * ( ( 1 - 2 ) ) ) ) ) * ( ( 1 - 2 ) ) ) ) ) ) ) ) ) ) = ( ( ( ( 1 - 2 ) * ( ( 1 - 2 ) ) * ( ( 1 - 2 ) ) ) * ( ( 1 - 2
rng ( ( ( ( ( ( ( ( ( ( ( n + 1 ) + 1 ) ) / 2 ) ) * ( 1 - 1 ) ) * ( 1 - 1 ) ) ) ) ) ) ) ) ) ) ) ) is REAL ;
the carrier of X = the carrier of X ;
ex p being Point of TOP-REAL 2 st p in { p } & q in { p } ;
set h = \raise .4ex \hbox .4ex \hbox \hbox \hbox \hbox } \chi \chi } , A , A , A , A , A , A , A , A , .4ex .4ex \hbox .4ex \hbox .4ex \hbox .4ex \hbox .4ex \hbox .4ex \hbox .4ex \hbox .4ex } } } , A , A , A , A , A , A , A , A , A , A , A , A , A , A , A , A
R ^ ( n + 1 ) = ( R ^ ( n + 1 ) ^ ( R ^ ( n + 1 ) ) ^ ( R ^ ( n + 1 ) ) ;
Sum ( ( ( ( Sum ( F ) ) ) (#) ( F ) ) (#) ( ( F ) ) ) ) ) ) is summable ;
set f = ( the addF of K ) . ( 0. K , 0. K ) ;
S . ( ( b ) = ( b . ( b . ( ) ) ) . ( b . b ) .= ( b . b ) . b ;
( p in LSeg ( q , p ) ;
dom ( f | n ) = Seg n ;
assume o = ( the connectives of S ) . ( ( the connectives of S ) . ( the connectives of S ) ;
set \varphi = ( ( l , 1 ) *^ *^ *^ \varphi ) *^ \varphi ( l *^ \varphi *^ \varphi ) ;
cluster p \ast q -> irreducible ;
( Y | ( X \ Y ) ) /\ ( ( ( Y | ( X \ Y ) ) ) = {} ;
defpred X [ set ] means ex X being set st X [ $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 , $1 ,
consider k such that for n holds k < n + 1 ;
Det ( M * ( K , n ) ) = 0. K ;
sqrt ( ( b - c ) ^2 - ( b - c / 2 ) ^2 < b - c ;
C . ( ( d + 1 ) mod ( d + 1 ) = ( d mod ( ( d + 1 ) ) mod ( ( d + 1 ) mod ( ( d -' 1 ) ) mod ( ( d -' 1 ) ) mod ( ( d -' 1 ) ) ) ;
assume that X is open and X is open and X is open ;
deffunc F ( Element of I , set , set , set , set , set , set , set , set , set , set , set , set , f = ( the carrier of I ) \ ( the carrier of I ) ;
t ^ <* t *> ^ <* 1 *> in succ t ;
( x \ y ) \ ( x \ y ) = ( 0. X ) \ ( x \ y ) .= 0. X ;
cluster X -> empty for Subset of [: X , Y :] ;
We We We We We We We We We We We We We We We A A for A , B being Subset of X holds A /\ B is open ;
len ( M * ( len M ) ) = len ( ( M * ( len M ) ) ;
v = \notin Carrier ( L ) ;
( ( ( ( ( ( ( m * n ) * m ) ) * m ) mod m ) ) mod m ) ) mod m ) ) mod m <> 0 ;
inf divset ( D , k ) = ( inf divset ( D , k ) ;
g . ( ( ( - 1 ) / 2 ) * ( 1 - 1 ) ) / 2 = ( - 1 / 2 ) / 2 ;
||. a / 2 .|| = ||. a / 2 .|| ;
f . ( x , y ) = ( f . ( x , y ) ;
ex w being Element of NAT st ( w in dom ( w ^ <* 1 *> ) ^ <* 1 *> ) & w in dom w ;
\llangle 1 , 1 , 2 , 3 , 4 , 5 , 5 , 6 , 5 , 6 , 6 , 5 , 6 , 6 , 6 , 5 , 6 , 6 , 6 , 6 , 5 , 6 , 6 , 5 , 6 , 6 , 5 , 6 , 5 , 6 , 6 , 5 , 6 , 6 , 6 , 6 , 5 , 6 , 6 , 6 ,
IC Exec ( i , s ) = succ IC Exec ( i , s ) .= succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ ( succ ( i + 1 ) ) .= succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ ( i + 1 ) ;
IC Comput ( P1 , s , 1 ) = IC Comput ( P1 , s , 1 ) .= IC Exec ( i1 , SCMPDS ) .= IC Exec ( i1 , SCMPDS ) .= succ ( i1 + 1 ) .= succ ( i1 + 1 ) .= succ ( i1 + 1 ) .= succ ( i1 + 1 ) .= i1 + 1 ) .= i1 + 1 .= i2 + 1 ;
IExec ( IExec ( B1 , Q , t ) . intpos i = IExec ( IExec ( B1 , Q , t ) . intpos i ) ;
LIN f , f , p ;
assume for x being element holds x in C implies x in C
inf ( f `| X ) = inf ( f `| X ) ;
let F be Function of X , Y ;
||. R /. ( m + 1 ) - R /. ( m + 1 ) .|| < R /. ( m + 1 ) ;
assume a in { q : q .| < r } ;
set p = [ p , q ] , q = [ p , q ] , p = [ p , q ] , q `2 ] , p = [ p , q ] , q `2 , q `2 ] , q `2 , q `2 ] , q `2 ] ;
consider x such that x in X and y in X and x in X ;
for y being Element of X holds y in X implies x in X
func @ -> -> -> @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
consider t being Element of S such that t = t . x and x <> t ;
dom ( x * ( x * y ) ) = Seg len ( x * y ) ;
consider x being element such that x in ]. y , x .[ and y < x ;
||. f /. x0 .|| = ||. f /. x0 .|| ;
( the carrier of X ) /\ ( the carrier of X ) = {} ;
assume that i in dom p and j in dom p and i in dom p ;
reconsider h = f | ( X \/ Y ) as Function of X , Y ;
x1 in the carrier of X ;
defpred P [ Element of NAT ] means ex M st f . $1 = ( f . $1 ) * f . $1 ;
T |^ ( ( s , t ) = s |^ ( ( s + t ) .= s |^ ( s + t ) .= s |^ ( s + t ) .= s |^ ( s + t ) .= s |^ ( s + t ) .= s |^ ( s + t ) .= s |^ ( s * t ) .= s |^ ( s * t ) .= s |^ ( s * t ) .= s
- ( - ( x - y ) ) = - ( - x - y ) ;
consider a , b being Point of I[01] such that a in B and b in B ;
f = id ( ( dom f , g ) , f = f | ( ( ( ( ( ( id A ) | ( ( ( B ) ) | ( B , A ) ) ) ;
let k be Nat ;
for x being element holds x in A implies x in A
consider u such that u = a * u ;
- 1 - ( ( ( - ( p `1 / ( 1 - sn ) ) / ( 1 - sn ) ) ) ^2 > 0 ;
( L . ( ( L . k ) ) /\ ( L . k ) ) = {} ;
set i = AddTo ( a , b ) ;
If B is universal , then B is universal & B is universal & B is universal ;
{ a where d is Element of D : a in D } ;
| ( ( ( ( ( \square | ( n + 1 ) ) | ( ( n + 1 ) ) | ( ( ( n + 1 ) ) | ( ( n + 1 ) ) | ( ( n + 1 ) ) ) ) ) is bounded ;
( ( - ( f | A ) ) | A ) | A is bounded ;
( G * ( ( ( ( ( ( ( ( ( ( G * ( len G ) ) ) ) ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ;
( ( ( Proj ( i , n ) * ( i , n ) ) * ( i , n ) = ( ( Proj ( i , n ) * ( i , n ) ) * ( i , n ) .= ( ( Proj ( i , n ) * ( i , n ) ) * ( i , n ) .= Proj ( i , n ) * ( i , n ) .= ( Proj ( i , n ) * ( i , n )
( f + ( ( ( reproj ( i , n ) ) * ( i + 1 ) ) ) * ( ( ( f | ( i + 1 ) ) (#) ( ( f | ( i + 1 ) ) (#) ( ( f | ( i + 1 ) ) (#) ( ( f | ( i + 1 ) ) (#) ( f | ( i + 1 ) ) ) ) ;
( the function of ( ( ( the function ) ) | ( ( the carrier of TOP-REAL 2 ) ) | ( the carrier of TOP-REAL 2 ) ) ) ) ) ) . x = ( the function of ( ( the carrier of TOP-REAL 2 ) | ( the carrier of TOP-REAL 2 ) ;
ex t being State of S st t = ( t . x ) . t ;
defpred P [ set ] means ex A being Ordinal st P [ A , $1 ] ;
consider y being element such that y in dom p and p . y = q . y ;
reconsider L = Carrier ( l , i ) as being_line ;
for c being Element of C holds c . d = ( id d ) . c ;
( f , n ) ^ ( f , p ) = f ^ p ;
( f * g ) * f = ( ( f * g ) * f ;
p in LSeg ( ( f , i ) ;
f + ( - f + ( - f ) ) = ( - - f ) - f ;
consider r such that r in rng ( f | X ) and for x st x in X holds ( f | X ) . x = ( f | X ) . x ;
( f . ( 1 - 1 ) / ( 1 - 1 ) in { r : r < 1 / 2 } ;
eval ( a , x ) = a * ( a * x ) .= a * a .= a * a .= a * a .= a * a .= a * a .= a * a .= a * a .= a * a .= a * a .= 0. L .= 0. L * a .= 0. L .= 0. L .= 0. L * a .= 0. L .= 0. L * a .= 0. L .= 0. L .= 0. L * a .= 0. L .=
z = eval ( t , x ) .= eval ( t , x ) .= eval ( t , x ) .= eval ( t , x ) .= eval ( t , x ) .= eval ( t , x ) .= eval ( t , x ) .= ( eval ( t , x ) ) ;
set H = the TopStruct of X ;
consider S such that S = S ^ T ^ T ;
assume that that x in dom f and f . x = g . x ;
- 1 <= ( - ( ( ( - ( ( 1 - 1 ) ) / ( 1 - ( 1 - 1 ) ) ) / ( 1 - 1 ) ) ) ;
Sum ( V ) is Sum ( V ) ;
let k , l be Nat ;
consider j such that j in dom ( a * ( b * a ) and ( a * a ) . j = ( a * b ) . j ;
( ( ( ( ( ( ( 1 - 1 ) ) | ( 1 - 1 ) ) | ( 1 - 1 ) ) | ( ( 1 - 1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . x c= ( 1 - 1 ) | ( ( 1 - 1 ) ) ;
consider p such that a = p * a * p + p * p ;
assume a <= b & b in dom f ;
cell ( C , n , m ) is non empty ;
{ A } in { S where S is Subset of T : S is open } ;
( T * T ) * ( b * T ) = ( b * T ) * ( b * T ) .= b * ( b * T ) .= b * ( b * T ) .= b * ( b * T ) .= b * ( b * T ) .= b * T * ( b * T ) .= b * ( b * T ) .= 0. L * ( b * T ) .= 0. L * ( b * T ) .= 0. L * ( b *
g . ( s , I ) = s . ( s , I ) ;
( ( ( ( ( ( ( to_power n ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( n + ) ) ) * ( ( n + 1 ) ) ) ) ) ) ) * ( ( ( ( n + 1 ) ) ) * ( ( ( n + 1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) * ( ( ( ( (
p => q => q => p => q => p => q => p => q => q => p => q => q => p => q => q => p => q => q => q => p => q => q => q => p => q => q => q => q => q => p => q => q => p => q => q => q => q => q => p => q => q => q => q => q => q => q => p => q =>
dom ( ( the Tran of [: the carrier of [: the carrier of [: the carrier of M , the carrier of M :] ) = the carrier of [: the carrier of M , the carrier of M :] ;
cluster f | ( X \/ Y ) -> constant for Function ;
for a being element holds X [ a , f . a ]
i = ( p + ( 1 - 1 ) ) * ( p + 1 ) ;
( l + 1 ) * ( ( l + 1 ) ) = ( ( ( l + 1 ) * ( l + 1 ) * ( l + 1 ) ) * ( ( l + 1 ) ) ;
CurInstr ( P1 , Comput ( P2 , s2 , 1 ) ) = CurInstr ( P2 , s2 ) ;
for n being Nat holds ( ( Partial_Sums ( s ) ) ^\ n ) ^\ n = ( Partial_Sums ( s ^\ n ) ^\ n ) . n ;
sin | ( ( ( r * ( 2 * ( 2 * ( 2 * ( 2 * r ) ) ) ) ) = ( ( 2 * ( 2 * r ) ) | ( 2 * ( 2 * r ) ) ) | ( 2 * ( 2 * r ) ) ) ) ;
set q = ( ( ( ( ( ( ( ( ( g | ( n + 1 ) ) ) | ( n + 1 ) ) | ( n + 1 ) ) ) | ( n + 1 ) ) ) ;
consider G being set such that G = F . ( n + 1 ) ;
consider G such that F = G . ( k + 1 ) and G . k = G . k ;
the Arity of S in the Sorts of S ;
Z c= dom ( ( ( ( ( id Z ) (#) ( ( ( ( ( ( ( ( - 2 ) * ( ( - 2 ) * ( ( - 2 ) * ( ( - 2 ) ) ) ) ) (#) ( ( ( ( - 2 ) * ( ( - 2 ) ) ) ) (#) ( ( ( - 2 ) * ( ( - 2 ) ) ) ) ) ) ) ) ) ) ;
for k being Nat holds ( ( r (#) f ) . k = ( r * f ) . k ;
assume that 1 < ( ( 1 - ( ( ( ( ( 1 - 1 ) ) ) ) / ( 1 - ( 1 - ( 1 - 1 ) ) ) ) ^2 ) ^2 > 0 ;
assume that f . a = f . a and f . a = f . a ;
consider r being Element of REAL such that r = ( ( q `1 ) * ( r * ( r * ( 1 - r ) ) * ( 1 - r ) ) ;
f /. i in L~ f ;
assume that x in the carrier of K and y in the carrier of K ;
assume f * ( ( ( proj ( i , n ) ) * ( i + 1 ) ) ) * ( ( ( proj ( i , n ) * ( i + 1 ) ) ) * ( ( proj ( i , n ) * ( ( i + 1 ) ) ) ;
rng ( M * ( ( ( M * ( n , m ) ) ) * ( M * ( n + m ) ) ) ) ) = ( ( ( M * ( n + m ) ) * ( M * ( n + m ) ) ) * ( M * ( n + m ) ) ) ) ;
assume z in the topology of T ;
consider l such that l < m ;
consider t be Element of REAL such that t = ( ( ( - 1 ) * ( t - 1 ) ) * ( t - 1 ) ) * ( t - 1 ) ;
assume that v in the carrier of Lin ( A ) and v in Lin ( A ) ;
consider a being Element of the carrier of A such that a is being_line and a <> 0. A ;
( ( ( ( ( ( - 1 ) * ( x - 1 ) ) * ( x - 1 ) ) ) * ( ( x - 1 ) ) ) ) ) * ( ( x / 2 ) ) ) ) ) ) ) = ( ( ( - 1 ) * ( x - 1 ) ) * ( x - 1 ) ) * ( x - 1 ) ) ;
cluster p * q -> FinSequence of D ;
defpred R [ element ] means ex x being element st x in dom ( ( f | $1 ) | $1 ) & x in ( ( f | $1 ) | $1 ) ;
( \widetilde L ) /\ ( \widetilde L ) = {} ;
i + 1 < len h ;
for n being Nat holds F . n = ( F . n ) * ( F . n ) ;
for r being Real holds s in [. s , r .] implies s in [. s , s .]
assume v in { G where G is Subset of T : G is open } ;
g be Function of A , REAL ;
min ( min ( x , y ) , min ( x , y ) ) = min ( min ( x , y ) , min ( min ( x , y ) , min ( x , y ) ) ;
consider q such that for n holds q . n = ( q . n ) * ( q . n ) ;
consider f such that for n holds f . n = ( f . n ) * ( f . n ) ;
reconsider BB = B as Subset of X ;
consider j such that j = n and j = n and n = j and j = n ;
consider x being element such that x in succ ( ( O \/ { 0 } } ) and x in { O } ;
( ( C * ( ( Talalalalalalalalal) | ( ( n + 1 ) ) | ( n + 1 ) ) ) ) is constant ;
dom ( ( ( X --> Y ) ) = ( ( X --> Y ) --> ( X --> Y ) ) | ( X \/ Y ) ;
S is compact implies S is compact
attr x is collinear means : Def2 : ex y being Point of T st x = y & y in S & x = y & y in S ;
consider X such that X in dom ( ( f | X ) | X ) and f | X is one-to-one ;
assume that that a * b is continuous and a * b is continuous and a * b is continuous ;
cos | ( ( ( 1 - 1 ) ) * ( ( 1 - 1 ) ) ) is continuous ;
defpred P [ Element of NAT ] means ex A being Subset of X st ( ( A \/ ( $1 \/ A ) ) ` ) ` = ( ( ( ( A \/ A ) ` ) ` ) ` ) ` ) ` ;
IC Comput ( P1 , s , 1 ) = IC Comput ( P1 , s , 1 ) .= IC Exec ( i1 , Comput ( P1 , s , 1 ) .= IC Exec ( i1 , 1 ) .= ( card I .= card I .= card I .= card I .= card I .= card I ;
f . ( ( x * y ) = f . ( x * y ) .= f . ( x * y ) ;
( M * ( ( ( ( ( ( M * ( n , m ) ) ) ) ) ) ) | ( ( n + m ) ) ) ) ) ) ) ) = ( ( M * ( ( n + m ) ) | ( ( n + m ) ) | ( ( n + m ) ) ) ;
the support support support ( L + ( A \/ B ) ) c= the support of L ;
assume that a in dom ( ( a * b ) * ( b * c ) ) and b in dom ( a * c ) ;
( the carrier of X ) /\ ( the carrier of X ) is open ;
then ( - 1 - 1 ) * ( ( - 1 ) / ( 1 - 1 ) ) ^2 >= 0 ;
reconsider s = p ^ q as Element of NAT ;
[ x , y ] in [: x , y :] ;
for F being Nat holds F . ( m + 1 ) = F . m ;
len ( G * ( i , j ) ) = len ( G * ( i , j ) ;
consider u such that u in W and v in W and u in W ;
consider F such that dom F = Seg n and for n st n in Seg n holds F . n = ( F . n ) * ( F . n ) ;
0 = 1 - 1 ;
consider n such that for x st x in Z holds ( ( ( ( ( ( x + n ) * ( x - n ) ) * ( x - n ) ) * ( x - n ) ) * ( ( x - n ) ) ) . x ;
for defined being defined holds ( defined defined defined defined : defined : the defined : defined : the defined : ( defined : defined : the defined : ( defined : defined : the defined : defined : ( defined is defined : defined : the defined : ( defined : defined : ( defined : defined : ( defined : defined : defined : ( defined : defined : ( defined : defined : ( defined : defined : ( defined : defined : ( defined : defined : ( defined : defined : ( defined : defined :
\bigsqcap ( B , S ) = \bigsqcap & inf ( B ) = {} ;
r / 2 < r / 2 ;
for x being element holds ( ( ( x in X ) \ Y ) /\ ( ( X \/ Y ) ) implies ( ( ( X \/ Y ) \/ Y ) implies ( ( ( X \/ Y ) \/ Y ) /\ ( ( X \/ Y ) ) ) /\ ( ( ( X \/ Y ) ) ) ) ) ) ) /\ ( ( X \/ Y ) ) = ( ( ( X \/ Y ) /\ ( ( X \/ Y ) ) )
2 * ( ( 1 - 1 ) / 2 * ( 1 - 1 ) / 2 * ( 1 - 1 ) = ( 1 - 1 ) * ( 1 - 1 ) ;
reconsider p = P as Point of TOP-REAL 2 ;
consider x being element such that x in ]. x , y .[ and y in ]. x , y .[ and x < t ;
for n being Nat holds ( ( ( ( ( ( ( ( ( q ) ) * ( n + 1 ) ) * ( q ) ) * ( n + 1 ) ) ) ) * ( n + 1 ) ) ) ) * ( n + 1 ) ) ) ) * ( ( n + 1 ) ) ) ) ) * ( n + 1 ) = ( ( ( q ) * ( q ) * ( n + 1 ) ) * ( n + 1 ) ) * ( ( n +
consider y being element such that y in the carrier of A and z in the carrier of A and y in the carrier of A ;
consider x being Element of X such that x = ( ( ( the Tran of A ) . x ) . x ;
let S be non empty ManySortedSign ;
[: a , b :] in [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of , the carrier of :] :] :] ;
reconsider m = ( ( ( n + 1 ) * ( n + 1 ) ) * ( n + 1 ) as Element of NAT ;
I I c= the carrier of SCM+FSA ;
( ( ( ( ( f /* s1 ) /* s1 ) (#) f ) /* s1 ) (#) f ) /* s1 = f /* s1 /* s1 ;
If A is being_line , then A is being_line & ( A is being_line & ( A is being_line implies A is being_line & ( A is being_line implies A is being_line ) & ( A is being_line implies A is being_line ) & ( A is being_line ) & ( A is being_line implies ( A is being_line ) & ( A is being_line ) & ( A is being_line ) implies ( A is being_line ) & ( A is being_line ) implies ( A is being_line ) implies ( A is being_line ) & ( ( A is being_line ) implies ( ( A is being_line ) & ( ( A is being_line ) implies (
func A -> Subset of C equals ( the Sorts of C ) \ ( the Sorts of C ) \ ( the Sorts of C ) ;
dom ( ( p * q ) = Seg len ( p * q ) ;
cluster \llangle x , y \rrangle -> \llangle x , y \rrangle ;
E |= { \forall _ { x } } => ( { x } } => ( { x } ) => ( ( x => ( { x } ) => ( x => ( x => ( x => ( x => ( { x } ) => ( x => ( x => ( x => ( ) => ( x => ) ) ) ) ) ) => ( x => ( x => ( x => ) ) ) ) ) ) ) ) ) => ( x => ( x => ( x => ( x => ( x => ( x => ( x => ( x => ( x => ( x
F .: ( F .: .: ( X , Y ) ) = F .: ( X , Y ) ;
R . ( ( ( ( ( ( ( m - - n ) ) * ( m - n ) ) * ( ( m - n ) ) * ( m - n ) ) ) ) ) ) ) ) ) ) ) ) ) / ( ( m - n ) ) ) / ( ( m - n ) ) ) ) / ( ( m - n ) * ( m - n ) ) ) / ( ( m - n ) ) ) ) ) ) ) / ( ( ( m - n ) ) ) ) / ( ( ( m - n ) ) ) / ( ( (
Int cell ( G , i , j ) misses UBD L~ f ;
IC Comput ( P1 , s , 1 ) = IC Comput ( P1 , s , 1 ) .= IC Exec ( i1 , i2 ) .= IC Exec ( i1 , i2 ) .= IC Exec ( i1 , i2 ) .= ( card I ) .= card I .= card I .= card I .= card I ;
sqrt ( 1 - ( ( ( ( ( ( ( 1 - ( ( 1 - ( 1 - ( 1 - 1 ) ) ) ) ) ) ) ^2 + ( ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( ) - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( ) - ( 1 - ( ) - 1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ^2 >= 1 ;
consider x being element such that x in dom ( ( ( g | ( a | ( n + 1 ) ) | ( n + 1 ) ) | ( n + 1 ) ) ) ;
dom ( ( r (#) ( f , A ) ) = [: A , A :] ;
reconsider d = ( ( ( ( y ) ) | ( ( z | ( ( z | ( z | ( z | ( ) ) ) | ( z | ( z | ( ( z | ( z | ( ( z | ( ( z | ( ) ) ) | ( ( z | ( ( z | ( ) ) ) ) | ( z | ( z | ( ) ) ) ) ) ) ) as Nat ;
for i being Nat holds A /\ ( B \/ C ) = {}
assume that that for x holds x in dom ( f | X ) ;
p in Cl A implies p in Cl A
for x being Element of REAL holds ( - ( x - y ) - ( x - y ) ) * ( ( - y ) ) * ( ( x - y ) ) = ( - y ) * ( ( x - y ) * ( x - y ) - ( ( - y ) * ( x - y ) ) ;
func func exp -> exp ( a *^ exp ( b *^ exp ( b *^ exp ( b *^ exp ( b *^ exp ( b *^ exp ( b *^ exp ) ) ) ) -> limit_ordinal ;
[: [: A , the carrier of [: A , the carrier of [: A , the carrier of [: A , the carrier of [: A , the carrier of [: A , the carrier of [: A :] :] :] :] , the carrier of [: A , the carrier of [: A , the carrier of [: A , the carrier of [: A , the carrier of [: A , the carrier of [: A :] :] :] :] :] ;
ex a being element st a in the carrier of S & b in the carrier of S & a in the carrier of S ;
||. ( v - ( x - y ) ) .|| < ||. v .|| ;
for Z being set holds Z in Y implies Z in Y
sup ( compactbelow compactbelow compactbelow ( x ) ) = sup compactbelow ( sup compactbelow x ) .= sup compactbelow ( sup ( compactbelow x ) .= sup compactbelow ( sup compactbelow x ) .= sup compactbelow ( sup compactbelow x ) ;
consider i such that i in dom f and f . i = ( f . i ) . i ;
consider p being FinSequence of D such that p ^ q = p ^ q ;
consider c such that c in the carrier of X and a <> c ;
set U = I \! U as Function ;
|. q .| = |. q .| ^2 ;
let x , y be Element of T ;
dom ( ( ( ( the Sorts of A ) ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) ) . ( ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( ( the Sorts of A
dom ( h | X ) = X /\ Y ;
for N being set , K being Field st N c= [: N , N :] holds [: N , N :] = [: N , N :]
( mod m mod m ) mod m = ( mod m ) mod m mod m .= ( m mod m ) mod m mod m mod m mod m .= ( m mod m ) mod m ) mod m mod m ) mod m ;
( - ( ( 1 - ( 1 - 1 ) ) * ( 1 - 1 ) ) ^2 >= 0 ;
If for r holds ( r (#) ( f | X ) ) | X is constant and for x being Point of TOP-REAL 2 holds f . x = r * ( f | X ) ;
( v | ( X /\ Y ) ) | ( X /\ Y ) is bounded ;
assume that a <> 0 and b <> 0 ;
consider i such that i in dom p and p . i = q . i ;
|. p .| ^2 + ( |. p .| ^2 + ( |. p .| ) ^2 + ( ( |. p .| ) ^2 + ( ( |. p .| ) ^2 ) ^2 ) ^2 ) ^2 ) ^2 ) ^2 ;
consider p such that p in { p } and q ^ <* p *> ;
( multcomplexcpfunc ( A ) . ( ( r , A ) ) = ( r (#) ( ( r (#) ( A * A ) * ( r * A ) ) ;
( ( ( ( ( ( ( TOP-REAL 2 ) | D ) ) | ( A ) ) | ( A /\ ( A /\ ( ( A /\ ( ( A /\ ( ( A /\ ( ( A /\ ( ( A /\ ( ( A /\ ( ( A ) ) ) | ( A ) ) ) ) ) ) ) ;
s |= |= |= All ( x3 , x2 ) implies |= |= All ( x3 , x2 , x3 )
card ( card ( b + 1 ) ) = card ( b + 1 ) ;
consider z being element such that z in X and z in X and y >= z ;
( UMP D ) /\ ( ( ( ( ( UMP D ) \/ D ) /\ ( ( ( ( ( ( ( UMP ) ) /\ D ) ) ) ) ) ) ) ) ) ) ) ) ) ;
lim ( ( ( f /* ( ( g - f ) /* ( f /* ( h ^\ n ) ) /* ( f /* ( g ^\ n ) ) ) ) ) ) ) ) ) = ( ( ( ( ( ( ( f /* ( f /* ( f /* ( g ^\ n ) /* ( /* ( g ^\ n ) /* ( g ^\ n ) ) ) ) /* ( ( ( g /* ( g ^\ n ) (#) f /* ( g ^\ n ) ) ;
P [ i + 1 ] ;
for r be Real holds ( ( for n holds ( ( for m st n >= m holds ||. seq . m - seq . m .|| * seq . m .|| * seq . m - seq . m .|| < r / 2 ;
let a , b be Point of TOP-REAL 2 , a , b be Point of TOP-REAL 2 ;
Z c= dom ( ( ( - ( ( ( 1 - 1 ) ) (#) ( f - ( 1 - 1 ) ) (#) f ) ) / ( ( 1 - 1 ) ) ) ) ;
ex j being Nat st x = y & y = z ^ <* x *> ;
for u being element st u in REAL holds r * u in REAL & r * v in REAL
A , B , C , C , D , C , D , C , C , C , D , C , C , C , D , C , A , C , C , Q , Q , Q , Q , Q , P , Q , Q , Q , P , Q , Q , Q , Q , P , Q , Q , P , Q , Q , Q , Q , Q , P , Q , Q , P , Q , Q , Q , Q , Q , Q , P , Q , P , Q , Q , Q , P , Q , Q , Q , Q , P , P , Q
- ( v - u ) = - u - v - u .= - - ( - u - v ) .= - ( - u - v ;
Exec ( i1 , succ succ succ succ succ succ succ succ succ succ i1 = succ succ succ i1 ;
consider f such that for x holds f . x = ( f . x ) * ( f . x ) ;
cluster \pi | ( D , D ) -> empty ;
card X = card X ;
Cage ( C , n ) in \widetilde L~ f ;
let T be DecoratedTree of T , T be DecoratedTree of T , p , q be decorated with with with with T ;
\llangle i , j \rrangle in Indices G ;
func func func func func func func ( m + n ) -> Nat means : Def2 : it : it divides m + n ;
rng F is finite ;
consider C being Subset of Lin ( A ) such that C = B \/ C ;
for V being Subset of T holds V is open
set X = { v : v in Lin ( B ) } ;
\measuredangle ( p , q ) = \measuredangle ( p , q ) ;
- ( ( 1 - ( 1 - 1 ) ) / ( ( 1 - 1 ) ) = ( - 1 / ( ( 1 - 1 ) ) / ( 1 - 1 ) ) ;
ex f being Function st f | ( ( n + 1 ) = f | ( n + 1 ) ;
assume that f | ( ( X , Y ) is constant and f | ( X , Y ) is constant ;
ex r being Real st ( ( ex x being Point of TOP-REAL 2 st r < 1 & x < r & r < 1 / 2 ;
assume that f is being_line and f is being_line and f is being_line and f is being_line ;
assume that i in dom ( G * ( i , 1 ) ) and ( ( G * ( i , 1 ) ) * ( ( ( G * ( i , 1 ) ) * ( i , 1 ) ) * ( ( ( ( G * ( i , 1 ) ) * ( ( ( ( ( ( ( ( ( ( the carrier of TOP-REAL 2 ) * ( ( ( the carrier of TOP-REAL 2 ) * ( ( the carrier of TOP-REAL 2 ) ) ) * ( ( ( the carrier of TOP-REAL 2 ) ) * ( ( the carrier of TOP-REAL 2 ) ) * ( ( ( the carrier of TOP-REAL 2 ) * ( ( the carrier
consider c such that c = ( c + b ) /. c and c /. c = ( c /. c ) /. ( c + 1 ) ;
u in Ball ( r , 1 ) ;
card ( X \/ Y ) = card ( X \/ Y ) .= card ( X \/ Y ) ;
If for M , M1 , M2 being Matrix of len M , K holds M1 = M2 * ( M * ( ( ( M1 + M2 ) * ( M * ( M1 + M2 ) ) ) - ( M2 * ( M1 * ( M2 * ( M2 * ( M2 ) ) ) ) ;
consider g such that for x st x in Z holds ||. g /. x - g /. x .|| < g /. x ;
assume x < ( - ( b - c ) / 2 ) * ( b - c ) ;
( G * ( i + 1 ) ) ^ <* i *> = ( G ^ ( i + 1 ) ^ <* i *> ) ^ ( G ^ ( i + 1 ) ) ^ ( G ^ ( i + 1 ) ) ^ ( G ^ ( i + 1 ) ) ) ) ^ ( G ^ ( i + 1 ) ) ) ) ) ) ) ) ) ) ^ ( G ^ ( i + 1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ^ ( ) ) ) ) ^ ( = ( G ^ ( i + 1 ) ^ ( G ^ ( G ^ ( i + 1 ) ^ ( ( G
for i , j being Nat st M < j & M * ( i + 1 ) < ( M * ( i + 1 ) * ( ( i + 1 ) ) * ( i + 1 ) + ( M * ( i + 1 ) ) * ( M * ( i + 1 ) ) ;
for i being Nat holds f . i <> f . i ;
assume that F = ( a \ b ) \ ( a \ b ) \ ( a \ b ) ;
( b * ( a * b ) - ( b * ( b - b ) ) * ( b - b ) ) * ( b - b * ( b - a ) ) ) = ( b - b ) * ( b - a ) ;
ex D being Subset of T st D = B & D in B ;
If for s being Real holds s is summable & s is summable & ( Partial_Sums ( s ) is summable implies Partial_Sums ( s is summable & Partial_Sums ( s ) is summable & Partial_Sums ( s is summable ) & Partial_Sums ( s is summable ) & Partial_Sums ( s is summable implies Partial_Sums ( s is summable & Partial_Sums ( s is summable ) . n = Sum ( s is convergent & ( Partial_Sums ( s is summable ) . n = Sum ( s . n ) ;
dom ( ( ( ( the carrier of TOP-REAL 2 ) | D ) | D ) ) = ( the carrier of TOP-REAL 2 ) | D ;
X , Z // ( ( the carrier of [: Z , Y :] , Z :] --> [: Z , Z :] :] --> id Z ;
( ( G * ( i + 1 ) ) * ( i + 1 ) `1 <= ( ( ( G * ( i + 1 ) ) * ( i + 1 ) ) * ( i + 1 ) ) * ( i + 1 ) ) ) * ( ( ( G * ( i + 1 ) ) * ( i + 1 ) ) ) ) ) ) ) ) * ( ( ( ( ( ( G * ( i + 1 ) ) * ( ( ( G * ( i + 1 ) ) * ( ( ( ( ( i + 1 ) ) ) * ( ( ( ( ( ( ( G * ( i + 1 ) ) ) * ( ( ( ( ( ( ( ( ( ( ( ( (
cluster M1 + M2 -> M1 for Matrix of K ;
consider a being element such that a in dom ( ( F | A ) and b in A ;
We We We We We We We We We We We We We We We We We (# -> -> -> -> -> -> Element -> -> -> -> Element -> -> -> -> -> -> -> -> -> Element -> Element -> Element -> -> Element -> Element -> -> -> -> -> -> -> -> -> -> -> -> Element -> Element -> -> Element -> Element -> -> Element -> -> Element -> -> Element -> -> Element -> -> -> Element -> Element -> -> Element -> -> Element -> Element -> Element -> -> -> -> -> -> -> Element -> -> -> -> -> -> -> Element -> Element -> -> -> -> -> -> -> -> -> -> Element -> Element -> -> Element -> Element -> -> Element -> -> Element -> -> Element -> -> Element -> Element -> -> Element -> Element -> -> -> -> -> ->
( b + b ) + c = b + b ;
cluster ( ( ( - 1 ) * ( i + 1 ) ) * ( i + 1 ) ) * ( ( i + 1 ) ) ) * ( i + 1 ) = ( ( ( - 1 ) * ( i + 1 ) ) * ( i + 1 ) ) * ( i + 1 ) ) ;
- 1 / ( ( ( ( ( ( - 1 ) ) / 2 ) * ( 1 - ( 1 - 1 ) ) / ( 1 - 1 ) ) ) ) ) ) ) ) ) / ( 1 - ( 1 - ( ( 1 - 1 ) ) / ( 1 - 1 ) ) ) ) ) ) ) / ( 1 - ( 1 - 1 ) ) ) ) < ( 1 - ( 1 - 1 ) / ( 1 - 1 ) ) / ( 1 - 1 ) ;
( ( a *' ) *' ) *' = a *' ;
assume that the TopStruct of T is open and the TopStruct of T = the TopStruct of T ;
assume that 1 <= ( q | ( ( n + 1 ) ) | ( ( n + 1 ) ) ;
2 * ( ( a + b ) - c * ( a * c ) ) ^2 >= 0 ;
M \models _ _ { M } => ( { x } ) => ( { x } ) => ( { x } ) ) => ( { x } ) ) => ( { x } ) ) ) ) ) ) ) ) ) ) ) ) ) ) ;
assume that f . x <> 0 and l . x > 0 ;
let e be set , W be Walk of G , e be set ;
( c is not empty implies not c is not empty
the indices of f = [: the carrier of TOP-REAL 2 , the carrier of TOP-REAL 2 :] ;
let G be finite Group , a be Element of G ;
Int Cl ( ( ( intloc ( 0 ) ) ) ) = \mathop { \rm intloc ( 0 , SCM+FSA ) ) ;
for f being Function of X , Y holds f ^ <* p *> ^ f ^ f ^ f ^ f ^ f ;
( ( ( ( ( ( p `1 ) / ( 1 - sn ) ) ^2 ) ^2 + ( ( ( 1 - sn ) ) ^2 ) ^2 ) ^2 ) ^2 ;
let x , y be Point of TOP-REAL 2 ;
for x being element holds ( ( ( x + y ) (#) ( x - y ) ) (#) ( ( x - y ) (#) ( x - y ) ) ) (#) ( x - y ) ) ) implies ( ( x - y ) (#) ( x - y ) ) (#) ( x - y ) ) ) (#) ( x - y ) = ( x - y ) (#) ( x - y ) (#) ( ( x - y ) ) (#) ( x - y ) ) (#) ( x - y ) ) (#) ( ( x - y ) (#) ( x - y ) (#) ( ( x - y ) ) (#) ( ( x - y ) (#) ( ( x - y ) ) (#) ( x - y ) (#) ( x - y ) ) (#) ( x
consider P being Subset of T such that P is open and P is open and P is open and P is open ;
( a '&' b ) '&' ( a '&' b ) = ( a '&' ( b '&' c ) '&' ( ( a '&' ( b '&' c ) '&' ( b '&' c ) ) '&' ( b '&' c ) ) '&' ( ( b '&' c ) '&' ( ( b '&' c ) '&' ( b '&' c ) ) '&' ( ( b '&' c ) '&' ( b '&' c ) ) '&' ( ( b '&' c ) '&' ( ( b '&' c ) '&' ( ( b '&' c ) ) '&' ( ( ( b '&' c ) ) ) '&' ( ( b '&' c ) '&' ( ( b '&' c ) ) '&' ( ( b '&' c ) ) '&' ( ( ( b '&' c ) ) '&' ( b '&' ( c '&' c ) ) '&' ( (
for e being set holds e in X implies e in X
for i being set holds ( for i be Nat holds P [ i + 1 ] implies P [ i + 1 ]
for v , w holds w <> ( v , w ) implies w , y // ( v , w ) , ( v , w )
card ( ( ( ( 1 - 1 ) / 2 ) ^2 ) ^2 = 1 - 1 ;
IC Exec ( i , s ) = succ IC Exec ( i , s ) .= succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ succ 0 .= succ 0 .= succ succ succ succ succ succ succ 0 .= succ succ succ succ succ 0 .= succ succ 0 .= succ succ succ succ 0 .= succ succ succ succ succ 0 .= succ succ succ succ 0 .= succ succ succ succ succ succ 0 .= succ succ succ succ 0 .= succ succ 0 .= succ succ 0 ;
len f = len f - 1 ;
for a , b being Real holds a < b & b < d & d < b & a < b & b < d & a < b / 2
let p be Point of TOP-REAL 2 ;
lim ( ( ( ( ( ( ( ( lim ( F + 1 ) ) (#) F ) (#) F ) ) (#) ( ( ( ( F + 1 ) (#) F ) (#) F ) (#) ( ( ( F + 1 ) (#) F ) (#) ( ( F + 1 ) (#) F ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) (#) ( = ( ( ( ( ( ( ( F (#) ( F (#) ( F (#) ( F (#) ( F (#) ( F (#) ( F (#) ( F (#) ( F (#) ( F (#) ( F (#) ( F (#) ( F (#) ( F (#) ( F (#) ( F (#) (
reconsider z = g /. n as Point of TOP-REAL 2 ;
[: [: f , f :] , f :] in [: [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: , the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: , the carrier of [: the carrier of [: the carrier of
consider G being Subset of X such that G = ( union G ) \ ( {} ) ;
CurInstr ( P1 , Comput ( P1 , s , 1 ) ) = CurInstr ( P1 , Comput ( P1 , s , 1 ) .= CurInstr ( P1 , Comput ( P1 , s , 1 ) ;
assume that a on A and b on A and a on A and c on C ;
consider T such that T is finite-ind and ind ind ind T is finite-ind and ind ind T is finite-ind and ind ind T is finite-ind and ind ind T c= ind T and ind ind T c= finite-ind and ind ind T c= ind T ;
for x1 , x2 , x3 being Point of X holds ( g | ( x1 /\ x2 ) ) | ( ( ( g | ( x2 /\ ) ) ) is bounded
( ( ( ( ( the carrier of TOP-REAL 2 ) | ( ( ( ( ( ( ( ( ( ) ) ) | ( ( ( TOP-REAL 2 ) | ( ( ( ) ) | ( ( ( TOP-REAL 2 ) | ( ( ( TOP-REAL 2 ) | ( ( 1 - 2 ) ) | ( ( 1 - 2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) | ( ( ( 1 - 2 ) ) ;
F . ( i + 1 ) = 0 mod ( n + 1 ) .= 0 ;
ex f being Function st f . n = ( f . n ) * f . n ;
func f | ( ( i + 1 ) ) -> FinSequence of REAL means : Def3 : for i being Nat st i in dom f holds it . i = f . i ;
not x in { x } \/ { x } ;
for n being Nat holds ( ( ( ( ( ( ( n + 1 ) * ( n + 1 ) ) * ( n + 1 ) ) * ( n + 1 ) ) ) * ( n + 1 ) ) ) * ( n + 1 ) ) ) ) * ( n + 1 ) in ( ( ( n + 1 ) * ( n + 1 ) ) * ( n + 1 ) ) * ( n + 1 ) ;
ex S being Element of SubWFF st S = SubWFF & S is VERUM ;
consider P such that Q = P * Q and P [ i , j ] ;
let T be non empty TopSpace , T be Subset-Family of T ;
assume that f | X is constant and f | X is constant and f | X is constant ;
defpred P [ set ] means ex F being set st F = F ^ <* $1 *> ;
ex i being Nat st that ( ( ( GoB f ) * ( i , j ) ) * ( i + 1 ) ) `1 < ( ( ( ( f ) ) * ( i + 1 ) ) * ( i + 1 ) ) ;
defpred U [ set , set , set , set , set , set , set , set , set , set , set , set , set such that $2 in F and ( $2 is open and for n being Nat st n in dom $2 holds F . n = F . n ;
for p being Point of TOP-REAL 2 st p in P & q in P & q in P & p in P & q in P & q in P & q in P & q in P & q in P & q in P & q in P & q in P & q in P & q in P & q in P & q in P & q q in P & q q Segment ( P , p , q ) ;
assume that f in M and g in M and M in M ;
ex p being Point of TOP-REAL 2 st ( ( p ) | D ) | D = p ;
assume for n holds |. d .| <= |. t . n .| ;
assume that s <> x and t <> 0. TOP-REAL 2 ;
consider r such that for x st x in X holds ||. ( f /. x ) - f /. x .|| < r / 2 ;
( p ^ q ) ^ ( p ^ q ) ) ^ q = ( p ^ q ) ^ q ;
assume that x in dom ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - 2 * * ( 2 * * ( 1 - 2 ) ) ) ) ) ) ) ) ) ) * ( ( 1 - 2 ) ) ) ) ) * ( 1 + 2 ) ) ) ;
assume that i in Seg n and j in Seg n and i in Seg n and j in Seg n ;
for i being Nat st i in dom ( ( ( n --> 0 ) mod n ) mod n holds not i divides n
( ( ( ( ( ( ( ( ( b ) ) 'imp' b ) 'imp' c ) 'imp' c ) '&' ( ( ( b 'imp' c ) 'imp' c ) 'imp' c ) '&' ( b 'imp' c ) ) '&' ( b 'imp' c ) '&' ( b 'imp' c ) '&' ( b 'imp' c ) '&' ( b 'imp' c ) '&' ( b 'imp' c ) ) ) 'imp' ( b 'imp' c ) ) ) ) 'imp' ( b 'imp' c ) ) ) ) ) ) ) ) ) ) 'imp' ( b 'imp' ( c 'imp' c ) ) 'imp' ( b 'imp' c ) ) 'imp' ( b 'imp' c ) 'imp' ( b 'imp' c ) 'imp' ( b 'imp' c ) ) 'imp' ( b 'imp' ( b 'imp' c ) ) 'imp' ( b 'imp' ( c 'imp' c ) ) 'imp' ( b ) ) ) 'imp' ( b 'imp' ( b 'imp' c ) ) 'imp' ( b 'imp' c ) 'imp' ( b 'imp' ( b 'imp' c ) 'imp' ( b 'imp' c ) 'imp' ( b
assume for x holds ( ( ( ( ( ( cos * ( ( cos ) ) ) * ( cos ) ) ) * ( cos ) ) ) ) * ( ( cos ) ) ) ) ) ;
consider R such that for n holds R . n = ( ( ( M * ( n + 1 ) ) . n ;
ex k be Nat st ( for x be Point of X st x in REAL holds ( ( ( proj ( i , n ) * ( x , x ) ) * ( ( ( proj ( i , n ) ) * ( x , x ) ) ) * ( ( proj ( i , n ) * ( x , x ) ) ) * ( ( x , x ) ) ;
x in { x } \/ { x } ;
( ( ( ( ( ( G * ( i , j ) ) ) * ( i , j ) ) ) * ( ( ( ( G * ( i , j ) ) * ( ( i , j ) ) ) ) * ( ( ( G * ( i + j ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) `2 = ( ( ( ( ( ( G * ( i + 1 , j ) * ( * ( i + 1 ) ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( G * ( i + 1 ) ) ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( i + 1 ) ) * ( ( ( ( i + 1 ) ) * ( ( i + 1 ) ) ) * ( ( (
( f * ( ( ( the carrier of TOP-REAL 2 ) | ( the carrier of TOP-REAL 2 ) ) | ( the carrier of TOP-REAL 2 ) ) ) ) ) = ( ( the carrier of TOP-REAL 2 ) | ( the carrier of TOP-REAL 2 ) ;
func p => q -> '&' q -> non empty for p , r being non empty FinSequence of T holds r '&' q = r '&' q ;
Fq = ( ( p + 1 ) * ( p + 1 ) ;
let A , B be Matrix of K , K ;
( ( ( Partial_Sums ( ( s ) ) ^\ n ) ^\ n ) ^\ n ) ^\ n = ( Partial_Sums ( s ^\ n ) ^\ n ) ^\ n ;
assume x in the carrier of A ;
defpred P [ set ] means ex f being Function st f = ( ( f ^ g ) ^ ( f ^ g ) ^ ( f ^ g ) ;
assume that 1 <= i and i <= len G and i + 1 and j + 1 <= width G and i + 1 <= len G ;
assume that that that that p in dom ( ( ( q ) ) and q `2 = ( ( ( q ) ) * ( - 1 ) / ( 1 - 1 ) ) / ( 1 - ( ( - 1 ) ) ) / ( 1 - ( ( 1 - ( 1 - ( 1 - ( 1 - 1 ) ) ) ) ) ) ;
ex x being Point of X st x in Ball ( p , x ) & dist ( p , x ) = dist ( p , x ) ;
defpred P [ Element of REAL ] means ex f st ( for x st x in Z holds ( ( ( ( - 1 ) * ( ( ( - 1 ) * ( ( ( - 1 / ( 2 * ( ( 2 * ( ( 2 * ( ( 2 - ( ( ( - ( ( ( 2 * ( ( - - ( ( 2 * ( ( - ( ( ( - ( ( ( 2 * ( ( - - ( ( ( - ( ( ( ( - - ( ( 1 - ( ( 1 - ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - ( ( 1 - ( ( - ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - ( ( ( ( ( ( - ( ( - - ( ( ( - ( ( ( - ( ( ( ( (
defpred P [ Element of NAT ] means ex r be Point of TOP-REAL 2 st f . $1 in Ball ( f . $1 , r ) ;
( f ^ mid ( f , i , j ) ) ^ mid ( f , i , j ) = mid ( f , i , j ) ^ mid ( f , i ) ;
1 / ( ( ( ( 1 - 1 ) ) * ( 1 - ( 1 - 1 ) ) / ( 1 - 1 ) ) ) = ( 1 - ( 1 - 1 ) * ( 1 - 1 ) ) * ( 1 - 1 ) ;
defpred P [ Element of the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of [: the carrier of :] , the carrier of :] , the carrier of :] ;
assume that f /. 1 in Ball ( u , r ) and f /. ( i + 1 ) in Ball ( u , r ) ;
defpred P [ set , set , set , set , set , set , set , set , ( ( ( ( ( ( for x , y being Element of REAL ) ) * ( $1 + 1 ) ) * ( $1 + 1 ) ) - ( $1 + 1 ) ) * ( $1 + 1 ) ) ;
for x being element holds x in dom ( F * ( i ) ) implies x in dom F * ( i + 1 )
( ( x * ( ( ( x * y ) ) " ) " ) " ) " ) " = ( ( ( x * y ) " ) " ) " " ) " ) " ) " ) " ) " ;
DataPart DataPart ( s1 +* DataPart s1 , DataPart sss1 ) = DataPart DataPart s1 ;
consider r such that for x st x in Z holds f . x = ( f . x ) / ( r / 2 ) / 2 ;
assume that X is non empty and Y is non empty ;
for l being Element of L holds ( l /\ ( L "/\" ( X ) ) = ( ( L "/\" ( X "/\" Y ) ) "/\" ( ( X "/\" Y ) ) )
ex m being Ordinal st m in Support p & p in Support p ;
( ( ( ( ( ( - 1 ) (#) f ) (#) f ) (#) f ) (#) f ) (#) f ) (#) ( ( ( ( - 1 ) (#) f (#) f ) (#) f ) (#) f ) (#) f ) (#) f ) (#) ( ( ( ( - 1 ) (#) f (#) f ) (#) f ) (#) f ) ) (#) f ) ) (#) f ) ) ) (#) ( ( ( ( - 1 ) (#) f (#) f ) (#) f ) (#) f (#) f ) (#) ( ( ( ( ( ( ( ( - f ) (#) f ) (#) f ) (#) f ) (#) ( ( ( ( - f ) (#) f ) ) (#) f ) ) ) (#) ( ( ( - f ) (#) f ) ) ) (#) ( ( ( ( ( - f ) (#) f ) (#) f ) (#) f ) (#) f ) (#) f ) (#) f ) (#) f ) (#) f ) (#) ( ( ( ( ( - f ) (#) f ) ) (#) f ) ) (#) f ) ) (#) f ) (#) f ) (#) f ) (#) f ) (#) ( f + f )
ex p being Element of [: A , A :] st p in [: A , A :] & q = p ^ q ;
mid ( f , i1 , i2 ) = mid ( f , i1 , i2 ) ;
( p ^ q ) ^ q = ( p ^ q ) ^ q ;
len mid ( f , D1 , j ) = len mid ( D2 , D1 , j ) ;
x * y = z * ( x * y ) .= z * z .= z * z .= z * z * z .= z * z .= z * z .= z * z ;
v ( v * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - - - 1 ) ) ) ) ) ) ) ) ) ) ) * ( ( ( ( - 1 ) ) * ( ( ( - 1 ) ) * ( ( ( - 1 ) ) * ( ( ( 1 - 1 ) ) ) ) ) ) ) ) * ( ( ( ( - 1 ) ) * ( ( 1 - 1 ) ) ) ) ) ) ) ) * ( ( ( ( 1 - 1 ) ) ) ) ) * ( ( ( 1 - 1 ) ) ) * ( ( ( 1 - 1 ) ) ) ) ) ) * ( ( ( ( ( ( ( 1 - 1 ) ) * ( ( ( ( 1 - 1 ) ) * ( ( ( ( ( ( ( ( ( ( ( ( ( 1 - 1 ) * ( ( ( ( ( ( ( ( 1 - 1 ) ) * ( ( ( ( ( 1 - 1 ) ) * ( ( ( 1 - 1 ) ) ) * ( ( ( ( 1 - 1 ) ) ) * ( ( ( 1 - 1 )
i * 0 = 0 ;
Sum ( L ) = Sum ( L * ( ( ( L * ( L * ( L * ( L * ) ) ) .= Sum ( L * ( L * ( L * ( L * ( L * ) ) ) .= Sum ( L * ( L * ( L * ( L * ( L * ) ) ) .= Sum ( L * ( L * ( L * ( ) ) ) .= Sum ( L * ( L * ( L * ( L * ( L * ( L * ( L * ( L * ( ) ) ) .= Sum ( L * ( L * ( L * ( ) ) ) .= Sum ( L * ( L * ( L * ( L * ( L * ( L * ( L * ( ( L * ( L * ( L * ( L * ( L * ( L * ( L * ( L * ( L * ( L * ( L * ( L * ( L * ( L * ( L * ( L * ( L * ( L * ( L * ( ) ) ) ) .= Sum ( L * ( L * ( L * ( L * ( L * ( L * (
ex Y being Subset of X st Y c= REAL & Y c= REAL & Y c= REAL & Y c= REAL ;
( the Go-board of f ) /. k = ( the Go-board of f ) /. k ;
( ( ( - 1 ) (#) ( ( ( - 1 ) ) (#) ( ( 1 - 1 ) (#) ( ( ( 1 - 1 ) (#) ( ( ( 1 - 1 ) (#) ( ( ( 1 - 1 ) (#) ( ( ( 1 - ) ) (#) ( ( ( 1 - 1 ) (#) ( ( ( 1 - 1 ) (#) ( ( 1 - 1 ) ) (#) ( ( 1 - 1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ^2 = ( ( ( 1 - 1 ) (#) ( ( ( 1 - 1 ) ) (#) ( ( ( ( 1 - 1 ) (#) ( ( ( 1 - 1 ) ) (#) ( ( ( 1 - 1 ) (#) ( ( ( 1 - 1 ) ) (#) ( ( ( ( 1 - 1 ) ) (#) ( ( ( ( 1 - 1 ) (#) ( ( ( 1 - 1 ) (#) ( ( ( ( 1 - 1 ) (#) ( ( ( 1 - 1 ) (#) ( ( ( 1 - 1 ) (#) ( ( ( 1 - 1 ) ) (#) ( ( ( 1 - 1 ) (#) ( ( ( ( 1 - 1
- ( ( ( - b ) / 2 ) / 2 < ( - b / 2 ) / 2 ;
assume inf X in X ;
( i * j ) * ( i , j ) = ( ( id Z ) * ( i , j ) .= ( ( id Z ) * ( i , j ) ) * ( i , j ) .= ( ( id Z ) * ( i , j ) .= id Z ;