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contradiction ;
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contradiction ;
contradiction ;
contradiction ;
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 ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
B ;
a <> c
T c= S
D c= B
c ;
b ;
X ;
b in D ;
x = e ;
let m ;
h is onto ;
N in K ;
let i ;
j = 1 ;
x = u ;
let n ;
let k ;
y in A ;
let x ;
let x ;
m c= y ;
F is one-to-one ;
let q ;
m = 1 ;
1 < k ;
G is prime ;
b in A ;
d divides a ;
i < n ;
s <= b ;
b in B ;
let r ;
B is one-to-one ;
R is total ;
x = 2 ;
d in D ;
let c ;
let c ;
b = Y ;
0 < k ;
let b ;
let n ;
r <= b ;
x in X ;
i >= 8 ;
let n ;
let n ;
y in f ;
let n ;
1 < j ;
a in L ;
C is boundary ;
a in A ;
1 < x ;
S is finite ;
u in I ;
z << z ;
x in V ;
r < t ;
let t ;
x c= y ;
a <= b ;
m in NAT ;
assume f is prime ;
not x in Y ;
z = +infty ;
k be Nat ;
K ` is being_line ;
assume n >= N ;
assume n >= N ;
assume X is reconsider ^2 ;
assume x in I ;
q is measurable ;
assume c in x ;
'not' p > 0 ;
assume x in Z ;
assume x in Z ;
1 <= k12 ;
assume m <= i ;
assume G is prime ;
assume a divides b ;
assume P is closed ;
O > 0 ;
assume q in A ;
W is not bounded ;
f is means : Def1 : f is one-to-one ;
assume A is dense ;
g is special ;
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 even ;
assume i in I ;
assume 1 <= k ;
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 s is negative ;
b `2 <= c `2 ;
A meets W ;
i `2 <= j `2 ;
assume H is universal ;
assume x in X ;
let X be set ;
let T be Tree ;
let d be element ;
let t be element ;
let x be element ;
let x be element ;
let s be element ;
k <= 5 ;
let X be set ;
let X be set ;
let y be element ;
let x be element ;
P [ 0 ]
let E be set , F be Subset-Family of E ;
let C be Category ;
let x be element ;
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 ;
a be Real ;
let x be element ;
let X be element ;
P [ 0 ]
let x be element ;
let x be element ;
let y be element ;
r in REAL ;
let e be element ;
n1 is invertible ;
Q halts_on s ;
x in \in \rbrack ;
M < m + 1 ;
T2 is open ;
z in b "\/" a ;
R2 is well-ordering ;
1 <= k + 1 ;
i > n + 1 ;
q1 is one-to-one ;
let x be trivial set ;
P3 is one-to-one ;
n <= n + 2 ;
1 <= k + 1 ;
1 <= k + 1 ;
let e be Real ;
i < i + 1 ;
p3 in P ;
p1 in K & p2 in K ;
y in C1 ;
k + 1 <= n ;
let a be Real , A be Subset of R^1 ;
X |- r => p ;
x in { A } ;
let n be Nat ;
let k be Nat ;
let k be Nat ;
let m be Nat ;
0 < 0 + k ;
f is_differentiable_in x ;
let x0 ;
let E be Ordinal ;
o o OperSymbol o1 ;
O <> O2 ;
let r be Real ;
let f be FinSeq-Location ;
let i be Nat ;
let n be Nat ;
Cl A = A ;
L c= Cl L ;
A /\ M = B ;
let V be RealUnitarySpace , W be Subspace of V ;
not s in Y |^ 0 ;
rng f <= w
b "/\" e = b ;
m = m1 ;
t in h . D ;
P [ 0 ] ;
assume z = x * y ;
S . n is bounded ;
let V be RealLinearSpace , W be Subspace of V ;
P [ 1 ] ;
P [ {} ] ;
C1 is Subset of X ;
H = G . i ;
1 <= i `2 + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
a! <= b ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , W be Subspace of V ;
s is trivial non trivial & s is trivial ;
dom c = Q ;
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be complete LATTICE , x be Element of T ;
the object of F is one-to-one
sgn x = 1 ;
k in support a ;
1 in Seg 1 & 1 <= len f ;
rng f = X ;
len T in X ;
vA2 < n ;
Spion1 is bounded ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let U ;
p0 `2 = c ;
j in dom h ;
let k ;
f | Z is continuous ;
k in dom G ;
UBD C = B ;
1 <= len M ;
p in LSeg ( x , r ) ;
1 <= j1 ;
set A = *> ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H is \rm \rrangle ;
assume that n1 <= m and n2 <= m ;
T is increasing ;
e2 <> e2 ;
Z c= dom g ;
dom p = X ;
H is proper ;
i + 1 <= n ;
v <> 0. V ;
A c= Affin A ;
S c= dom F ;
m in dom f ;
let X0 be set ;
c = sup N ;
R is_connected union M ;
assume not x in REAL ;
Im f is complete ;
x in Int y ;
dom F = M ;
a in On W ;
assume e in [: A ( ) , B ( ) :] ;
C c= Cu ;
m-5 <> {} ;
let x be Element of Y ;
let f be \Omega Chain , p be Element of f ;
not n in Seg 3 ;
assume X in f .: A ;
assume that p <= n and p <= m ;
assume not u in { v } ;
d is Element of A ;
A |^ b misses B ;
e in v `2 ;
- y in I ;
let A be non empty set , B be set ;
P0 = 1 ;
assume r in F . k ;
assume f is simple ;
let A be infinite countable set ;
rng f c= NAT ;
assume P [ k ] ;
f6 <> {} & f6 <> {} ;
let o be Ordinal ;
assume x is sum of squares ;
assume not v in { 1 } ;
let I1 ;
assume that 1 <= j and j < l ;
v = - u & u = v - v ;
assume s . b > 0 ;
d1 in Y ;
assume t . 1 in A ;
let Y be non empty TopSpace , x be Point of Y ;
assume a in ]. s , t .[ ;
let S be non empty Poset ;
a , b // b , a ;
a * b = p * q ;
assume x , y are_the space ;
assume x in Omega ( f ) ;
[ a , c ] in X ;
m-14 <> {} ;
M + N c= M + M ;
assume M is \rrangle /. /. M ;
assume f is with_indeeeddddPL ;
let x , y be element ;
let T be non empty TopSpace ;
b , a // b , c ;
k in dom Sum p ;
let v be Element of V ;
[ x , y ] in T ;
assume len p = 0 ;
assume C in rng f ;
k1 = k2 & k2 = k2 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
n + i >= n + 1 ;
assume Re y = 0 ;
k1 <= j1 & k2 <= k ;
f | A is IC ;
f . x > b ;
assume y in dom h ;
x * y in B1 ;
set X = Seg n ;
1 <= i2 + 1 ;
k + 0 <= k + 1 ;
p ^ q = p ;
j |^ y divides m ;
set m = max A ;
[ x , x ] in R ;
assume x in succ 0 ;
a in sup phi ;
Cj ;
q2 c= C1 & q2 in C2 ;
a2 < c2 & c2 < c2 ;
s2 is 0 -started ;
IC s = 0 ;
s6 = s4 & s6 = s4 ;
let V ;
let x , y be element ;
let x be Element of T ;
assume a in rng F ;
x in dom T ` ;
let S be redefine of L ;
y " <> 0 ;
y " <> 0 ;
0. V = u-vector w ;
y2 , y , w is_collinear ;
R8 is open ;
let a , b be Real , x be Real ;
let a be Object of C ;
let x be Vertex of G ;
let o be object of C , a be Object of C ;
r '&' q = P \lbrack l , l \rbrack ;
let i , j be Nat ;
let s be State of A , P be Subset of A , Q be Subset of A ;
s3 . n = N ;
set y = ( x - y ) / 2 ;
mi in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in C0 ;
V1 is non empty ;
let x be Element of X ;
0 <> f . g2 ;
f2 /* q is convergent ;
f . i is_measurable_on E ;
assume \xi in NH ;
reconsider i = i as Ordinal ;
r * v = 0. X ;
rng f c= INT & f is one-to-one ;
G = 0 .--> goto 0 ;
let A be Subset of X ;
assume that A9 is dense and A is dense ;
|. f . x .| <= r ;
let x be Element of R ;
let b be Element of L ;
assume x in W-19 ;
P [ k , a ] ;
let X be Subset of L ;
let b be object of B ;
let A , B be category ;
set X = Vars ( C ) ;
let o be OperSymbol of S ;
let R be connected non empty Poset ;
n + 1 = succ n ;
xY c= Z1 & Z c= Z1 ;
dom f = C1 & dom g = C2 ;
assume [ a , y ] in X ;
Re ( seq ) is convergent ;
assume a1 = b1 & b1 = b2 ;
A = Int ( A ) ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be Instruction of S , s be State of S ;
assume r2 > x0 & x0 < r2 ;
let Y be non empty set , x be Element of Y ;
2 * x in dom W ;
m in dom g2 & n <= len g2 ;
n in dom ( g1 | X ) ;
k + 1 in dom f ;
not the still not bound in { s } ;
assume x1 <> x2 & x2 <> x3 ;
v2 in V1 & v2 in V1 ;
not [ b `1 , b `2 ] in T ;
i-35 + 1 = i ;
T c= *> ( T ) ;
( l - 1 ) * ( l - 1 ) = 0 ;
let n be Nat ;
( t `2 ) ^2 = r ;
Amax : f is_integrable_on M & f is_integrable_on M ;
set t = Bottom t ;
let A , B be real-membered set ;
k <= len G + 1 ;
[: C , D :] misses [: D , D :] ;
Product ( seq ) is non empty ;
e <= f or f <= e ;
cluster ordinal for sequence ;
assume c2 = b2 & c2 = b3 ;
assume h in [. q , p .] ;
1 + 1 <= len C ;
not c in B . m1 ;
cluster R .: X -> empty ;
p . n = H . n ;
assume that xseq is convergent and lim seq = lim seq ;
IC Comput ( P3 , s3 , k ) = 0 ;
k in N or k in K ;
F1 \/ F2 c= F ;
Int ( G1 ) <> {} & Int ( G2 ) <> {} ;
( z `1 ) ^2 = 0 ;
p0 <> p1 & p2 <> p2 ;
assume z in { y , w } ;
MaxADSet ( a ) c= F ;
ex_sup_of ]. s , t .[ , S ;
f . x <= f . y ;
let T be up-complete non empty reflexive RelStr ;
q |^ m >= 1 ;
a >= X & b >= Y ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one implies G is one-to-one
A \/ { a } \not c= B ;
0. V = 0. V .= 0. V ;
let I be cluster cluster as as Instruction Instruction Instruction of S , s be Element of S ;
f-24 . x = 1 / ( x ^2 ) ;
assume z \ x = 0. X ;
C4 = 2 to_power n ;
let B be SetSequence of Omega ;
assume X1 = p .: D ;
n + l2 in NAT & n + 1 in NAT ;
f " P is compact ;
assume x1 in [: REAL , REAL :] ;
p1 = K1 & p2 = ( sn -FanMorphE ) . p2 ;
M . k = <*> ( REAL ) ;
phi . 0 in rng phi ;
sup MML is closed
assume z0 <> 0. L ;
n < N7 . k ;
0 <= ( seq . 0 ) * ( seq . 0 ) ;
- q + p = v ;
{ v } is Subset of B ;
set g = f /. 1 , h = f /. 2 ;
[: R , S :] is stable implies R is stable
set RR = Vertices R , RR = Vertices R ;
p0 `1 c= s2 & p `2 <= p `2 ;
x in [. 0 , 1 .] ;
f . y in dom F ;
let T be Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott TopAugmentation of S ;
ex_inf_of the carrier of S , S ;
sup { a } = sup { b } ;
P , C , K is_collinear ;
assume x in F ( s , r , t ) ;
2 to_power i < 2 to_power m ;
x + z = x + z + q ;
x \ ( a \ x ) = x ;
||. x-y .|| <= r ;
assume that Y c= field Q and Y <> {} and Y <> {} ;
a ~ , b opp are isomorphic ;
assume a in [: A ( ) , B ( ) :] ;
k in dom ( q | k ) ;
p is FinSequence of S ;
i -' 1 = i-1 - 1 ;
f | A is one-to-one ;
assume x in f .: [: X , Y :] ;
i2 - i1 = 0 & i2 = 0 ;
j2 + 1 <= i2 & j2 + 1 <= i2 ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster strict strict strict strict Rrng ;
|. q .| > 0 ;
|. p4 .| = |. p .| ;
s2 - s1 > 0 & s2 - s1 < s2 - s1 ;
assume x in { Gik } ;
W-min C in C & E-max C in C ;
assume x in { Gik } ;
assume i + 1 = len G ;
assume i + 1 = len G ;
dom I = Seg n & dom I = Seg n ;
assume that k in dom C and k <> i ;
1 + 1-1 <= i + 1-1 ;
dom S = dom F & dom S = dom F ;
let s be Element of NAT ;
let R be ManySortedSet of A ;
let n be Element of NAT ;
let S be non empty non void void void void void void void void void distributive ;
let f be ManySortedSet of I ;
let z be Element of COMPLEX , p be Element of COMPLEX ;
u in { \boldmath g } , { g } } ;
2 * n < ( 2 * n ) ;
x , y be set ;
B-11 c= V1 & B-15 c= V1 ;
assume I is_closed_on s , P ;
U = U , U = U , E = U , F = U , f = U , g = U , h = U , f = U , h = U , i =
M /. 1 = z /. 1 ;
x11 = x22 & x22 = x22 ;
i + 1 < n + 1 + 1 ;
x in { {} , <* 0 *> } ;
f7 <= f6 & f7 <= len f ;
let l be Element of L ;
x in dom ( F . n ) ;
let i be Element of NAT ;
r8 is ( len r ) -element ;
assume <* o2 , o *> <> {} ;
s . x |^ 0 = 1 ;
card ( K1 ) in M & card ( K1 ) in M ;
assume that X in U and Y in U ;
let D be Subset-Family of Omega ;
set r = - { k + 1 } ;
y = W . ( 2 * x ) ;
assume dom g = cod f & cod g = cod f ;
let X , Y be non empty TopSpace , f be Function of X , Y ;
x in A ++ B ;
|. <*> A .| . a = 0 ;
cluster strict SubSublattice L -> strict ;
a1 in B . s1 & a2 in B . s2 ;
let V be finite VectSp of F , v be Element of V ;
A * B on B implies A on B
f-3 = [: NAT , NAT :] --> 0 ;
let A , B be Subset of V ;
z1 = P1 . j & z2 = P1 . j ;
assume f " P is closed ;
reconsider j = i as Element of M ;
let a , b be Element of L ;
assume q in A \/ ( B "\/" C ) ;
dom ( F * C ) = o ;
set S = ( INT , X ) --> 0. X ;
z in dom ( A --> y ) ;
P [ y , h . y ] ;
{ x0 } c= dom f & { x0 } c= dom f ;
let B be non-empty ManySortedSet of I , A be non-empty ManySortedSet of I ;
sqrt ( PI / 2 ) < Arg z ;
reconsider z9 = 0 as Nat ;
LIN a , d , c ;
[ y , x ] in [: II , the carrier of L :] ;
( Q ) * ( 1 , 3 ) = 0 ;
set j = x0 gcd ( m , m ) ;
assume a in { x , y , c } ;
j2 - ( j - 1 ) > 0 ;
I -TruthEval phi = 1 ;
[ y , d ] in F-8 ;
let f be Function of X , Y ;
set A2 = ( B C ) * ( A * B ) ;
s1 , s2 being Element of R ;
j1 -' 1 = 0 & j1 -' 1 = 1 ;
set m2 = 2 * n + j ;
reconsider t = t as bag of n ;
I2 . j = m . j ;
i |^ s , n are_relative_prime ;
set g = f | [: DL , DL :] ;
assume that X is lower and 0 <= r ;
( ( p1 `1 ) / p1 `1 ) ^2 = 1 ;
a < ( p3 `1 ) ^2 + ( p3 `2 ) ^2 ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= i1 -' 1 & i1 + 1 <= len f ;
1 <= i1 -' 1 & i1 + 1 <= len f ;
i + i2 <= len h & i + 1 <= len h ;
x = W-min ( P ) & x = E-max ( P ) ;
[ x , z ] in X ~ ;
assume y in [. x0 , x .] ;
assume p = <* 1 , 2 , 3 *> ;
len <* A1 *> = 1 & len <* A1 *> = 1 ;
set H = h . ( g2 . n ) ;
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 ** h1 , h2 = h2 ** h2 ;
assume x in X0 /\ ( X1 union X2 ) ;
||. h .|| < d1 & ||. h .|| < d ;
not x in the carrier of f & not x in the carrier of f ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k -' l = kk ;
<* p , q *> /. 2 = q ;
let S be Subset of the lattice of Y ;
let P , Q be empty one-to-one s of P ;
Q /\ M c= union ( F | M )
f = b * canFS ( S ) ;
let a , b be Element of G ;
f .: X <= f . sup X ;
let L be non empty reflexive transitive RelStr , x be Element of L ;
Sc is x -\leq x -One ;
let r be non positive Real ;
M , v |= ( x , y ) |= H ;
v + w = 0. ( V ) ;
P [ len F ] implies P [ F ( ) ]
assume InsCode ( i ) = 8 & InsCode ( i ) = 8 ;
the non zero Element of M = 0 & the carrier of M = 0 ;
cluster z * seq -> summable ;
let O be Subset of the carrier of C ;
||. f .|| | X is continuous ;
x2 = g . ( j + 1 ) ;
cluster empty for Element of U ;
reconsider l1 = lm1 as Nat ;
v4 is Vertex of r2 & not v in dom ( r (#) ( v | X ) ) ;
T3 is SubSpace of T2 & the carrier of T2 = the carrier of T2 ;
Q1 /\ Q1 <> {} & Q1 /\ [#] ( TOP-REAL 2 ) <> {} ;
k be Nat ;
q " is Element of X & q " is Element of X ;
F . t is set of the carrier of M ;
assume that n <> 0 and n <> 1 ;
set e = EmptyBag n , f = EmptyBag n , g = EmptyBag n ;
let b be Element of Bags n ;
assume for i holds b . i is commutative ;
x is root implies x in ( p ) . ( x , y )
not r in ]. p , q .[ ;
let R be FinSequence of REAL , x be Element of R ;
S7 does not destroy b1 & not I does not destroy b1 ;
IC SCM+FSA <> a & IC SCM+FSA <> b ;
|. |[ x , y ]| - |[ x , y ]| .| >= r ;
1 * ( seq . n ) = seq . n * seq . n ;
let x be FinSequence of NAT , n be Nat ;
let f be Function of C , D , g be Function of C , D ;
for a holds 0. L + a = a
IC s = s . NAT .= s . NAT .= s . NAT ;
H + G = F- ( GG ) ;
C1 . x = x2 & C1 . x = y2 ;
f1 = f . ( f1 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . ( f2 . (
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u + W ;
{ a1 } = { a2 } & { a2 } = { a2 } ;
a1 , b1 _|_ b , a ;
a3 , o _|_ o , a3 ;
II is_reflexive & CI is_reflexive implies for x , y being Element of I holds x in ( the carrier of C ) . ( x , y )
II is_reflexive transitive in CC & for x st x in CC holds x in C
sup ( rng H1 ) = e & sup ( rng H1 ) = e ;
x = ( ( a * b ) * ( a * c ) ) * ( a * c ) ;
|. p1 .| / ( 1 + 1 ) >= 1 ;
assume j2 -' 1 < j2 -' 1 ;
rng s c= dom f1 & rng s c= dom ( f1 + f2 ) ;
assume support a misses support b & not a in support b ;
let L be associative non empty doubleLoopStr , x be Element of L ;
s " + 0 < n + 1 ;
p . c = ( f . 1 ) `1 ;
R . n <= R . ( n + 1 ) ;
Directed I1 = I1 & card I1 = card I1 + 2 ;
set f = + ( x , y , r ) ;
cluster Ball ( x , r ) -> bounded ;
consider r being Real such that r in A ;
cluster non empty for NAT -defined Function ;
let X be non empty directed Subset of S ;
let S be non empty full SubRelStr of L ;
cluster <* L1 . N , \subseteq \rangle -> complete for non trivial Real ;
sqrt ( 1 - a ) = a ;
( q . {} ) `1 = o ;
n - ( i -' 1 ) > 0 ;
assume ( 1 / 2 ) ^2 <= 1 ^2 ;
card B = k + 1 - 1 ;
x in union ( rng ( f | A ) ) ;
assume x in the carrier of R & y in the carrier of S ;
d ;
f . 1 = L . F ;
the carrier of G = { v } & v in { v } ;
let G be *> ;
e , v6 , v6 , R ;
c . ( i0 + 1 ) in rng c ;
f2 /* q is divergent_to+infty & f2 /* q is divergent_to+infty ;
set z1 = - z2 , z2 = - z2 , z2 = - z2 , z2 = - z2 , z2 = - z2 , z2 = - z2 , z2 = - z2 , z2 = - z2 , z2
assume w is an lllas string of S , G ;
set f = p \! \mathop { t } , g = p \! \mathop { t } , h = p ;
let c be Object of C ;
assume ex a st P [ a ] ;
let x be Element of REAL m , m be Element of REAL m ;
let II be Subset-Family of X ;
reconsider p = p as Element of NAT ;
let v , w be Point of X ;
let s be State of SCM+FSA , P , s be State of SCM+FSA ;
p is FinSequence of SCM+FSA & p in dom ( p +* q ) ;
stop I c= P3 , s4 = P3 , P4 = P3 , P4 = P3 , P4 = P3 , P4 = P3 , P4 = P3 , P4 = P3 , P4 = P3 , P4 = P3 , P4
set ci = ( f /. i ) `1 , ci = ( f /. i ) `1 , ci = ( f /. i ) `1 , ci = ( f /. i ) `1 , c
w ^ t ^ w ^ s ^ t ^ w ^ t ^ w ^ t ^ w ^ t ^ w ^ t ^ w ^ t ^ w ^ w ^ t ^ w ^ w ^ t ^
W1 /\ W = W1 /\ W2 & W2 /\ W1 = W2 /\ W1 ;
f . j is Element of J . j ;
let x , y be Element of T2 , f be Function of T2 , T2 ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 & c <> 0 ;
ord x = 1 & x is positive ;
set g2 = lim ( seq , x0 ) , g1 = lim ( seq , x0 ) ;
2 * x >= 2 * ( 1 / 2 ) ;
assume ( a 'or' c ) . z <> TRUE ;
f (*) g in Hom ( c , c9 ) ;
Hom ( c , c9 + c ) <> {} ;
assume 2 * Sum ( q | m ) > m ;
L1 . ( F1 . ( F1 . n ) ) = 0 ;
the carrier of X \/ R1 = the carrier of X & the carrier of R1 = the carrier of R2 ;
( ( ( sin * sin ) `| Z ) . x ) <> 0 ;
( ( ( ( id Z ) (#) ( ( exp_R * f ) ) `| Z ) ) . x ) ^2 > 0 ;
o1 in ( X1 /\ X2 ) /\ ( X2 /\ Y2 ) ;
e , v6 , v6 , R ;
r3 > ( 1 / 2 ) * 0 ;
x in P .: ( F -ideal ) ;
let J be closed non empty Subset of R , I be Subset of R ;
h . p1 = f2 . O & h . O = g2 . I ;
Index ( p , f ) + 1 <= j ;
len ( q | M ) = width M & width ( q | M ) = width M ;
the carrier of K c= A & K c= A ;
dom f c= union rng ( F . n ) ;
k + 1 in support ( cluster support ( n ) ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y `2 ] in InnerVertices R & [ x `1 , y `2 ] in R ;
i = D1 or i = D2 or i = D1 . i ;
assume a mod n = b mod n ;
h . x2 = g . x1 .= f . x2 ;
F c= 2 -tuples_on the carrier of X & F is one-to-one ;
reconsider w = |. s1 .| as Real_Sequence of REAL , REAL ;
sqrt ( 1 / m * r ) < p ;
dom f = dom ( I . f ) .= dom ( I . f ) ;
[#] ( ( TOP-REAL 2 ) | K1 ) = [#] ( ( TOP-REAL 2 ) | K1 ) ;
cluster - x -> ExtReal for ExtReal ;
then { d } c= A ;
cluster [: TOP-REAL n , TOP-REAL n :] -> finite-ind ;
let w1 be Element of M ;
let x be Element of dyadic ( n ) ;
u in W1 & v in W2 & u in W2 implies u in W2
reconsider y = y as Element of L2 ( ) ;
N is full SubRelStr of ( T |^ the carrier of S ) ;
sup { x , y } = c "\/" c ;
g . n = n / ( 1 / ( n + 1 ) ) .= n ;
h . J = EqClass ( u , J ) . M ;
let seq be sequence of X ;
dist ( x , y ) < r / 2 ;
reconsider mm = m - 1 as Element of NAT ;
x- x0 < r1 - x0 & r1 < x0 + r2 ;
reconsider P ' = P as strict Subgroup of N ;
set g1 = p * idseq ( q `1 , q `2 ) , g2 = q `2 * idseq ( q `2 , q `2 ) ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is lower ;
D2 . I in { x } & D2 . I in { x } ;
cluster -> subcondensed for Subset of T ;
let P be compact non empty Subset of TOP-REAL 2 , p , q be Point of TOP-REAL 2 ;
Gik in LSeg ( \pi , 1 ) /\ LSeg ( \pi , 1 ) ;
let n be Element of NAT , m be Element of NAT ;
reconsider ST = S as Subset of T ;
dom ( i .--> X ) = { i } ;
let X be non-empty ManySortedSet of S ;
let X be non-empty ManySortedSet of S ;
not op _ 1 c= { [ {} , {} ] }
reconsider m = mm as Element of NAT ;
reconsider d = x as Element of [: C , D :] ;
let s be 0 -started State of SCMPDS , P , s be State of SCMPDS ;
let t be 0 -started State of SCMPDS , Q ;
b , b , x , y is_collinear & x , y , z is_collinear ;
assume that i = n \/ { k } and j = k \/ { k } ;
let f be PartFunc of X , Y ;
NN1 >= sqrt ( sqrt ( c ^2 + c ^2 ) ) ;
reconsider t7 = T7 as Point of TOP-REAL n ;
set q = h * p ^ <* d *> ;
z2 in U . ( y2 , z2 ) /\ Q2 . ( x2 , y2 ) ;
A |^ 0 = { <* E *> } |^ 0 .= { <* E *> } ;
len W2 = len W + 2 & len W2 = len W + 1 ;
len h2 in dom h2 & len h2 in dom h2 & len h2 = len h2 ;
i + 1 in Seg ( len s2 ) & i + 1 <= len s2 ;
z in dom ( g1 | dom f ) /\ dom ( g2 | dom f ) ;
assume p2 = E-max ( K ) & p3 = E-max ( K ) ;
len G + 1 <= i1 + 1 ;
f1 * f2 - f2 * f1 is convergent & lim ( f1 * f2 ) = f2 * f1 - f2 * f1 ;
cluster seq + seq seq is summable for Real_Sequence ;
assume j in dom ( M1 * M2 ) ;
let A , B , C be Subset of X ;
let x , y , z be Point of X , p be Point of X ;
b ^2 - ( 4 * a * c ) >= 0 ;
<* x-' y *> ^ <* y *> divides x ;
a , b in { a , b } ;
len p2 is Element of NAT & len p2 = len p1 & len p1 = len p2 ;
ex x being element st x in dom R & y = R . x ;
len q = len ( K (#) G ) & len q = len ( K (#) G ) ;
s1 = Initialize Initialized Initialized s , p1 = p1 +* p2 , p2 = p2 +* stop I ;
consider w be Nat such that q = z + w ;
x ` is ` & x ` is a ` ;
k = 0 & n <> k or k > n ;
then X is discrete for X being Subset of X ;
for x st x in L holds x is FinSequence
||. f /. c .|| <= r1 & ||. f /. c .|| <= r1 ;
c in ]. p , q .[ & not c in { p } ;
reconsider V = V as Subset of the \mathbin { 0 } , 1 } ;
let N , M be empty net of L ;
then z >= waybelow x & z >= compactbelow x ;
M \lbrack f , g .] = f & M [. g , f .] = g ;
( ( ( ( ( 1 / 2 ) ) |^ ( 1 + 1 ) ) ) ) * ( ( 1 / 2 ) ) = TRUE ;
dom g = dom f & dom g = dom f & rng g c= dom f ;
mode \mathbb Shift is Walk of G ;
[ i , j ] in Indices M & [ i , j ] in Indices M ;
reconsider s = x " as Element of H ;
let f be Element of Subformulae p ( ) , x be Element of Subformulae p ( ) ;
F1 ( a1 , - a2 ) = G1 ( a1 , a2 ) & F1 ( a1 , a2 ) = G2 ( a1 , a2 ) ;
cluster rectangle ( a , b , r ) -> compact ;
let a , b , c , d be Real ;
rng s c= dom ( 1 / 2 ) & rng s c= dom ( 1 / 2 ) ;
curry ( FK , k ) . ( n + k ) is additive ;
set k2 = card ( B . ( card A ) ) ;
set G = coprod ( X ) ;
reconsider a = [ x , s ] as Object of G ;
let a , b be Element of [: M , M :] , M be Subset of [: M , M :] ;
reconsider s1 = s as Element of S0 , s2 = t as Element of S0 ;
rng p c= the carrier of L & p . ( len p ) = 0. L ;
let d be Subset of the Sorts of A ;
( x | x ) = 0 iff x = 0. W
I-21 in dom stop I & card I + 2 < card J ;
let g be continuous Function of X | B , Y ;
reconsider D = Y as Subset of TOP-REAL n ;
reconsider i0 = len p1 - 1 as Integer ;
dom f = the carrier of S & rng f c= the carrier of S ;
rng h c= union ( ( the carrier of J ) --> NAT )
cluster All ( x , H ) -> strict ;
d * N1 / ( 1 - p ) > N1 * 1 / ( 1 - p ) ;
]. a , b .[ c= [. a , b .] ;
set g = f " ( ( f " ) | D1 ) , f = f " ( f " ( f " ) ) ;
dom ( p | ( ( m + 1 ) , m ) ) = REAL ;
3 + - 2 <= k + - 2 ;
tan is_differentiable_in ( ( tan * arccot ) `| Z ) . x ;
x in rng ( f /^ ( p .. f ) ) ;
let f , g be FinSequence of D ;
[: p , S1 :] in the carrier of [: S1 , S2 :] & [: S1 , S2 :] in the carrier of S1 ;
rng f " = dom f & rng f c= dom ( f " ) ;
( the Source of G ) . e = v & ( the Source of G ) . e = v ;
width G -' 1 < width G & width G -' 1 < width G ;
assume v in rng ( S | E1 ) ;
assume x is root or x is root or x is root & x is root ;
assume 0 in rng ( g2 | A ) & g2 | A is bounded ;
let q be Point of TOP-REAL 2 , a , b be Real ;
let p be Point of TOP-REAL 2 , a , b be Real ;
dist ( O , u ) <= |. p2 .| + 1 ;
assume dist ( x , b ) < dist ( a , b ) ;
<* S7 *> is in the carrier of C-20 & <* C7 *> is in the carrier of C-20 ;
i <= len ( G | ( len G -' 1 ) ) ;
let p be Point of TOP-REAL 2 , a , b be Real ;
x1 in the carrier of [: I[01] , I[01] :] & x2 in the carrier of I[01] ;
set p1 = f /. i , p2 = f /. j ;
g in { g2 : r < g2 & g2 < x0 + r } ;
Q2 = Snc " . ( Q . ( Q . ( Q . ( Q . ( Q . ( Q . ( Q . ( Q . ( Q . ( Q . ( Q . ( Q . ( Q . ( Q
( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) * ( 1 / 2 ) ) ) ) ) is summable ;
- p + I c= - p + A + - B ;
n < LifeSpan ( P1 , s1 ) + 1 ;
CurInstr ( p1 , s1 ) = i & CurInstr ( p1 , s1 ) = i ;
A /\ Cl { x } <> {} ;
rng f c= ]. r , r + 1 .[ ;
let g be Function of S , V ;
let f be Function of L1 , L2 , L2 be Function of L2 , L2 ;
reconsider z = z as Element of InclPoset ( L ) ;
let f be Function of S , T ;
reconsider g = g as Morphism of c opp , b opp ;
[ s , I ] in S ~ & [ s , I ] in [: > , succ A :] ;
len ( the connectives of C ) = 4 & len ( the connectives of C ) = 5 ;
let C1 , C2 be SubFunctor of C , C2 ;
reconsider V1 = V as Subset of X | B , V1 = V | B ;
pred p is valid means : Def1 : All ( x , p ) is valid ;
assume that X c= dom f and f .: X c= dom g and f .: X c= dom g ;
H |^ a is Subgroup of H & a is Subgroup of H ;
let A1 be : A1 : A1 on E & A2 on E & A1 on E ;
p2 , p3 , p2 is_collinear & q2 <> p2 & p2 <> p3 & p2 <> p4 & p2 <> p1 ;
consider x being element such that x in v ^ K and x in v ^ K ;
not x in { 0. TOP-REAL 2 } & not x in { 0. TOP-REAL 2 } ;
p in [#] ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( 1 + 1 ) ) ) ) ) ;
0 in M . EE & M . EE < M . EE ;
^ ( c / ( d / c ) ) = c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . s2 ) & a2 = ( F . s2 ) . y ;
cluster -> with_an \rbrace -| for non empty Poset ;
set i1 = the Element of NAT , i2 = the Element of NAT , j2 = the Element of NAT ;
let s be 0 -started State of SCM+FSA , P , s be State of SCM+FSA ;
assume y in ( f1 \/ f2 ) .: ( A \/ B ) ;
f . ( len f ) = f /. len f .= f /. len f ;
x , f . x '||' f . x , f . y ;
pred X c= Y means : Def1 : cos ( X ) c= cos ( Y ) ;
let y be upper Subset of Y , x , z be Element of Y ;
cluster -> of ( x , y ) -valued for non empty finite sequence ;
set S = <* Bags n , i1 *> , i2 = <* i1 *> , z = <* i2 *> , i1 = <* i1 *> , i2 = <* i2 *> , i2 = <* i1 *> , S = <* i2 *> , S = <* i1
set T = [. 0 , 1 / 2 .] , G = [. 1 / 2 , 1 .] ;
1 in dom mid ( f , 1 , 1 ) & len mid ( f , 1 , 1 ) = len f ;
sqrt ( 4 * PI ) < sqrt ( 2 * PI ) ;
x2 in dom f1 /\ dom ( f | X ) ;
O c= dom I & { {} } = { {} } ;
( the Source of G ) . x = v & ( the Source of G ) . x = v ;
{ HT ( f , T ) } c= Support f & Support ( f ) c= Support f ;
reconsider h = R . k as Polynomial of n , L ;
ex b being Element of G st y = b * H ;
let x , y , z be Element of G ` ;
h19 . i = f . ( h . i ) ;
( p `1 ) ^2 = ( p1 `1 ) ^2 + ( p1 `2 ) ^2 ;
i + 1 <= len Cage ( C , n ) ;
len <* P *> = len P & len <* P *> = len P ;
set NN = the Element of the LSeg ( N , m ) , NN = the Element of N ;
len g-2 + ( x + 1 ) - 1 <= x ;
a on B & b on B & a on B & b on C & a on C & b on C ;
reconsider r-12 = r * I . v as FinSequence of REAL ;
consider d such that x = d and a _|_ d and a [= c ;
given u such that u in W and x = v + u ;
len f /. ( \downharpoonright n ) = len \mathbb n & len f = len \mathbb n ;
set q2 = ( E-max C ) .. ( ( E-max C ) .. ( ( E-max C ) .. ( ( E-max C ) .. ( the Cage of C ) ) ) ) ;
set S = Carrier ( S1 , S2 ) , T = Carrier ( S2 , S2 ) ;
MaxADSet ( b ) c= MaxADSet ( P /\ Q ) ;
Cl ( G . q1 ) c= F . r2 ;
f " D meets h " ( h " V ) ;
reconsider D = E as non empty directed Subset of L1 ;
H = the_left_argument_of H & H = the_left_argument_of H implies H = H
assume t is Element of ( \mathfrak F ) . ( X , X ) ;
rng f c= the carrier of S2 & f . 0 = the carrier of S2 & f . 1 = the carrier of S2 ;
consider y being Element of X such that x = { y } ;
f1 . a1 , b1 . a1 = b1 . a1 , c1 = c1 . b1 ;
the carrier' of G = E \/ { E } ;
reconsider m = len \langle - k *> as Element of NAT ;
set S1 = LSeg ( n , UMP C ) , S2 = LSeg ( n , UMP C ) ;
[ i , j ] in Indices M1 & [ i , j ] in Indices M1 & [ i , j ] in Indices M2 ;
assume that P c= Seg m and M is not contradiction and M is not \rm \times ;
for k st m <= k holds z in K . k ;
consider a being set such that p in a and a in G ;
L1 . p = p * L /. ( 1 / p ) ;
p0 . i = p1 . i & p2 . i = p2 . i ;
let PA , G be a_partition of Y , z be set ;
pred 0 < r & r < 1 implies 1 < r & r <= 1 ;
rng \lbrace proj ( a , X ) . i } = [#] ( X ) ;
reconsider x = x , y = y as Element of K ;
consider k such that z = f . k and n <= k ;
consider x being element such that x in X \ { p } ;
len ( canFS ( s ) ) = card ( s ) .= card ( s ) ;
reconsider x2 = x1 , y2 = x2 as Element of L2 ( ) ;
Q in FinMeetCl ( ( the topology of X ) \ { 0 } ) ;
dom ( f | Y ) c= dom ( u | Y ) & dom ( f | Y ) c= dom ( u | Y ) ;
pred n divides m & m divides n implies n = m ;
reconsider x = x as Point of [: I[01] , I[01] :] , R^1 ;
a in support \vert the carrier of T2 ( T2 , T2 ) ;
not y9 in the still of f & not y9 in dom ( f . 0 ) ;
Hom ( ( a \times b ) \times c ) <> {} & Hom ( a , c ) <> {} ;
consider k1 such that p " < k1 and p . k1 < p . k1 ;
consider c , d such that dom f = c \ d and f . c = d ;
[ x , y ] in dom g ~ & [ y , z ] in dom g ;
set S1 = that x = that x in @ then x in @ ( x , y , z ) ;
l1 = m2 & l2 = i2 & l2 = j2 & l2 = i2 & l1 = j2 & l2 = i2 ;
x0 in dom ( u | A9 ) /\ ( ( u | A9 ) ^ ( v | A9 ) ) ;
reconsider p = x as Point of TOP-REAL 2 , p = x as Point of TOP-REAL 2 ;
[: I[01] , I[01] :] = [: [: [: I[01] , I[01] :] , [: I[01] :] :] ;
f . p4 <= f . p1 & f . p2 <= f . p2 ;
( ( F . x ) `1 ) ^2 <= ( F . x ) ^2 ;
( x - ( y - z ) ) / ( 2 * ( y - z ) ) = ( ( y - z ) / ( 2 * ( y - z ) ) ) / ( 2 * ( y - z ) ) ;
for n being Element of NAT holds P [ n ] ;
let J , K be non empty Subset of I ;
assume 1 <= i & i <= len <* a " *> ;
0 |-> a = <*> ( the carrier of K ) & 0 <= 1 ;
X . i in 2 -tuples_on B . i \ B . i ;
<* 0 *> in dom ( e --> [ 1 , 0 ] ) ;
then P [ a ] & P [ succ a ] ;
reconsider s/. i = sN . i as terminal of D . i , X ;
( - i -' 1 ) <= len \mathbb j - 1 ;
[#] S c= [#] ( T ) & the TopStruct of T = the TopStruct of T ;
for V being strict Subspace of V holds V in W1 iff V in W2
assume k in dom mid ( f , i , j ) ;
let P be non empty Subset of TOP-REAL 2 , p , q be Point of TOP-REAL 2 ;
let A , B be Matrix of K , K , n1 , n2 be Nat ;
- a * ( - b ) = a * b - b * c ;
for A being being_line Subset of A9 holds A // A & A // K implies A // K
( id o2 ) in <* o2 , o2 *> & ( id o2 ) . ( o1 , o2 ) = <* o2 , o1 *> ;
then ||. x .|| = 0 & x = 0. X ;
let N1 , N2 be strict Subgroup of G , N be strict normal Subgroup of G ;
j >= len ( ( g | D1 ) ^ <* D2 *> ) ;
b = Q . ( len Q - 1 ) ;
f2 * f1 /* s is divergent_to+infty & f2 * f1 is divergent_to+infty ;
reconsider h = f * g as Function of N4 , G ;
assume that a <> 0 and Polynom ( a , b , c ) >= 0 ;
[ t , t ] in the InternalRel of A & [ t , t ] in the InternalRel of A ;
( v |-- E ) | n is Element of T7 ;
{} = the carrier of L1 + L2 & L1 + L2 = the carrier of L2 + L2 ;
Directed I is_halting_on Initialized s , P & Initialized s , P & Initialized s , P ;
Initialized ( p +* I ) = Initialize ( ( p +* I ) +* I ) , p = p +* I , p2 = p +* I , p3 = p +* I , E = p +* I , E = p +* I , E = p +* I ,
reconsider N2 = N1 as strict net of R2 , R2 be strict net of R2 , R2 ;
reconsider Y = Y as Element of <* Ids L , \subseteq the carrier of L ;
"/\" ( { p } , { p } ) <> p ;
consider j being Nat such that i2 = i1 + j and j <= len f ;
not [ s , 0 ] in the carrier of S2 & [ s , 0 ] in the carrier of S2 ;
m-5 in ( B '&' C ) \ D ;
n <= len ( ( P + Q ) ^ ( len P + 1 ) ) ;
( x1 - x2 ) / ( x1 - x2 ) = ( x2 - x1 ) / ( x2 - x1 ) ;
InputVertices S = { x1 , x2 , x3 , x4 , x5 , x5 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7
let x , y be Element of FTTTTTT1 ( n ) ;
p = |[ p `1 , p `2 ]| & p `1 = |[ p `1 , p `2 ]| ;
g * 1_ G = h " * g * h " .= h " * g " * h " .= h " * g " * h " ;
let p , q be Element of V , C be Subset of V ;
x0 in dom ( x1 - x2 ) /\ dom ( x2 - y2 ) ;
( R qua Function ) " = R " * ( R * ( R * ( R * S ) ) ) ;
n in Seg len ( f /^ ( len f -' 1 ) ) ;
for s being Real st s in R holds s <= s2 & t <= s2 ;
rng s c= dom ( f2 * f1 ) & rng s c= dom ( f2 * f1 ) ;
synonym ex \mathop { \rm ex X , Y st X in = { Y } & Y in \mathop { \rm Fin X } ;
1_ K * 1_ K = 1_ K & 1_ K * 1_ K = 1_ K ;
set S = Segm ( A , P1 , Q1 ) , Q1 = Segm ( A , P1 , Q1 ) ;
ex w st e = ( w - f ) / ( w - f ) & w in F ;
curry ( PZ , k ) # x is convergent ;
cluster open -> open for Subset of T ;
len f1 = 1 .= len ( f1 ^ <* x *> ) .= len f1 + len <* x *> .= len f1 + 1 ;
sqrt ( i * p ) < sqrt ( 2 * p ) ;
let x , y be Element of \rm Sub Sub Sub ( U0 ) ;
b1 , c1 // b9 , c2 & b1 , c1 // c1 , c2 ;
consider p being element such that c1 . j = { p } and c2 . j = p ;
assume that f " { 0 } = {} and f is total and f is total and f is total ;
assume IC Comput ( F , s , k ) = n ;
Reloc ( J , card I + card J + 3 ) does not destroy a ;
Macro ( card I + 1 ) does not destroy c ;
set m1 = LifeSpan ( P3 , s3 ) , m2 = LifeSpan ( P3 , s3 ) , P4 = LifeSpan ( P3 , s3 ) , P4 = Comput ( P3 , s3 , 1 ) , P4 = P3 , P4 = P3 , P4 = P3 , P4 = P3
IC Comput ( P , s , k ) in dom Initialize ( ( intloc 0 ) .--> 1 ) ;
dom t = the carrier of SCM R & dom t = the carrier of SCM R & dom t = the carrier of S ;
( E-max L~ f ) .. f = 1 & ( E-max L~ f ) .. f = 1 ;
let a , b be Element of V , C be Subset of V ;
Cl ( union F ) c= Cl ( union F ) ;
the carrier of X1 union X2 misses ( A1 \/ A2 ) & the carrier of X1 misses ( A1 \/ A2 ) ;
assume not LIN a , f . a , g . a , g . b ;
consider i be Element of M such that i = d6 . i and i in d ;
then Y c= { x } or Y = { x } or Y = { x } ;
M , v / ( y , x ) |= ( H / ( y , x ) ) ;
consider m being element such that m in Intersect ( F . m ) and x = Intersect ( F . m ) ;
reconsider A1 = support u1 , A2 = support v1 as Subset of X ;
card ( A \/ B ) = k-1 + ( 2 * 1 ) ;
assume that a1 <> a3 and a2 <> a3 and a3 <> a4 and a1 <> a4 and a3 <> a4 and a1 <> a4 and a1 <> a3 and a1 <> a3 and a3 <> a4 and a1 <> a4 and a1 <> a3 and a1 <> a3 and a1 <> a3 and a1 <> a3
cluster s \! \mathop { \rm \hbox { - } such that s is string of S ;
Carrier ( L2 ) /. n2 = ( L2 ) . n2 .= ( L2 ) . n2 ;
let P be compact non empty Subset of TOP-REAL 2 , p , q be Point of TOP-REAL 2 ;
assume that r-7 in LSeg ( p1 , p2 ) and r <= 1 and p1 in LSeg ( p2 , p3 ) ;
let A be non empty compact Subset of TOP-REAL n , p be Point of TOP-REAL n ;
assume [ k , m ] in Indices ( ( D * ) * ( i , j ) ) ;
0 <= ( ( 1 / 2 ) |^ p ) * ( ( 1 / 2 ) |^ p ) ;
( F . N ) | E8 . x = +infty ;
pred X c= Y & Z c= V implies X \ V c= Y \ Z ;
( y - z ) * ( z - w ) <> 0. I & ( y - z ) * ( z - w ) <> 0. I ;
1 + card ( ( card X + 1 ) \ { x } ) <= card u + card ( X + 1 ) ;
set g = z \circlearrowleft ( E-max L~ z ) , h = z .. z , i = ( E-max L~ z ) .. z ;
then k = 1 & p . k = <* x , y *> . k ;
cluster C -for ( the carrier of C ) --> ( the carrier of C ) -> total ;
reconsider B = A as non empty Subset of TOP-REAL n , C be Subset of TOP-REAL n ;
let a , b , c be Function of Y , BOOLEAN ;
L1 . i = ( i .--> g ) . i .= g . i .= g . i ;
Plane ( x1 , x2 , x3 , x4 ) c= P & P [ x1 , x2 , x3 ] ;
n <= indx ( D2 , D1 , j1 ) + 1 ;
( ( g2 ) . O ) `1 = - 1 & ( g2 ) . I = 1 ;
j + p .. f -' len f <= len f + 1 ;
set W = W-bound C , S = N-bound C , E = E-bound C , N = E-bound C , N = E-bound C , N = E-bound C , S = E-bound C , N = E-bound C , S = E-bound C , N = E-bound C , S = E-bound
S1 . ( a , e ) = a + e .= a + e .= a + e ;
1 in Seg width ( M * ( Line ( M , 1 ) ) ) ;
dom ( i (#) Im f ) = dom ( Im f ) /\ dom ( Im f ) ;
| ( x , x ) = W . ( a , ( a , p ) ) ;
set Q = |= ( All ( g , f , h ) ) ;
cluster -> SubSorts for ManySortedSet of U1 ;
attr F = { A } means : Def1 : F is discrete ;
reconsider z9 = *> as Element of product G ;
rng f c= rng f1 \/ rng f2 & rng ( f ^ ) c= rng ( f ^ ) ;
consider x such that x in f .: A and x in f .: C ;
f = <*> ( the carrier of F_Complex ) & f is FinSequence of ( the carrier of F_Complex ) ;
E , j |= All ( x , All ( x , H ) ) ;
reconsider n1 = n as Morphism of o1 , o2 , o2 be Morphism of o2 , o2 ;
assume that P is idempotent and R is idempotent and P \circ R = R and P \circ R = R and P \circ R = R and P .: R = R ;
card ( ( B2 \/ { x } ) \ { x } ) = k-1 + 1 ;
card ( ( x \ B1 ) /\ ( B \ B2 ) ) = 0 & card ( ( x \ B1 ) /\ ( B \ B2 ) ) = 0 ;
g + R in { s : g-r < s & s < g + r } ;
set q9 = ( q , <* s *> ) -\cal S , q = ( q , <* s *> ) -\cal S , r = ( q , <* s *> ) -\cal S , s = ( q , <* s *> ) -\cal S , s
for x being element st x in X holds x in rng f1 & x in X
h2 /. ( i + 1 ) = h2 . ( i + 1 ) ;
set mw = max ( B , max ( B , max ( C , NAT ) ) ) ;
t in Seg width ( I ^ <* n *> ) & t in dom ( I ^ <* n *> ) ;
reconsider X = dom f , C = C as Element of Fin NAT , f = ( the carrier of C ) --> NAT ;
IncAddr ( i , k ) = halt SCM+FSA .= ( l , k ) .--> ( l . k ) ;
( E-max L~ f ) .. f <= ( q `2 ) .. f & ( q `2 ) .. f <= ( q `2 ) .. f ;
attr R is condensed means : Def1 : for R being Subset of R holds ( Cl R ) ` is condensed & Cl R is condensed ;
pred 0 <= a & b <= 1 & a * b <= 1 implies a * b <= 1 ;
u in ( ( ( c /\ ( ( d /\ e ) /\ f ) /\ e ) /\ f ) /\ j ) /\ f ;
u in ( ( ( c /\ ( ( ( d /\ e ) /\ f ) /\ b ) /\ f ) /\ j ) /\ j ) /\ j ;
len C + ( - 2 ) >= 9 + ( - 2 ) ;
not x , z , y is_collinear & x , z , y is_collinear ;
a |^ n1 + 1 = a |^ n1 * a |^ n1 ;
<* \underbrace ( 0 , \dots , 0 , 0 ) *> in Line ( x , a ) ;
set y9 = <* y , c *> , f = <* y , c *> ;
F2 /. 1 in rng Line ( D , 1 ) & F /. len F = Line ( D , 1 ) ;
p . m Joins r /. m , r /. ( m + 1 ) , G ;
( p `1 ) / ( ( f /. i1 ) / ( f /. i2 ) ) = ( f /. i1 ) / ( f /. i2 ) ;
( W-min ( X \/ Y ) ) `1 = W-bound ( X \/ Y ) & ( E-max ( X \/ Y ) ) `1 = W-bound ( X \/ Y ) ;
0 + ( p `2 / p `1 ) ^2 <= 2 * r + ( p `2 / p `1 ) ^2 ;
x in dom g & not x in g " { 0 } ;
f1 /* ( seq ^\ k ) is divergent_to+infty & f2 /* ( seq ^\ k ) is divergent_to+infty ;
reconsider u2 = u as VECTOR of \mathop { \rm P\mathop { \rm contradiction } ( X ) , X = u as VECTOR of \mathop { \bf 0. } } ;
p \! \mathop { \rm \hbox { - } count ( Sgm ( X ) ) = 0 & p = ( Sgm ( X ) ) . p ;
len <* x *> + 1 < i + 1 & i <= len c + 1 ;
assume that I is non empty and { x } /\ { y } = { 0. I } ;
set i2 = card I + 4 + card J + 2 + 1 , j2 = ( card I + 2 ) .--> ( card J + 3 ) ;
x in { x , y } & h . x = {} & h . y = {} & h . x = {} ;
consider y being Element of F such that y in B and y <= x ` ;
len S = len ( the charact of ( A ) ) & len ( the charact of ( A ) ) = len ( the charact of ( A ) ) ;
reconsider m = M , i = I , n = N as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . j ;
set Nmin = \rrangle and Gmin ( Gmin , Gmin ( Gmin ( G , H ) ) ;
rng F c= the carrier of gr ( { a } , the carrier of G ) ;
Comput ( P , s , K ) . ( K , n ) is a and ( P . ( K , n ) ) `1 is a and ( P . ( K , n ) ) `1 is which ;
f . k , f . ( mod n ) in rng f ;
h " ( P ) /\ [#] ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( P ) ) ) = f " ( ( TOP-REAL 2 ) | ( P ) ) ;
g in dom f2 \ ( f2 " { 0 } ) \ ( f2 " { 0 } ) ;
gj2 X /\ dom f1 = g1 " { g1 } & g1 in ( g1 " { 0 } ) /\ ( g2 " { 0 } ) ;
consider n being element such that n in NAT and Z = G . n ;
set d1 = dist ( x1 , y1 ) , d2 = dist ( x2 , y2 ) , d2 = dist ( x2 , y2 ) ;
b `1 + sqrt ( 1 + ( b `1 / b ) ^2 ) < ( 1 + sqrt ( 1 + ( b `1 / b ) ^2 ) ;
reconsider f1 = f as VECTOR of the carrier of X , Y be bounded bounded bounded Function of X , Y ;
pred i <> 0 means i ^2 mod ( i + 1 ) = 1 ;
j2 in Seg len ( g2 . i2 ) & j2 <= len ( g2 . i2 ) ;
dom ( i + 1 ) = dom ( ( i + 1 ) * ( i + 1 ) ) .= dom ( ( i + 1 ) * ( i + 1 ) ) .= dom ( ( i + 1 ) * ( i + 1 ) ) ;
cluster sec | ]. PI / 2 , PI .[ -> one-to-one ;
Ball ( u , e ) = Ball ( f . p , e / 2 ) ;
reconsider x1 = x0 as Function of S , II ( ) , II ( ) ;
reconsider R1 = x , R2 = y , R2 = z as Relation of L , L ;
consider a , b being Subset of A such that x = [ a , b ] ;
( <* 1 *> ^ p ) ^ <* n *> in RL & <* n *> ^ ( <* n *> ^ p ) in RL ;
S1 +* S2 +* S2 = S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2
( ( ( ( ( id Z ) (#) ( ( arccot ) ^ ) ) (#) ( ( arccot ) ^ ) ) ) `| Z ) = dom ( ( ( ( id Z ) ^ ) (#) ( ( arccot ) ^ ) ) `| Z ) ;
cluster empty -> [. 0 , 1 .] -valued for Function of C , REAL ;
set C7 = 1GateCircStr ( <* z , x *> , f3 ) , strict non-empty non-empty MSAlgebra over S ;
E8 . a2 = E8 . a2 & E8 . a3 = ( n + 2 ) -T . a2 ;
( ( arctan * arccot ) `| Z ) . x = ( ( arctan * arccot ) `| Z ) . x ;
sup A = ( PI * 2 ) / 2 & inf A = 0 & inf A = 0 ;
F . ( dom f , - F . cod f ) is \emptyset & F . ( cod f , cod f ) = F . ( cod f , cod f ) ;
reconsider p9 = q9 as Point of ( TOP-REAL 2 ) | K1 , e = ( sn -FanMorphE ) | K1 , f = ( sn -FanMorphE ) | K1 , g = ( sn -FanMorphE ) | K1 , h = ( sn -FanMorphE ) | K1 , e = ( sn -FanMorphE ) | K1 , f = ( sn -FanMorphE ) | K1 ,
g . W in [#] ( Y | ( the carrier of X ) ) & g . W c= [#] ( Y | ( the carrier of X ) ) ;
let C be compact non vertical Subset of TOP-REAL 2 , p , q be Point of TOP-REAL 2 ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) \/ LSeg ( f , j ) ;
rng s c= dom f /\ ]. - r , x0 .[ & rng s c= ]. x0 - r , x0 .[ ;
assume x in { idseq 2 , Rev ( idseq 2 ) } ;
reconsider n2 = n , m2 = m - 1 , m2 = n - 1 as Element of NAT ;
for y being ExtReal st y in rng seq holds g <= y
for k st P [ k ] holds P [ k + 1 ] ;
m = m1 + m2 .= m1 + m2 .= m1 + m2 + m2 .= m1 + m2 + m2 + m2 ;
assume for n holds H1 . n = G . n -H . n ;
set BB = f .: ( ( the carrier of X1 ) \/ the carrier of X2 ) ;
ex d being Element of L st d in D & x << d ;
assume R " ( a ) c= R " ( b ) & R " ( a ) c= R " ( a ) ;
t in ]. r , s .[ or t = r or t = s or t = s ;
z + v2 in W & x = u + ( z + v2 ) ;
x2 |-- ( x2 , y2 ) iff P [ x2 , y2 ] ;
pred x1 <> x2 means x1 <> 0 & x2 <> 0 & |. x1 - x2 .| > 0 ;
assume p2 - p1 , p3 - p2 - p3 - p1 , p4 - p1 - p2 - p3 - p1 - p1 - p1 - p1 - p1 - p1 - p1 - p1 - p1 - p1 - p1 - p1 - p1 - p1 - p1 - p1 - p1 - p1 - p1 - p1 - p1 - p1 - p1 - p1 - p1 - p1 - p1
set q = \HM { \rangle } ^ <* 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not'
let f be PartFunc of REAL-NS 1 , REAL-NS n , i , j be Nat ;
( n mod ( 2 * k ) ) ! = n mod k .= k mod ( 2 * k ) ;
dom ( T * ( succ t ) ) = dom ( ( succ t ) . ( <* 0 *> ) ) ;
consider x being element such that x in wc iff x in c & x in X ;
assume ( F * G ) . x3 = v . x3 ;
assume the carrier of D1 c= the carrier of D2 & the carrier of D2 c= the carrier of D2 & the carrier of D2 = the carrier of D2 ;
reconsider A1 = [. a , b .] as Subset of [: REAL , REAL :] ;
consider y being element such that y in dom F and F . y = x ;
consider s being element such that s in dom o and a = o . s ;
set p = W-min L~ Cage ( C , n ) , q = E-max L~ Cage ( C , n ) ;
n1 -' len f + 1 <= len g + 1 & len g + 1 <= len g ;
ConsecutiveDelta ( q , O1 ) = [ u , v , a ] ;
set C-2 = ( \mathclose { [ ' ] } ) . ( k + 1 ) ;
Sum ( L * p ) = 0. R * Sum ( p ) .= 0. V * Sum ( p ) .= 0. V ;
consider i being element such that i in dom p and t = p . i ;
defpred Q [ Nat ] means 0 = Q ( ) implies $1 = ( f . $1 ) * ( f . $1 ) ;
set s3 = Comput ( P1 , s1 , k ) , P4 = Comput ( P2 , s2 , k ) , P4 = Comput ( P2 , s2 , k ) , P4 = Comput ( P2 , s2 , k ) , P4 = Comput ( P2 , s2 , k ) , P4 = Comput ( P2 , s2
let l be variable , A , B be Element of k -tuples_on NAT , x be Element of l , y be Element of k -tuples_on NAT ;
reconsider U = union ( ( G . n ) ` ) , U = ( G . n ) ` as Subset-Family of ( ( G . n ) ` ) ;
consider r such that r > 0 and Ball ( p `1 , r ) c= Q ` ;
( h | ( n + 2 ) ) /. ( i + 1 ) = p9 . ( i + 1 ) ;
reconsider B = the carrier of X1 , C = the carrier of X2 as Subset of X ;
p0 = <* - ( c - 1 ) , ( - c ) * ( - c ) *> ;
synonym f is real-valued for rng f c= NAT & rng f c= NAT & f is one-to-one ;
consider b being element such that b in dom F and a = F . b ;
x10 < card ( ( X + Y ) \ ( X + Y ) ) + card ( Y + Y ) ;
pred X c= B1 , then for A , B being set st X c= succ B & A in succ B holds A in succ B ;
then w in Cl Ball ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( x , y , z ) ;
pred 1 <= len s means : Def1 : for i being Element of NAT holds ( the _ of s ) . ( i + 1 ) = s . ( i + 1 ) ;
fY. c= f . ( k + ( n + 1 ) ) ;
the carrier of { 1_ G } = { 1_ G } & the carrier of G = { 1_ G } ;
pred p '&' q in TAUT ( A ) means : Def1 : q '&' p in TAUT ( A ) ;
- ( t `2 / t `1 ) < ( - t `1 ) / t `1 ;
U . 1 = U /. 1 .= ( U /. 1 ) `1 .= ( U /. 1 ) `1 .= ( U /. 1 ) `1 .= ( U /. 1 ) `1 ;
f .: ( the carrier of x ) = the carrier of x & f .: ( the carrier of x ) = the carrier of x ;
Indices ( ( O * A ) @ ) = [: Seg n , Seg n :] & [: Seg n , Seg n :] = [: Seg n :] ;
for n being Element of NAT holds G . n c= G . ( n + 1 )
then V in M .: { x } ;
ex f being Element of F-9 st f is invertible & f is invertible & f is invertible & f is invertible & f is invertible & f is invertible ;
[ h . 0 , h . 3 ] in the InternalRel of G & [ h . 2 , h . 3 ] in the InternalRel of G ;
s +* Initialize ( ( intloc 0 ) .--> 1 ) = s3 +* ( ( intloc 0 ) .--> 1 ) ;
|[ w1 , v1 ]| `1 <> 0. TOP-REAL 2 & |[ w1 , v1 ]| `1 <> 0. TOP-REAL 2 ;
reconsider t = t as Element of INT -tuples_on INT ( ) ;
C \/ P c= [#] ( ( G \ A ) \ A ) & C /\ ( ( G \ A ) \ A ) = {} ;
f " V in ( ( TOP-REAL 2 ) | D ) /\ ( ( TOP-REAL 2 ) | D ) ;
x in [#] ( ( the carrier of ( T ) ) /\ the carrier of ( T ) ) ;
g . x <= h1 . x & h . x <= h1 . x & h1 . x <= h2 . x ;
InputVertices S = { xy , yz , yz , zx , cin } & InputVertices S = { xy , yz , yz , zx } ;
for n being Nat st P [ n ] holds P [ n + 1 ] ;
set R = Line ( M , i ) * Line ( M , i ) ;
assume that M1 is being_line and M2 is being_line and M1 is being_line and M2 is being_line and M2 is being_line and M1 is being_line and M2 is being_line ;
reconsider a = f0 . i0 - 1 as Element of K ;
len ( ( Len F1 ) ^ ( Len F2 ) ) = Sum ( Len F1 ) + Sum ( Len F2 ) .= Sum ( Len F1 ) + Sum ( ( len F2 ) + len ( len F2 ) ) ;
len ( ( the \mathbb R ) * ( i , j ) ) = n & len ( ( the multF of R ) * ( i , j ) ) = n ;
dom max ( - f , - g ) = dom ( f + g ) /\ dom ( g + h ) ;
( the Sorts of seq ) . n = sup ( Y1 /\ Y2 ) & ( the Sorts of seq ) . n = sup ( Y1 /\ Y2 ) ;
dom ( p1 ^ p2 ) = dom ( f ^ <* p1 *> ) .= dom ( f ^ <* p2 *> ) .= dom ( f ^ <* p1 *> ) ;
M . [ 1 , y ] = 1 / ( 1 / ( 1 - y ) ) .= 1 / ( 1 - y ) .= y ;
assume that W is non trivial and W .last() c= the carrier of G2 and W is the carrier of G2 and W is open ;
C6 * ( i1 , i2 ) = G1 * ( i1 , i2 ) .= G1 * ( i1 , i2 ) ;
C8 |- 'not' All ( x , p ) 'or' 'not' All ( x , p ) ;
for b st b in rng g holds lower_bound rng fD1 <= b implies lower_bound rng fD1 <= b
- ( ( ( q `1 / |. q .| - cn ) / ( 1 + sn ) ) ^2 ) = 1 - ( q `1 / |. q .| - cn ) ^2 ;
( LSeg ( c , m ) \/ { l } ) \/ ( { l } \/ { k } ) c= R ;
consider p be element such that p in LSeg ( x , p ) and p in L~ f and x = [ p , p ] ;
Indices ( X @ ) = [: Seg n , Seg n :] & [: Seg n , Seg n :] = [: Seg n , Seg n :] ;
cluster s => ( q => p ) => ( s => ( s => p ) ) ) => ( ( s => ( s => p ) ) => ( ( s => ( s => p ) ) => ( ( s => ( s => p ) ) => ( s => ( s => p ) ) ) ) ) is
Im ( ( Partial_Sums F ) . m ) + ( Partial_Sums F ) . m ) is_measurable_on E ;
cluster f . ( x1 , x2 ) -> ( f . ( x1 , x2 ) ) -valued for Element of D ;
consider g being Function such that g = F . t and Q [ t , g ] ;
p in LSeg ( NW-corner ( Z ) , NW-corner ( Z ) ) \/ LSeg ( NW-corner ( Z ) , NW-corner ( Z ) ) ;
set R8 = R |^ 1 / ]. b , +infty .[ ;
IncAddr ( I , k ) = SubFrom ( da , db ) .= IncAddr ( da , \mathop { \rm succ } ) .= halt SCM+FSA .= halt SCM+FSA ;
seq . m <= ( ( the Sorts of seq ) . k ) . ( ( the Sorts of seq ) . k ) ;
a + b = ( a ` ) ` + ( b ` ` ` ` ` ` ` ) ` .= ( a ` ` + ( b ` ` ) ` ) ` ;
id ( X /\ Y ) = id X /\ id ( X /\ Y ) .= id X /\ id Y ;
for x being element st x in dom h holds h . x = f . x
reconsider H = U \/ U2 as non empty Subset of U2 , a = U \/ ( { a } \/ { a } ) ;
u in ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
consider y being element such that y in Y and P [ y , inf B ] ;
consider A being finite stable stable Subset of R such that card A = card ( R * R ) and A in A ;
p2 in rng ( f |-- p1 ) \ rng <* p1 *> & rng <* p1 *> c= rng ( f | p1 ) ;
len s1 - 1 > 0 & len s2 - 1 > 0 & len s2 - 1 > 0 ;
( ( E-max C ) .. ( ( E-max C ) .. ( ( E-max C ) .. ( ( E-max C ) .. ( ( E-max C ) ) ) ) ) ) .. ( ( E-max C ) .. ( ( E-max C ) .. ( ( E-max C ) ) ) ) ;
Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) ;
f . a1 ` = f . a1 ` .= ( a1 ` ) ` .= ( a1 ` ) ` .= ( a1 ` ) ` ;
( seq ^\ k ) . n in ]. - r , x0 .[ & ( seq ^\ k ) . n in ]. x0 - r , x0 .[ ;
gg . i0 = g . i0 .= G . i0 .= G . i0 .= G . i0 ;
the InternalRel of S is transitive & the InternalRel of S is transitive implies the InternalRel of S is transitive
deffunc F ( Ordinal , Ordinal ) = phi . ( $1 + 1 ) , $2 = phi . ( $1 + 1 ) ;
F . ( s1 . a1 ) = F . s2 . a1 .= s2 . a1 .= s2 . a1 .= s2 . a1 ;
x `2 = A . ( o . a ) .= Den ( o , A ) . ( a . a ) ;
Cl ( f " P1 ) c= f " ( Cl P1 ) & f " P1 c= f " ( Cl P1 ) ;
FinMeetCl ( ( ( the topology of S ) \ { 0 } ) \/ the topology of T ) c= the topology of T ;
synonym o is OperSymbol means : Def2 : o <> \ast & o <> \ast & o <> * & o <> * & o <> * & o <> * & o <> * & o <> * & o <> * & o <> * & o <> * & o <> * & o <> * & o <> * & o <> * & o <> * & o <> *
assume that X = ( Y + Z ) |^ n and card X <> card Y and card X <> card Y ;
the non empty finite for s , t being Element of ( the carrier of S ) * ;
LIN a , a1 , d or b , c // b1 , c1 or a , c // a1 , c1 ;
e / 2 . 1 = 0 & e / 2 . 2 = 1 & e / 2 . 3 = 0 & e / 2 . 3 = 1 ;
EE in SE & EE in SE & EE in { NE } ;
set J = ( l , u ) -TruthEval ( u , v ) ;
set A1 = 1GateCircStr ( a9 , b9 , c ) , A2 = Following ( a9 , b9 , c9 ) ;
set c9 = [ <* cin , dp *> , and2 ] , A2 = [ <* cin , bm *> , and2 ] , zx = [ <* cin , -39 *> , and2 ] , zx = [ <* cin , bm *> , <* cin *> ] ;
x * z * x " in x * ( z * N ) * ( z * N ) " ;
for x being element st x in dom f holds f . x = g2 . x
Int cell ( GoB f , 1 , width GoB f -' 1 ) c= RightComp f \/ RightComp f \/ RightComp f \/ RightComp f ;
U is_an_arc_of Cage ( C , n ) , W-min L~ Cage ( C , n ) & Cage ( C , n ) /. len ( Cage ( C , n ) ) = Cage ( C , n ) /. len ( Cage ( C , n ) ) ;
set f-17 = f .: @ g , fd = f .: @ g ;
attr S1 is convergent means : Def1 : for S2 , S2 , S1 , S2 , S2 , S2 , S2 , S2 , S1 , S2 , S2 , S2 , S2 , S2 , S2 , S2 , S1 , S2 , S2 , S2 , S2 , S1 , S2 , S2 , S2 , S2 , S1 be convergent ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a + a .= a ;
cluster -> \mathclose -> \mathclose -> \mathclose -> \mathclose for non empty RelStr ;
consider d being element such that R reduces b , d and R reduces c , d and R reduces c , d ;
not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) ;
( z + a ) + x = z + ( a + y ) .= z + ( a + y ) .= z + ( a + y ) ;
len ( l \lbrack ( a , A ) .--> x ) = len l & len ( l .--> x ) = len l ;
t4 *> ^ ( {} , rng ( t ^ <* N *> ) ) -valued ( X , rng ( t ^ <* N *> ) ) -valued ( X , rng ( t ^ <* N *> ) ) -valued ( X , rng ( t ^ <* N *> ) ) -valued ( X , rng ( t ^ <* N *> ) ) -valued
t = <* F . t *> ^ ( C ^ q ) ^ ( C ^ q ) ;
set p0 = W-min L~ Cage ( C , n ) , p1 = W-min L~ Cage ( C , n ) , p2 = W-min L~ Cage ( C , n ) , p2 = W-min L~ Cage ( C , n ) , p1 = W-min L~ Cage ( C , n ) , p2 = W-min L~ Cage ( C , n ) , p2 = Cage ( C
k9 -' ( i + 1 ) = ( k + 1 ) - ( i + 1 ) ;
consider u being Element of L such that u = u ` and u in D and u in D and u in D ;
len ( ( width ( ( A |-> a ) |-> ( A --> b ) ) ) |-> ( A --> b ) ) = width ( ( A --> a ) --> ( A --> b ) ) ;
F3 . x in dom ( ( G * the_arity_of o ) . x ) ;
set c2 = the carrier of H2 , c2 = the carrier of H2 , c2 = the carrier of H2 , c2 = the carrier of H2 , c2 = the carrier of H2 , c2 = the carrier of H2 ;
set [: H1 , H2 :] = the carrier of H1 , the carrier of H2 :] ;
( Comput ( P , s , 6 ) ) . intpos ( m + 6 ) = s . intpos ( m + 6 ) ;
IC Comput ( P3 , t , k ) + 1 = ( card I + 1 ) + 1 ;
dom ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( A ) ) ) ) ) ) ) ) * ( ( ( ( ( ( A ) ) * ( ( ( A ) ) * ( ( ( A ) ) * ( ( ( A ) ) * ( ( ( A ) ) * ( ( A ) ) * ( ( A )
cluster <* l *> ^ phi -> ( 1 + 1 ) string of S ;
set b5 = [ <* bm , bm *> , <* bm , bm *> ] , [ <* bm , bm *> , <* cin *> ] , [ <* cin , bm *> , <* cin *> ] ] ;
Line ( Segm M , P , Q ) . x = L * Sgm Q . x .= Sgm Q . x ;
n in dom ( ( ( the Sorts of A ) * ( the_arity_of o ) ) . n ) ;
cluster f1 + f2 -> continuous for PartFunc of REAL , REAL n , REAL n ;
consider y be Point of X such that a = y and ||. x - y .|| <= r ;
set x3 = t1 . DataLoc ( s . a , 2 ) , x2 = P . DataLoc ( s . a , 2 ) , x3 = Comput ( P , s , 2 ) , x4 = Comput ( P , s , 2 ) , P4 = Comput ( P , s , 2 ) , P4 = P3 ;
set p0 = stop I , p1 = P +* stop I , p2 = Comput ( P3 , s3 , 1 ) , P4 = P3 ;
consider a being Point of D2 such that a in W1 and b = g . a and a in W2 and b = g . a ;
{ A , B , C , D } = { A , B , C } \/ { C , D } ;
let A , B , C , D , E , F , J , M , N , N , M , N , N , N , F , M , N , N , N , N , F , M , N , N , N , M , N , N , N , N , F , M ;
|. p2 .| ^2 - ( p2 `1 / p2 `1 ) ^2 >= 0 & ( p2 `1 / p2 `2 ) ^2 >= 0 ;
l -' 1 + 1 = ( l * l1 + 1 ) + ( m + 1 ) ;
x = v + ( a * w1 + ( b * w2 ) ) + ( c * w2 ) ;
the TopStruct of L = ( the Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott of L ) ;
consider y being element such that y in dom H1 and x = H1 . y and x = H1 . y ;
ft \ { n } = ( Free ( { v } , H ) ) \ ( { v } \/ { n } ) ;
for Y being Subset of X st Y is summable holds Y is summable iff Y is not empty
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
for s being FinSequence holds len ( the { F } ) = len ( the { F } ) & len ( the { F } ) = len s
for x st x in Z holds ( exp_R * f ) is_differentiable_in x & ( exp_R * f ) . x > 0
rng ( h2 * ( f - g ) ) c= the carrier of ( ( TOP-REAL 2 ) | K1 ) & rng ( h2 * ( f - g ) ) c= the carrier of ( ( TOP-REAL 2 ) | K1 ) ;
j + ( len f -' len f ) <= len f + ( len f -' len f ) - len f ;
reconsider R1 = R * I as PartFunc of REAL , REAL n , REAL n , REAL n ;
C8 . x = s1 . a .= C7 . x .= C7 . x .= C7 . x .= ( C . a ) . x .= ( C . a ) . x .= ( C . a ) . x .= ( C . a ) . x .= ( C . a ) . x .= ( C . a ) . x ;
power F_Complex . ( z , n ) = 1 .= ( x |^ ( n + 1 ) ) * ( x |^ ( n + 1 ) ) .= ( x |^ ( n + 1 ) ) * ( x |^ ( n + 1 ) ) ;
t in ( the connectives of C ) . ( ( the connectives of C ) . t ) ;
support ( f + g ) c= support f \/ support ( g ) & support ( f + g ) c= support f \/ support ( g ) ;
ex N st N = j1 & 2 * Sum ( ( r | N ) | N ) > N & 2 * Sum ( ( r | N ) | N ) > N ;
for y , p st P [ p ] holds P [ All ( y , p ) ]
{ [ x1 , x2 ] } is Subset of [: X1 , X2 :] & { x1 , x2 } is Subset of [: X1 , X2 :] ;
h = ( i , j ) |-- ( id B , id B ) . ( i , j ) .= H . ( i , j ) .= H . ( i , j ) ;
ex x1 being Element of G st x1 = x & x1 * N c= A & x1 * N c= A ;
set X = ( ( ( |. q .| ) ) . ( ( |. q .| ) . O ) ) , Y = ( ( |. q .| ) . O ) , Z = ( ( |. q .| ) . O ) , Y = ( ( |. q .| ) . O ) , Z = ( ( |. q .| ) . O ) , X = ( ( |. q .| ) . O ) , Y = (
b . n in { g1 : x0 - r < g1 & g1 < x0 + r } ;
f /* s1 is convergent & f /. ( lim s1 ) = lim ( f /* s1 ) ;
the lattice of the lattice of T = the lattice of the lattice of T & the lattice of T = the lattice of the lattice of T & the carrier of T = the carrier of T ;
'not' ( a . x ) '&' b . x 'or' a . x = FALSE ;
2 = len ( ( q ^ <* r1 *> ) ^ ( q ^ <* r1 *> ) ) + len ( q ^ <* r1 *> ) ;
( ( 1 / a ) (#) ( ( ( ( ( ( ( ( ( ( ( ( 1 / a ) ) ) * ( ( ( ( ( ( ( ( a ) ) / a ) ) * ( ( ( ( a + b ) ) * ( ( ( a + b ) / ( ( a + b ) ) * ( ( a + b ) / ( (
set K1 = ( upper ( f , A ) ) || [: A , A :] & ( the Sorts of A ) || [: A , A :] = ( the Sorts of A ) || [: A , A :] ;
assume e in { ( w1 - w2 ) / 2 : w1 in F & w2 in G & w1 in F & w1 in G } ;
reconsider d7 = dom a `1 , d6 = dom F `1 , d6 = dom F `2 , d6 = dom F `2 , d6 = dom F `2 , d6 = dom F `2 , d6 = dom F `2 , d6 = dom G `2 , d6 = I `2 , d6 = I `2 , d7 = I `2 , d7
LSeg ( f /^ ( j -' 1 ) , j ) = LSeg ( f , j ) \/ LSeg ( f , j ) ;
assume X in { T . g2 : h . g2 = ( N . g2 ) `1 } ;
assume that Hom ( d , c ) <> {} and <* f , g *> * ( f , g ) = <* f , g *> and <* f , g *> * ( g , f ) = <* f , g *> * ( g , f ) ;
dom S[. S , n .] = dom S /\ Seg n .= dom ( L . n ) .= dom ( L . n ) .= dom ( L . n ) .= dom ( L . n ) .= dom ( L . n ) ;
x in ( H |^ a ) |^ a implies ex g st x = g |^ a & g in H |^ a
a * ( ( - 1 ) * a ) = a ^2 - ( 0 ) * a .= a ;
D2 . j in { r : lower_bound A <= r & r <= ( D1 . i ) `2 } ;
ex p being Point of TOP-REAL 2 st p = x & P [ p ] & P [ p ] ;
for c holds f . c <= g . c implies f ^ @ c ^ g ^ f ;
dom ( ( f1 (#) f2 ) | X ) /\ X c= dom ( f1 (#) f2 ) /\ X ;
1 = ( p * p ) / ( p * q ) .= p * ( q * p ) .= p * ( q * p ) .= p * ( q * p ) ;
len g = len f + len <* x *> .= len f + len <* y *> .= len f + len <* y *> .= len f + 1 ;
dom ( F | [: N1 , S1 :] ) = dom ( F | [: N1 , S1 :] ) .= [: N1 , S1 :] ;
dom ( f . t ) * I . t = dom ( f . t ) * ( g . t ) ;
assume a in ( "\/" ( ( T |^ the carrier of S ) , F ) ) .: D ;
assume that g is one-to-one and ( the carrier' of S ) /\ rng g c= dom ( g * f ) and rng g c= dom ( g * f ) and g is one-to-one and rng g c= dom ( g * f ) and g is one-to-one and rng g c= dom ( g * f ) ;
( ( x \ y ) \ z ) \ ( ( x \ y ) \ z ) = 0. X ;
consider f such that f * f ' = id b and f = id a and f = id b and f = id a and f = id b ;
( ( ( ( ( ( ( ( ( ( ( ( ( 1 / 2 ) ) ) ) ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 1 / 1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
Index ( p , co ) <= len ( ( LS * LS ) .. LS ) + 1 ;
t1 , t2 , t2 be Element of ( ( the Sorts of T ) . s , ( the Sorts of T ) . s ) ;
( ( Frege ( ( Frege ( F ) ) ) . h ) ) . h <= ( ( Frege ( ( Frege ( F ) ) . h ) ) . j ) . j ;
then P [ f . i0 , f . i0 ] & F ( f . i0 , f . i0 ) < j ;
Q [ ( D . x ) `1 , F . ( D . x ) ] ;
consider x being element such that x in dom ( F . s ) and y = F . s and y = ( F . s ) . x ;
l . i < r . i & [ l . i , r . i ] in the InternalRel of G . i ;
the Sorts of A2 = ( the Sorts of S2 ) +* ( the Sorts of S2 ) +* ( the Sorts of S2 ) +* ( the Sorts of S2 ) +* ( the Sorts of S2 ) +* ( the Sorts of S2 ) +* ( the Sorts of S2 ) +* ( the Sorts of S2 ) +* ( the Sorts of S2 ) +* ( the Sorts of S2 ) +* ( the Sorts of S2
consider s being Function such that s is one-to-one and dom s = NAT and rng s = { 0 } and rng s = { 1 } and s is convergent and rng s = { 1 } and s is convergent and rng s = { 1 } ;
dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( a1 , a ) ;
( ( for n holds C /. ( len C ) ) `1 = ( ( C /. ( len C ) ) `1 ) `1 ;
q <= ( ( UMP C ) / 2 ) / 2 & q <= ( ( UMP C ) / 2 ) / 2 ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} & LSeg ( f | i2 , j ) /\ LSeg ( f | i2 , j ) = {} ;
given a being ExtReal such that a <= I and A = ]. a , I .] and a in A and b in A and a = f . ( a , I ) ;
consider a , b being Complex such that z = a and y = b and z = a + b and a = b ;
set X = { b |^ n where b is Element of NAT : b in n } , Y = { b } , Z = { b } , Z = { a } , Y = { b } , Z = { a } , Y = { b } , Z = { b } , Z = { a } , X = { b } , Y = { b } , Z =
( ( x * y ) * z ) \ ( x * y ) = 0. X ;
set xy = [ <* xy , yz *> , [ <* yz , yz *> , <* yz , yz *> ] , [ <* yz , yz *> , [ <* yz , yz *> , <* yz , yz *> ] ] ] ;
Carrier ( l ) /. len ( l ) = Carrier ( l ) . len ( l ) .= Carrier ( l ) . len ( l ) ;
sqrt ( ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 ) = 1 - ( q `2 / |. q .| - sn ) ^2 ;
sqrt ( ( ( p `1 / |. p .| - sn ) / ( 1 + sn ) ) ^2 ) < 1 ^2 / ( ( p `1 / |. p .| - sn ) / ( 1 + sn ) ) ^2 ;
( ( ( ( ( S \/ Y ) \ { x } ) \/ ( S \/ { y } ) ) \ { x } ) \/ ( ( S \/ Y ) \ { y } ) \ { x } ) = ( S \/ ( S \/ { y } ) \ { x } ) ;
( ( seq - seq ) . k ) . k = ( seq . k - seq . k ) . ( seq . k ) ;
rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , u0 ) & rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , u0 ) ;
the carrier of X = the carrier of X & the carrier of X = the carrier of X & the carrier of X = the carrier of X & the carrier of X = the carrier of X ;
ex p3 st p3 = p4 & |. p3 .| = r & |. p3 .| = r & |. p3 .| = r ;
set [: h , A :] = [: [: [: the carrier of X , the carrier of X :] , the carrier of X :] ;
R |^ ( 0 * n ) = \mathop { Ireal ( X , X ) , R |^ ( 0 + 1 ) } .= R |^ ( n + 1 ) ;
Partial_Sums ( ( curry ( F . n ) ) ) . ( n + 1 ) is nonnegative & ( ( curry ( F . n ) ) . ( n + 1 ) ) is nonnegative ;
f2 = C7 . ( ( E8 , the carrier of V ) --> ( [ V , the carrier of V ] , [ V , the carrier of V ] ) ) ;
S1 . b = s1 . b .= s2 . b .= S2 . b .= S2 . b .= S2 . b .= S2 . b .= S2 . b .= S2 . b ;
p2 in LSeg ( p2 , p1 ) /\ LSeg ( p1 , p2 ) /\ LSeg ( p2 , p2 ) /\ LSeg ( p1 , p2 ) ;
dom ( f . t ) = Seg n & dom ( I . t ) = Seg n & dom ( I . t ) = Seg n ;
assume o = ( the connectives of S ) . 11 & the connectives of S = ( the connectives of S ) . 12 ;
set phi = ( l1 , l2 ) -TruthEval ( 1 , 1 ) , phi = ( X , m ) -TruthEval ( 1 , m ) , N = ( X , m ) -TruthEval ( 1 , m ) , N = ( X , m ) -TruthEval ( 1 , m ) , N = ( X , m ) -TruthEval ( 1 , m ) , N = ( m , m ) -TruthEval ( 1 , m ) , N =
synonym p is_sequence T for p , T means : Def2 : HT ( p , T ) = 0. L & HT ( p , T ) = 0. L ;
( Y1 `1 ) / ( 1 + 1 ) = ( ( Y1 `1 ) / ( Y1 `2 ) ) / ( Y1 `2 ) & ( Y1 `2 ) / ( Y1 `2 ) <> {} ;
defpred X [ Nat , set ] means P [ $1 , $2 , $1 ] means P [ $1 , $2 , $1 ] ;
consider k being Nat such that for n being Nat st k <= n holds s . n < x0 + g ;
Det I |^ ( ( m -' n ) -' 1 ) = 1_ K & Det I |^ ( m -' n ) = 1_ K ;
sqrt ( - b ^2 - sqrt ( b ^2 + c ^2 ) ) * ( - b * a ) < 0 ;
CC . d = CC . ( dC . d mod ( CC . d mod ( CC . d mod ( CC . d mod ( CC . d mod ( CC . d mod ( CC . d ) ) ) ) ) ) ;
attr X1 is dense means : Def1 : X1 is dense & X2 is dense & X1 /\ X2 is dense implies X1 /\ X2 is dense & X2 /\ X1 is dense & X1 /\ X2 is dense & X1 /\ X2 is dense implies X1 meet X2 is dense ;
deffunc F6 ( Element of E , Element of I ) = ( $1 * ( $2 , $1 ) ) * ( $2 , $1 ) ;
t ^ <* n *> in { t ^ <* i *> : Q [ i , T . ( i + 1 ) ] } ;
( x \ y ) \ x = ( x \ x ) \ y .= 0. X .= 0. X .= 0. X ;
for X being non empty set holds X is Basis of [: <* X , \subseteq :] , <* X , \subseteq :]
synonym A , B means : Def1 : Cl ( A \/ B ) misses Cl ( B \/ C ) ;
len ( M @ ) = len p & width ( M @ ) = width ( M @ ) & width ( M @ ) = width ( M @ ) ;
J = { x where v is Element of K : 0 < v & v < 1 } ;
( Sgm ( m ) ) . d - ( Sgm ( m ) ) . e <> 0 ;
lower_bound divset ( D2 , k + k2 ) = D2 . ( k + k2 ) - D2 . ( k + k2 ) ;
g . r1 = - 2 * r1 + 1 & dom h = [. 0 , 1 .] & rng h = [. 0 , 1 .] ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `2 ;
ex w st w in dom ( B1 ^ <* s *> ) & <* s *> ^ w = <* 1 *> ^ w ^ w ;
[ 1 , {} , <* d1 *> ] in ( { [ 0 , {} ] } \/ ( { <* d1 *> , <* d2 *> } ) \/ ( { <* d1 *> , <* d2 *> } ) ) \/ ( { <* d1 *> , <* d2 *> } ) ;
IC Exec ( i , s1 ) + n = IC Comput ( P2 , s2 , n ) + n .= ( IC Comput ( P2 , s2 , n ) ) + n ;
IC Comput ( P , s , 1 ) = IC Comput ( P , s , 1 ) .= IC Comput ( P , s , 1 ) .= IC Comput ( P , s , 1 ) .= IC Comput ( P , s , 1 ) ;
( IExec ( W6 , Q , t ) ) . intpos i = t . intpos i & ( Initialize ( ( Initialize t ) . intpos i ) ) . intpos i = t . intpos i ;
LSeg ( f /^ ( len f -' 1 ) , i ) misses LSeg ( f /^ ( len f -' 1 ) , j ) ;
assume for x , y being Element of L st x in C holds x <= y or y <= x or x <= y ;
|( f , f .|| . x = f . ( sup C ) - f . ( sup C ) .= f . ( sup C ) - f . ( sup C ) ;
for F , G being FinSequence st rng F misses rng G & rng F c= rng G holds F ^ G is one-to-one
||. R /. L . h .|| < e1 * ( K + 1 ) * ( K + 1 ) ;
assume a in { q where q is Element of M : dist ( z , q ) <= r & r <= q & q <= 1 } ;
set p4 = [ 2 , 1 ] .--> [ 2 , 1 ] , p2 = [ 2 , 1 ] , p4 = [ 2 , 1 ] , p1 = [ 2 , 1 ] , p2 = [ 2 , 1 ] , p2 = [ 2 , 1 ] , p4 = [ 2 , 1 ] , p2 = [ 2 , 1 ] , p4 = [ 3 , 1 ] , p4 = [ 3 , 1 ] ,
consider x , y being Subset of X such that [ x , y ] in F and x in d and y in d and x in d and y in d and x in d and y in d and x in d and y in d ;
for y , x , y being Element of [: REAL , REAL :] st y in Y & x in X holds y <= x & y <= y
func |. p ^ q .| -> variable means : Def1 : for p , q being variable holds it . ( p , q ) = min ( p , q ) & it . ( p , q ) = min ( p , q ) ;
consider t being Element of S such that x , y , z is_collinear and x , z , t is_collinear and x , z , t is_collinear and x , z , t is_collinear ;
dom ( x1 ) = Seg len ( x1 ^ x2 ) & len ( x1 ^ x2 ) = len ( x1 ^ x2 ) & len ( x1 ^ x2 ) = len ( x1 ^ x2 ) ;
consider y2 being Real such that x2 = y2 and 0 <= y2 and y2 <= 1 and 0 <= y2 and y2 <= 1 and x2 <= 1 and y2 <= 1 and y2 <= 1 and |. y2 - x2 .| = 1 ;
||. f | X /* s1 .|| = ||. f .|| | ( X /\ dom f ) .= ||. f .|| | X ;
( the InternalRel of A ) ~ = {} or ( the InternalRel of A ) ~ /\ ( the carrier of A ) = {} \/ {} .= {} .= {} .= {} ;
assume that i in dom p and for j being Nat st j in dom q holds P [ i , j ] and for i being Nat st i in dom p holds P [ i , p . j ] and P [ i , j ] ;
reconsider h = f | [: X , Y :] as Function of [: [: X , Y :] , [: X , Y :] , [: Y , X :] ;
u1 in the carrier of W1 & u2 in the carrier of W2 & v2 in the carrier of W2 & v1 = the carrier of W2 & v2 = the carrier of W2 & v1 = the carrier of W2 & v2 = the carrier of W2 & v2 = the carrier of W2 implies v1 = v2
defpred P [ Element of L ] means M <= f . $1 & $1 <= len f implies f . $1 <= f . $1 ;
not ( u , a , v ) = s * x + ( - s * x ) .= b * x + ( - s * x ) .= b * x + ( - s ) * x .= b * x + ( - s ) * x .= b * y ;
- ( - ( - x ) ) = - ( - x ) + - ( - x ) .= - ( - x ) + - ( - x ) .= - ( - x ) + - x .= - x ;
given a being Point of Gu such that for x being Point of Gu holds a in A & x in B ;
fthesis = [ [ dom ( f . ( dom f ) , cod ( f . ( cod f ) ) ] , [ cod f , cod f ] ] , [ cod f , cod f ] ] ] ;
for k , n being Nat st k <> 0 & k < n & n <= k holds k divides n & k divides n implies k divides n
for x being element holds x in A |^ d iff x in ( ( A ` ) ` ) `
consider u , v being Element of R , a being Element of A such that l /. i = u * a and v in A ;
- ( ( - p `1 ) / ( 1 + ( p `2 / p `1 ) ^2 ) ) / ( 1 + ( p `2 / p `1 ) ^2 ) > 0 ;
Carrier ( LU ) . k = ( Carrier ( L ) ) . k & F . k in dom ( ( Carrier ( L ) ) . k ) ;
set i2 = AddTo ( a , i , - n ) , i1 = AddTo ( a , i , - n ) , i2 = AddTo ( a , i , - n ) , i2 = AddTo ( a , i , - n ) , i2 = AddTo ( a , i , - n ) , i1 = [ i1 , - n ] ;
attr B is universal means : Def1 : for B , S2 being non empty Subset of \mathopen ( B , S2 ) holds S1 = S2 & S2 = S2 & S2 = S2 & S1 = S2 & S2 = S2 & S2 = S2 ;
a9 "/\" D = { a "/\" d where d is Element of N : d in D } ;
||. ( \square - REAL ) * ( ( q - REAL ) * ( q - REAL ) ) .|| >= ||. ( q - REAL ) * ( q - REAL ) ) ;
( - f ) . ( sup A ) = ( - f ) . ( sup A ) .= ( - f ) . ( sup A ) ;
( G * ( len G , k ) ) `1 = ( G * ( len G , k ) ) `1 .= ( G * ( len G , k ) ) `1 ;
( Proj ( i , n ) ) . 3 = <* ( proj ( i , n ) ) . 3 *> .= <* ( proj ( i , n ) ) . 3 *> ;
f1 + f2 * reproj ( i , x ) * reproj ( i , x ) is_differentiable_in ( ( ( ( ( ( ( i - 1 ) * y ) * reproj ( i , x ) ) * reproj ( i , x ) ) ) . reproj ( i , x ) ) . ( x - y ) ;
pred ( ( ( - tan ) (#) ( ( tan * ( tan * ( tan * ( tan * ( tan * ( tan * ( tan * ( tan * ( tan * ( tan * ( tan * ( tan * ( tan * ( tan * ( tan * ( tan * ( tan * ( tan * ( tan * ( tan * ( tan * ( tan * ( tan * ( tan * ( tan * ( tan *
ex t being SortSymbol of S st t = s & h1 . {} = h2 . {} & for x being set st x in dom h1 holds h1 . x = h2 . {} ;
defpred C [ Nat ] means ( Pseq . $1 ) `1 is -functor & ( for n holds Pseq . n = A ) & ( for n holds A in ( A ) . n ) `1 ;
consider y being element such that y in dom ( ( p . i ) `1 ) and ( q . i ) `2 = ( p . i ) `1 and ( ( p . i ) `1 ) `2 = ( p . i ) `2 ;
reconsider L = product ( { x1 } +* ( index B ) , l ) as Basis of product A , product ( Carrier B ) ;
for c being Element of C holds T . ( id D ) = id D iff ex d being Element of D st T . ( id D ) = id D & ( T . ( id D ) ) . ( id D ) = id D
Comput ( f , n , p ) . ( len f + 1 ) = ( f | n ) . len f .= f . len f .= f . len f .= f . len f .= f . len f ;
( f * g ) . x = f . ( g . x ) & ( f * h ) . x = f . ( h . x ) ;
p in { 1 / 2 * ( G * ( i + 1 , j ) + G * ( i + 1 , j + 1 ) ) ;
f `2 - p = ( f | ( n , L ) ) *' ( f | ( n , L ) ) .= ( f . ( n , L ) ) *' ( f . ( n , L ) ) .= f . ( n , L ) ;
consider r being Real such that r in rng ( f | divset ( D , j ) ) and r < m + s ;
f1 . [ ( |. r2 .| ) * ( ( |. r2 .| ) * ( ( |. r2 .| ) * ( ( |. r2 .| ) * ( ( |. r2 .| ) * ( ( |. r2 .| ) * ( ( |. r2 .| ) * ( ( |. r2 .| ) * ( ( |. r2 .| ) * ( ( |. r2 .| ) * ( ( |. r2 .| ) * ( ( |. r2 .| ) * ( ( |. r2 .| )
eval ( a | ( n , L ) , x ) = ( a | ( n , L ) ) . x .= a . x .= a . x ;
z = DigA ( ( tt ) , ( x , m ) ) . z .= DigA ( ( ( ( n , m ) , ( n + 1 ) ) , ( n + 1 ) ) . z ) .= DigA ( ( ( n , m ) , ( n + 1 ) ) . z ) .= DigA ( ( ( n , m ) , ( n + 1 ) ) . z ) ;
set H = { Intersect ( S ) where S is Subset-Family of X : S c= G & S c= G } ;
consider S19 be Element of D ( ) , d being Element of D ( ) such that S = S19 ^ <* d *> and S = ( S ^ <* d *> ) ^ <* d *> ;
assume that x1 in dom f and x2 in dom f and f . x1 = f . x2 and f . x2 = f . x2 and f . x2 = f . x2 ;
- 1 <= ( ( q `1 / |. q .| - sn ) / ( 1 + sn ) ) / ( 1 + sn ) ;
not 0. V in A & Sum ( l ) = 0. V & Sum ( l ) = 0. V & Sum ( l ) = 0. V & Sum ( l ) = 0. V ;
let k1 , k2 , k2 , x4 , \overline ( I , J ) , \overline ( I , J ) , \overline ( I , J ) , \overline ( I , J ) = card ( I , J ) , card ( J , card I ) , \overline ( I , J ) = card ( I , card J ) , card ( J , card J ) ) , card I = card J + 2 ;
consider j being element such that j in dom a and j in dom g and x = g " . j and x = a . j ;
H1 . x1 c= H1 . x2 or H1 . x2 c= H1 . x2 or H1 . x2 c= H1 . x2 or H1 . x2 c= H1 . x2 & H1 . x2 c= H1 . x2 ;
consider a being Real such that p = such that p = e * p1 + ( a * p2 ) and 0 <= a and a <= 1 and a <= 1 and b <= 1 and a <= 1 and a <= 1 and a <= 1 and b <= 1 ;
assume that a <= c and c <= b and [ a , b ] in dom f and [ a , b ] in dom g and [ a , b ] in dom g and g . ( a , b ) = g . ( a , b ) ;
cell ( Gauge ( C , m ) , len Gauge ( C , m ) , 1 , width Gauge ( C , m ) -' 1 ) is non empty ;
Ain { ( S . i ) `1 where i is Element of NAT : i in dom ( S . i ) } ;
( T * b1 ) . y = L * ( b2 * b1 ) .= ( F * b1 ) . y .= ( F * b2 ) . y .= ( F * b2 ) . y ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x . y .| ;
( ( log ( 2 , k ) ) / ( 2 |^ ( k + 1 ) ) ) ^2 >= ( ( log ( 2 , k ) ) / ( 2 |^ ( k + 1 ) ) ) ^2 ;
then that p => q in S and not x in the carrier of ( the carrier of S ) and not x in S and not x in S and p => q in S ;
dom ( the initial of r-10 ) misses dom ( the initial of r-10 ) & dom ( the initial of r-10 ) misses dom ( the multF of r-10 ) ;
synonym f is extended integer means : Def1 : for for x being set st x in rng f holds x is integer ;
assume for a being Element of D holds f . { a } = a & f . { a } = f . a & f . { a } = f . a ;
i = len p1 .= len p3 + len <* x *> .= len p3 + len <* x *> .= len p3 + len <* x *> .= len p3 + 1 .= len p3 + 1 ;
( l , 3 ) = ( g . ( k + 1 ) ) * ( k + 1 ) + ( g . ( k + 1 ) ) * ( k + 1 ) ;
CurInstr ( P2 , Comput ( P2 , s2 , l ) ) = halt SCM+FSA .= halt SCM+FSA .= halt SCM+FSA .= halt SCM+FSA .= IC SCM+FSA .= IC SCM+FSA .= IC SCM+FSA .= 0 ;
assume for n be Nat holds ||. ( seq . n ) - ( seq . n ) .|| <= ( seq . n ) - ( seq . n ) ;
sin ( \HM { the } \HM { function } \HM { cos } ) = sin r * ( sin ( \HM { the } \HM { function } \HM { cos } ) ) .= 0 ;
set q = |[ g1 `1 / ( |. g2 .| - sn ) , g2 `2 / ( |. g2 .| - sn ) ]| ;
consider G being sequence of S such that for n being Element of NAT holds G . n in GJ . n ;
consider G such that F = G and ex G1 , G2 st G1 in Sand G2 in Sand G = ( the Sorts of G1 ) . ( len G ) ;
the root of ( ( the Sorts of C ) * the Arity of C ) . ( ( the Sorts of C ) . ( ( the Arity of C ) . ( o . s ) ) ) = ( the Sorts of ( the Sorts of C ) . s ) . ( ( the Sorts of C ) . ( ( the Sorts of C ) . s ) . ( ( the Arity of C ) . s ) ;
Z c= dom ( ( exp_R * ( ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( exp_R * ( 1 / 2
for k being Element of NAT holds rD . k = ( ( Im f ) . k ) * ( ( Im f ) . k ) * ( ( Im f ) . k ) )
assume that - 1 < sn and sn < 1 and q `1 / |. q .| >= sn and |. q .| >= 0 and |. q .| = 1 and |. q .| = 1 and |. q .| = 1 and |. q .| = 1 and |. q .| = 1 ;
assume that f is continuous and a < b and f . a = g and f . b = c and f . a = d and f . b = c and f . c = d and f . d = c and f . c = d and f . d = d and g . d = c and g . d = d and g . c = d and g . d = c ;
consider r being Element of NAT such that s1 = Comput ( P1 , s1 , r ) and r <= q and r <= 1 and q <= 1 and r <= 1 ;
LE f /. ( i + 1 ) , L~ f , L~ f , L~ f , L~ f , L~ f , L~ f ;
assume that x in the carrier of K and y in the carrier of K and inf { x , y } in the carrier of K and inf { x , y } in the carrier of K and inf { x , y } in the carrier of K and inf { x , y } in the carrier of K ;
assume f /^ i1 , i2 , j2 ) in ( ( proj ( F , i2 ) ) * ( ( proj ( F , i2 ) ) * ( ( proj ( F , i2 ) ) * ( ( proj ( F , i2 ) ) * ( ( the Sorts of A ) * ( ( the Sorts of A ) * ( the Sorts of A ) ) ) ) ) . ( i2 + 1 ) ;
rng ( ( Flow M ) ~ ) c= ( the carrier of M ) & ( ( Flow M ) ~ ) ~ c= ( the carrier of M ) ~ ;
assume z in { ( the carrier of G ) \times { t } where t is Element of T : t in { t } } ;
consider l be Nat such that for m be Nat st l <= m holds ||. ( s1 . m ) - ( lim s1 ) .|| < g ;
consider t be VECTOR of product G such that mt = ||. Dt .|| and ||. t .|| <= 1 and ||. t .|| <= 1 and ||. t .|| <= 1 ;
assume that the \square degree v = 2 and v ^ <* 0 *> in dom p and v ^ <* 1 *> in dom p and v ^ <* 1 *> in dom p and v ^ <* 1 *> in dom p and v ^ <* 1 *> in dom p ;
consider a being Element of the points of X1 , A such that a on the Points of X1 and not a on A and not a on A and not a on A and not a on A and not a on A ;
( - x ) |^ ( k + 1 ) * ( ( - x ) |^ ( k + 1 ) ) = 1 / ( ( - x ) |^ ( k + 1 ) ) ;
for D being set st for i st i in dom p holds p . i in D holds p . i in D & p . i in D
defpred R [ element ] means ex x , y st [ x , y ] = $1 & P [ x , y ] & P [ y , x ] ;
L~ f2 = union { LSeg ( |[ 0 , 0 ]| , |[ 0 ]| , |[ 1 ]| , |[ 1 , 1 ]| ) \/ { |[ 1 , 1 ]| } ;
i -' len ( h2 -' 1 ) + 2 - 1 < i -' len h11 + 2 -' 1 + 1 ;
for n be Element of NAT st n in dom F holds F . n = |. ( n -' 1 ) .|
for r , s1 , s2 , s3 st r in [. s1 , s2 .] & s1 <= s2 & s2 <= s3 holds s1 <= s2 & s2 <= s3
assume v in { G where G is Subset of T2 : G in ( B ) & G c= ( A \/ B ) & G c= ( A \/ B ) & G c= ( A \/ B ) & G c= ( A \/ B ) ;
let g be Element of A , X be Element of Z , Y be Element of Z holds ( ( A , B ) --> ( A , C ) ) . ( b , c ) <> 0 & ( A , B ) --> ( A , C ) <> 0 ;
min ( g . [ x , y ] , k ) . [ y , z ] = ( min ( g . [ y , z ] , k ) ) . [ y , z ] ;
consider q1 be sequence of CCarrier ( C ) such that for n holds P [ n , q1 . n ] & P [ n , q1 . ( n + 1 ) ] ;
consider f being Function such that dom f = NAT and for n being Element of NAT holds f . n = F ( n ) and f . ( n + 1 ) = F ( n ) ;
reconsider B-6 = B /\ C , Z = O /\ Z , O = ( B \/ C ) \ Z as Subset of B ;
consider j be Element of NAT such that x = ( the multF of n ) . j and 1 <= j and j <= n and n <= n and 1 <= j and j <= n and n <= n and n <= len f and 1 <= j and j <= n ;
consider x such that z = x and card ( x . O2 ) in card ( ( x . O2 ) . O ) and x in L1 and x in L2 . ( ( x . O1 ) . O ) and x in L1 . ( ( x . O ) . O ) ;
( C * \mathop { \rm ST4 } ( k , n2 ) ) . 0 = C . ( ( ( ( ( ( ( ( ( k + 1 ) , n2 ) ) * ( k + 1 ) ) * ( ( k + 1 ) + 1 ) ) * ( ( k + 1 ) + 1 ) ) ) . 0 ) ;
dom ( X --> rng f ) = X & dom ( X --> f ) = X & dom ( X --> f ) = dom ( X --> f ) ;
( not ( ex C st C in L~ Cage ( C , n ) ) & ( for i st i in dom Cage ( C , n ) holds ( ( Cage ( C , n ) ) /. i ) `1 <= ( ( Cage ( C , n ) ) /. i ) `1 ) `1 ) `1
synonym x , y means : Def1 : for x , y being Element of S st { x , y } = y or { x , y } c= l ;
consider X be element such that X in dom ( f | ( n + 1 ) ) and ( f | ( n + 1 ) ) . X = Y ;
assume that not ( for x , y being Element of L st x = y & y = a holds ex a , b being Element of L st a = x & b = y & a << b & a << b & b << a & a << b ;
( ( 1 / 2 ) (#) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - 1 ) ) ) ) ) ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 1 / 1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) `| REAL ) ) ) ;
defpred P [ Element of omega ] means ( ( the partial of A1 ) * ( the Sorts of A2 ) ) . ( $1 + 1 ) = A1 . ( $1 + 1 ) & ( the Sorts of A1 ) . ( $1 + 1 ) = ( the Sorts of A2 ) . ( $1 + 1 ) ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 1 ) .= IC Comput ( P , s , 1 ) .= 6 + 1 .= 6 + 1 .= 6 + 1 .= 6 + 1 ;
f . x = f . ( g1 . ( g1 . x ) ) * f . ( g2 . x ) .= ( ( g1 . x ) * ( g2 . x ) ) * ( g2 . x ) .= ( ( g1 . x ) * ( g2 . x ) ) * ( g2 . x ) .= ( f . x ) * ( f . x ) ;
( M * ( ( F . n ) ) ) . n = M . ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . n ) .= M . ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
the carrier of L1 + L2 c= ( ( the carrier of L1 ) \/ the carrier of L2 ) \/ ( the carrier of L2 ) & the carrier of L1 + L2 c= ( the carrier of L1 ) \/ ( the carrier of L2 ) ;
pred a , b , c , x , y , z is_collinear means : Def1 : for x , y , z being Element of o holds x , y , z , x is_collinear & x , z , y is_collinear & y , z , z is_collinear & x , z , z is_collinear & x , y , z , z is_collinear ;
( the Sorts of s ) . n <= ( the Sorts of A ) . ( n + 1 ) * ( the Sorts of A ) . ( n + 1 ) ;
pred - 1 <= r & r <= 1 implies ( ( ( 1 - r ) (#) ( ( 1 - r ) * ( 1 - r ) ) ) `| Z ) = - 1 / ( ( 1 - r ) * ( 1 - r ) )
s8 in { p ^ <* n *> where p is Nat : p ^ <* n *> in T1 & p ^ <* n *> in T1 } ;
|[ x1 , x2 , x3 ]| . 2 - |[ x1 , x2 ]| . 3 - |[ x1 , x2 ]| . 3 - x1 . 3 + x2 . 3 - x1 . 3 - x1 . 3 - x1 . 3 = x1 . 3 - x2 . 3 - x1 . 3 ;
attr for m being Nat holds F . m is nonnegative & ( Partial_Sums F ) . m is nonnegative implies ( Partial_Sums F ) . m is nonnegative & ( Partial_Sums F ) . m is nonnegative ;
len ( ( the multF of G ) . z ) = len ( ( the multF of G ) . ( y , z ) ) .= len ( ( the multF of G ) . ( y , z ) ) .= len ( ( the multF of G ) . ( y , z ) ) ;
consider u , v being VECTOR of V such that x = u + v and u in W1 and v in W2 and u in W2 and v in W2 and u in W2 and v in W2 and u in W2 and v in W2 and u in W2 ;
given F be FinSequence of NAT such that F = x and dom F = n and rng F c= { 0 , 1 } and for k st k in dom F holds Sum ( F ) = k and Sum ( F ) = k and Sum ( F ) = k ;
0 = ( 1 * a2 ) * u2 ^2 iff 1 = ( ( 1 - a2 ) * ( ( 1 - a3 ) * ( ( 1 - a2 ) * ( ( 1 - a3 ) * ( 1 - a3 ) ) ) ) ^2 ;
consider n be Nat such that for m be Nat st n <= m holds |. ( f # x ) . m - lim ( f # x ) .| < e ;
cluster non empty for \mathclose { \rm c } } is Boolean non empty for RelStr of ( { 0 , 1 } ) , ( { 0 , 1 } ) , ( { 1 , 2 } ) , ( { 1 , 2 } ) ) , ( { 2 , 3 } ) ) , ( { 1 , 3 } ) , ( { 1 , 2 } ) ) is Boolean
"/\" ( B , L ) = Bottom ( B ) .= ( the carrier of S ) "\/" ( the carrier of S ) .= "/\" ( ( the carrier of S ) \/ the carrier of S ) .= "/\" ( ( the carrier of S ) \/ the carrier of S ) .= "/\" ( ( the carrier of S ) \/ the carrier of S ) .= "/\" ( ( the carrier of S ) \/ the carrier of S ) ;
sqrt ( ( r / 2 ) ^2 + ( r / 2 ) ^2 ) <= sqrt ( ( r / 2 ) ^2 + ( r / 2 ) ^2 ) + sqrt ( ( r / 2 ) ^2 + ( r / 2 ) ^2 ) ;
for x being element st x in A /\ dom ( f `| X ) holds ( f `| X ) . x >= r2
2 * r1 - ( 2 * |[ a , c ]| - |[ a , c ]| ) = 0. TOP-REAL 2 & 2 * r1 - ( 2 * |[ a , c ]| - |[ a , c ]| ) = 0. TOP-REAL 2 ;
reconsider p = P /. ( \square , 1 ) , q = a " * ( ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( q `2 / q `1 ) ) ) ) / ( 1 + 1 ) ) ) ) ) ) ) ) as FinSequence of K ;
consider x1 , x2 being element such that x1 in uparrow s and x2 in uparrow t and x = [ x1 , x2 ] and x = [ x1 , x2 ] and y = [ x1 , x2 ] ;
for n be Nat st 1 <= n & n <= len q1 holds q1 . n = ( ( lower ( g , M1 ) ) . n ) * ( ( the Sorts of L~ g ) . n )
consider y , z being element such that y in the carrier of A and z in the carrier of A and i = [ y , z ] and i = [ y , z ] ;
given H1 , H2 being strict Subgroup of G such that x = H1 and y = H2 and H1 = H2 and H2 = H2 and H1 = H2 and H2 = H1 and H1 = H2 and H2 = H2 and H1 = H2 and H2 = H2 and H1 = H2 ;
for S , T , d being non empty RelStr , d being Function of T , S st T is complete holds d is monotone & d is monotone & d is monotone & d is monotone
[ a + i , b ] in ( the carrier of ( ( the carrier of V ) --> ( a , b ) ) ) \times ( the carrier of V ) ;
reconsider m5 = max ( len F1 , len ( p . n ) * ( <* x *> ^ <* x *> ) ) as Element of NAT ;
I <= width GoB f & J <= width GoB f & for i , j st 1 <= i & i < j & j <= width GoB f & f /. i = ( GoB f ) * ( i , j ) holds ( GoB f ) * ( i , j ) = ( GoB f ) * ( i , j )
f2 /* q = ( f2 /* ( f1 /* s ) ) ^\ k .= ( ( f2 * f1 ) /* s ) . k .= ( ( f2 * f1 ) /* s ) . k .= ( ( f2 * f1 ) /* s ) . k .= ( ( f2 * f1 ) /* s ) . k ;
attr A1 \/ A2 is linearly-independent means : Def1 : for A , B being Subset of V st A misses B & B misses A & A misses B holds Lin ( ( A1 \/ A2 ) \/ ( ( A1 \/ A2 ) \ ( A \/ B ) ) = Lin ( A1 \/ A2 ) & Lin ( A \/ B ) = Lin ( A ) + Lin ( B )
func A -Ccarrier C -> set means : Def1 : union { A where A is Element of R : A in it & A in C & it = { A where A is Element of R : A in C & A in C } ;
dom ( Line ( Line ( v , i + 1 ) , m ) ^ ( ( Line ( v , m ) ) @ ) ) = dom ( F ^ <* 0. F_Complex *> ) .= dom ( F ^ <* 0. F_Complex *> ) ;
cluster [ x , ( x - y ) , x - y ] -> real & [ x , ( x - y ) ] in the InternalRel of G & [ x , ( x - y ) ] in the InternalRel of G ;
E , ( All ( x2 , x1 ) ) |= All ( x2 , x2 , x1 ) '&' All ( x2 , x2 , x1 ) '&' ( ( x1 , x2 ) '&' ( x1 , x2 ) ) '&' ( ( x1 , x2 ) '&' ( x1 , x2 ) ) '&' ( ( x1 , x2 ) '&' ( x1 , x2 ) ) '&' ( ( x1 , x2 ) '&' ( x1 , x2 ) ) '&' ( ( x1 , x2 ) '&' ( x1 , x2 ) ) ) ;
F .: ( id X , g . x ) = F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( id X , g . x ) .= F . ( x , g . x ) ;
R . ( h . m ) = F . ( x0 + h . m ) - ( h . m ) + ( h . m ) + ( h . m ) + ( h . m ) ;
cell ( G , ( ( X -' 1 , Y ) + 1 ) , ( X -' 1 ) ) \ ( ( X + Y ) \ ( X + 1 ) ) meets ( ( ( X + Y ) \ ( X + 1 ) ) \ ( ( X + Y ) \ ( X + Y ) ) ;
IC Comput ( P2 , s2 , LifeSpan ( P2 , s2 ) ) = IC Comput ( P2 , s2 , LifeSpan ( P2 , s2 ) ) .= ( card I + card J + 2 ) .= card I + card J + 2 .= card I + card J + 2 .= card I + card J + 2 .= card J + card J + 2 ;
sqrt ( ( - ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 ) ) > 0 & sqrt ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 ) > 0 ;
consider x0 being element such that x0 in dom a and x0 in dom a and x0 in dom g and y = g . ( k + 1 ) and x0 = a . ( k + 1 ) and x0 = a . ( k + 1 ) ;
dom ( ( r1 (#) ( A . m ) ) (#) ( A . m ) ) = dom ( ( A . m ) (#) ( A . m ) ) .= dom ( ( A . m ) (#) ( A . m ) ) .= dom ( ( A . m ) (#) ( A . m ) ) .= dom ( ( A . m ) (#) ( A . m ) ) .= dom ( ( A . m ) (#) ( A . m ) ) .= dom ( ( A . m ) (#) ( A . m ) ;
dZ . [ y , z ] = ( ( ( y - z ) / ( y - z ) ) / ( y - z ) ) / ( y - z ) ;
pred for i being Nat holds C . i = A . i /\ B . i & C . i c= ( C . i ) /\ ( C . i ) ;
assume that x0 in dom f and f . x0 in { x0 } and f . x0 in dom f and for x st x in dom f holds ||. f . x - f . x0 .|| <= |. ( f . x - f . x0 ) .| ;
p in Cl A implies for K being Basis of p , Q being Subset of T st K in K holds A meets Q & A meets Q & A meets Q & A meets Q & A meets Q
for x be Element of REAL n st x in Line ( x1 , x2 ) holds |. y1 - y2 .| <= |. y1 - y2 .|
func <* a *> -> Ordinal means : Def1 : a in it & a in it iff a in it & a in it & a in it & a in it & a in it & a in it & a in it & a in it & b in it & a in it & b in it ;
[ a1 , a2 ] in ( ( the carrier of A ) \/ ( the carrier of B ) ) & [ a1 , a2 ] in ( the carrier of A ) & [ a1 , a2 ] in the carrier of A ;
ex a , b being element st a in the carrier of S1 & b in the carrier of S2 & x = [ a , b ] & [ a , b ] in the InternalRel of S2 & [ a , b ] in the InternalRel of S2 & [ a , b ] in the InternalRel of S2 ;
||. ( vseq . n - vseq . m ) .|| * ||. vseq . m - vseq . m .|| < ( e / 2 ) * ||. x .|| ;
then for Z being set st Z in { Y where Y is Element of I7 : F ( ) c= Z & Z in F ( ) } holds z in Z ;
sup ( compactbelow [ s , t ] ) = [ sup ( ( the carrier of S ) --> { [ s , t ] } , ( the carrier of S ) --> { [ s , t ] } ) ] ;
consider i , j being Element of NAT such that i < j and [ y , f . i ] in [: I , J :] and [ f . i , f . j ] in [: I , J :] and [ f . i , f . j ] in [: I , J :] ;
for D being non empty set , p , q being FinSequence of D st p c= q for n being Nat st n <= len p holds p ^ q = q ^ p
consider e being Element of the carrier of X such that c9 , a9 // a9 and a , a9 // a9 and not a , a9 // a9 and not a , a9 // a9 , b9 and not a , a9 // a9 , b9 and a , a9 // a9 , b9 and a , a9 // a9 , b9 and a , a9 // a9 , b9 ;
set U = I \! \mathop { + } , C = I \! \mathop { + } ;
|. q2 .| = ( ( ( |. q2 .| ) ) ^2 + ( ( |. q2 .| ) ^2 ) + ( ( |. q2 .| ) ^2 ) .= |. q2 .| ) ^2 + ( ( |. q2 .| ) ^2 + ( ( |. q2 .| ) ^2 ) .= |. q2 .| ;
for T being non empty TopSpace , x , y being Element of [: the topology of T , the topology of T :] holds x "\/" y = x "\/" y & x "/\" y = x & x "/\" y = y & x "/\" y = x "\/" y
dom ( ( ( the charact of U1 ) * ( the charact of U2 ) ) * ( the charact of U2 ) ) = dom ( ( the charact of U2 ) * ( the charact of U2 ) ) & dom ( ( the charact of U2 ) * ( the charact of U2 ) ) = dom ( the charact of U2 ) ;
dom ( h | X ) = dom h /\ X .= dom ( ( |. h .| ) | X ) .= dom ( ( |. h .| ) | X ) .= dom ( ( |. h .| ) | X ) .= dom ( ( |. h .| ) | X ) .= dom ( ( |. h .| ) | X ) .= dom ( ( |. h .| ) | X ) .= dom ( ( |. h .| ) | X ) ;
for N1 , N1 , N2 being Element of [: the carrier of [: the carrier of G , the carrier of G :] holds rng ( h . N1 ) = [: the carrier of G , the carrier of G :]
( mod ( u , m ) + mod ( v , m ) ) . i = ( mod ( u , m ) ) . i + ( mod ( v , m ) ) . i ;
- ( q `1 / |. q .| - cn ) < - ( q `1 / |. q .| - cn ) or q `1 / |. q .| >= cn & q `2 / |. q .| - cn & q `2 / |. q .| - cn ) >= 0 ;
pred r1 = ( f . 0 ) * ( f . 1 ) & r2 = ( f . 1 ) * ( f . 2 ) & r1 = ( f . 2 ) * ( f . 3 ) ;
xseq . m is bounded Function of X , the carrier of Y & vseq . m = ( vseq . m ) * ( vseq . m ) & vseq . n = ( vseq . m ) * ( vseq . n ) ;
pred a <> b & b <> c & angle ( a , b , c ) = PI & angle ( a , b , c ) = PI & angle ( a , b , c ) = PI & angle ( a , b , c ) = PI & angle ( a , b , c ) = PI & angle ( a , b , c ) = PI ;
consider i , j being Nat , r being Real such that p1 = [ i , r ] and p2 = [ i , s ] and r < 1 and r < 1 and r < 1 and s < 1 and r < 1 and r < 1 and r < 1 ;
|. p .| ^2 - ( 2 * |. p .| ) ^2 + ( 2 * |. p .| ) ^2 = |. p .| ^2 + ( 2 * |. p .| ) ^2 ;
consider p1 , q1 being Element of [: [: X , Y :] such that y = p1 ^ q1 and q1 = ( p ^ q ) ^ <* q1 *> and p1 ^ q1 = ( p ^ q ) ^ <* q1 *> ;
L1 . ( r1 , r2 ) = ( ( 1 - r ) * ( ( 1 - r ) * ( ( 1 - r ) * ( ( 1 - r ) * ( ( 1 - r ) * ( ( 1 - r ) * ( ( 1 - r ) * ( ( 1 - r ) * ( 1 - r ) ) ) ) ) / 2 ) ;
( ( UMP A ) . ( w , ( proj2 A ) . ( w , ( proj2 A ) . ( w , ( ( proj2 A ) . ( w , ( w , ( w , ( w , ( w , ( w , ( w , ( w , ( w , ( w , ( w , A ) ) ) ) ) ) ) ) ) ) ) ) is non empty & ( ( ( ( w , ( w , ( w , ( w , ( w , ( w , ( w , ( w , ( w , ( w , ( w , ( w , ( w , ( w ,
s , ( ( H . k ) |= H1 ) '&' ( ( H . k ) '&' ( H . ( k + 1 ) ) ) iff s |= All ( H1 , H2 ) & s |= All ( H2 , H1 )
len ( ( s + t ) / 2 ) = card ( support ( b1 + b2 ) ) + 1 .= card ( support ( b1 + b2 ) ) .= card ( support ( b1 + b2 ) ) + card ( support ( b1 + b2 ) ) .= card ( support ( b1 + b2 ) ) .= card ( support ( b1 + b2 ) ) .= card ( support ( b1 ) ) ;
consider z being Element of L1 such that z >= x and z >= y and for z being Element of L1 st z >= y holds z >= y & z >= x ;
LSeg ( UMP D , |[ ( ( ( ( ( ( ( ( ( ( ( ( D ) ) ) ) ) ) / 2 ) / 2 , ( ( ( ( D ) ) / 2 ) ) / 2 ) ) / 2 , ( ( ( ( D ) / 2 ) / 2 ) ) / 2 ]| ) /\ D = { ( ( ( D ) / 2 ) / 2 ) / 2 , ( ( D ) / 2 ) / 2 } } ;
lim ( ( ( f `| N ) `| N ) / ( f `| N ) /* b ) = lim ( ( f `| N ) / ( f `| N ) ) / ( f `| N ) ) ;
P [ i , pr1 ( f , i ) . ( f . i , pr1 ( f , i ) . ( f . i ) ] ;
for r be Real st 0 < r ex m be Nat st for k be Nat st m <= k holds ||. ( seq . k ) - ( seq . k ) .|| < r
for X being set , P , Q being a_partition of X , a , b being set st x in P & a in P & b in P & a in Q & b in Q & a in Q & b in Q & a in Q holds a = b
Z c= dom ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 1 / 2 ) ) ) ) ) ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( A A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) implies Z = dom ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
ex j being Nat st j in dom ( l ^ <* x *> ) & j < i & i = ( l ^ <* x *> ) . j & ( i + 1 ) + 1 = ( l ^ <* x *> ) . j & ( i + 1 ) + 1 = ( l + 1 ) . j ;
for u , v being VECTOR of V , r being Real st 0 < r & r < 1 holds r * u + ( r * v ) in _|_ N
for A , B , C , D , E , F , G , G being Function of A , B holds Cl ( A \/ B ) = Cl ( A \/ B ) & Cl ( A \/ B ) = Cl ( A \/ B ) & Cl ( A \/ B ) = Cl ( A \/ B ) & Cl ( A \/ B ) = Cl ( B \/ C )
- Sum <* v , u , w *> = - ( v + u ) + ( v + u ) .= - ( v + u ) + ( u + w ) .= - ( v + u ) + ( v + w ) .= ( v + u ) + ( w + w ) .= ( v + u ) + ( w + w ) .= ( v + u ) + ( w + w ) ;
( Exec ( a := b , s ) ) . IC SCM R = ( Exec ( a := b , s ) ) . IC SCM R .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s ;
consider h being Function such that f . a = h and dom h = I and for x being element st x in I holds h . x in ( the carrier of J ) . x and h . x = ( the carrier of J ) . x ;
for S1 , S2 being non empty reflexive RelStr for D being non empty Subset of S1 , S1 , S2 being non empty Subset of S2 holds S1 = S2 & S2 = S1 & S1 is directed & S1 is directed & S2 is directed & S1 is directed & S1 is directed & S2 is directed & S1 is directed
card X = 2 implies ex x , y st x in X & y in X & not x in X & not y in X & not x in X or y in Y & not x in X or y in Y & not x in X & y in Y & not x in Y ;
E-max L~ Cage ( C , n ) in rng ( Cage ( C , n ) ) & E-max L~ Cage ( C , n ) in rng ( Cage ( C , n ) ) ;
for T , T , p , q being FinSequence holds ( T , p ) -with T , q , r , s , t being Element of dom T st p in dom T & q in dom T holds ( T , q ) -tree ( p , t ) = T . q
[ i2 + 1 , j2 ] in Indices G & [ i2 + 1 , j2 ] in Indices G & f /. k = G * ( i2 , j2 ) & f /. k = G * ( i2 + 1 , j2 ) ;
cluster -> not of lcm ( k , n ) -> not zero & k divides ( k + 1 ) & ( k divides ( k + 1 ) implies ( k divides n ) & ( k divides ( k + 1 ) ) & ( k divides ( k + 1 ) ) & ( k divides ( k + 1 ) ) & ( k divides ( k + 1 ) ) & ( k divides ( k + 1 ) ) implies k divides ( k + 1 )
dom F " = the carrier of X1 & rng F = the carrier of X2 & rng F = the carrier of X2 & F " { 0 } = the carrier of X1 & F " { 0 } = the carrier of X2 & F " { 0 } = the carrier of X1 & F " { 0 } = the carrier of X2 & F " { 0 } = the carrier of X2 ;
consider C being finite Subset of V such that C c= A and card C = n and the carrier of V = Lin ( B ) and C = Lin ( C ) and C = Lin ( B ) and C = Lin ( C ) and C = Lin ( B ) and C = Lin ( B ) ;
V is prime implies for X , Y being Subset of [: the topology of T , the topology of T :] st X /\ Y c= V & X c= Y holds X c= Y or Y c= V or X c= Y
set X = { F ( v1 ) where v1 is Element of B ( ) , v2 is Element of C ( ) : P [ v1 ] & P [ v1 ] } , Y = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } ;
angle ( p1 , p3 , p4 ) = 0 .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p3 , p4 ) .= angle ( p2 , p4 ) .= angle ( p2 , p4 ) ;
- sqrt ( ( - ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 ) = - ( q `2 / |. q .| - sn ) ) / ( 1 + sn ) .= - ( q `2 / |. q .| - sn ) / ( 1 + sn ) .= ( q `2 / |. q .| - sn ) / ( 1 + sn ) ;
ex f being Function of I[01] , TOP-REAL 2 st f is continuous one-to-one & rng f = P & f . 0 = p1 & f . 1 = p2 & f . 1 = p2 & f . 1 = p4 & f . 0 = p2 & f . 1 = p4 & f . 1 = p2 & f . 1 = p2 ;
pred f is_partial differentiable on GoB f means : Def1 : for u , v , u1 , v1 st u in dom f & v in dom f & u in dom f holds SVF1 ( f , u , 3 ) . ( u + 1 ) - diff ( f , u ) . ( v + 1 ) = ( proj ( 2 , 3 ) ) . ( u + 1 ) - diff ( f , u ) . ( v + 1 ) ;
ex r , s st x = |[ r , s ]| & G * ( len G , 1 ) `1 < r & r < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < s & s < G * ( 1 , 1 ) `2 ;
assume that f is special and 1 <= len G and 1 <= width G and G * ( t , width G ) `2 >= ( G * ( t , width G ) `2 ) `2 and G * ( t , width G ) `2 >= ( G * ( t , width G ) `2 and ( G * ( t , width G ) `2 >= ( G * ( t , width G ) `2 ) `2 ;
pred i in dom G means : Def1 : r * ( f * reproj ( i , x ) ) = r * ( reproj ( i , x ) ) . ( f . ( x - x0 ) ) ;
consider c1 , c2 being bag of o1 such that ( <* c1 , c2 *> ) /. k = <* c1 , c2 *> and ( <* c1 , c2 *> ) /. k = <* c2 , c2 *> and ( <* c1 , c2 *> /. k = <* c2 , c2 *> . k and c2 = <* c2 , c2 *> . k ;
u0 in { |[ r1 , s1 ]| : r1 < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < s1 & s1 < G * ( 1 , 1 ) `2 } ;
Cl ( X ^ Y ) . k = the carrier of X . ( k2 + 1 ) .= ( C . ( k2 + 1 ) ) .= C . ( ( C . ( k2 + 1 ) ) . k ) .= C . ( ( C . ( k2 + 1 ) ) . k ) .= C . ( ( C . ( k2 + 1 ) ) . k ) .= C . ( ( C . ( k2 + 1 ) ) . k ) .= C . ( ( C . ( k2 + 1 ) ) . ( ( C . ( k2 + 1 ) ) .= C . ( ( C . ( k2 + 1 ) ) . ( ( C
pred len M1 = len M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & M1 = M2 - M2 - M3 = M3 - M3 - M3 M1 ;
consider g2 be Real such that 0 < g2 and for y being Point of S st y in { y where y is Point of S : ||. y - x0 .|| < g2 & ||. y - x0 .|| < g2 & g2 . y - x0 < g2 . y } c= N2 and g2 . y in N2 ;
assume x < ( - b + sqrt ( a , b ) ) / 2 or x > - ( - b ) / 2 ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ ( G1 ^ G2 ) ) . i & ( G1 '&' G2 ) . i = ( <* 3 *> ^ ( G1 ^ G2 ) ) . i & ( G1 ^ G2 ) . i = ( <* 3 *> ^ ( G1 ^ G2 ) ) . i ;
for i , j st [ i , j ] in Indices ( M3 + M1 ) holds ( M2 + M3 ) * ( i , j ) < M2 * ( i , j )
for f being FinSequence of NAT , i being Element of NAT , f being Element of NAT st f in dom f & i <= len f holds f divides f /. i & f /. ( i + 1 ) = Sum ( f | i )
assume F = { [ a , b ] where a , b is set : a in BX & b in BX & a in BX & b in BX & a <> b & a in BX & b in BX & a in B & b in B & a in B & b in B & a in B & b in B & a in B & b in B & a in B & b in B ;
b2 * q2 + ( b3 * q2 ) * q2 + ( b3 * q2 ) * q2 + ( b3 * q2 ) * q2 = 0. TOP-REAL n + ( - ( a1 * q2 ) * q2 ) * q2 .= ( - ( a1 * q2 ) ) * q2 + ( - ( p1 * q2 ) ) * q2 ;
Cl ( F ) = { D where D is Subset of T : ex B being Subset of T st D = Cl ( B ) & B in F & A in B & B in F & B in F & A = Cl ( B ) } ;
attr seq is summable means : Def1 : seq is summable & seq is summable & seq is summable & lim seq = lim seq & seq is summable & lim seq = lim seq & lim seq = lim seq & lim seq = lim seq & lim seq = lim seq & lim seq = lim seq ;
dom ( ( ( cn -FanMorphE ) | D ) ) = ( ( ( ( cn -FanMorphE ) | D ) ) /\ D .= ( ( ( ( cn -FanMorphE ) | D ) ) /\ D ) /\ D .= ( ( ( ( cn -FanMorphE ) | D ) ) /\ D ) /\ D .= ( ( ( cn -FanMorphE ) | D ) ) /\ D .= ( ( ( cn 2 ) | D ) /\ D ) /\ D .= ( ( ( cn 2 ) | D ) /\ D ) /\ D .= ( ( ( ) ) /\ D ) /\ D .= ( ( ( ( ( cn 2 ) ) /\ D ) /\ D ) /\ D ) /\ D ) /\ ( ( ( ( ( ( ( ( ( (
[ X \to Z ] is full full SubRelStr of ( ( Omega Z ) |^ the carrier of S ) |^ the carrier of S & [ X , Z ] is full SubRelStr of ( ( Omega Z ) |^ the carrier of S ) |^ the carrier of S ;
( G * ( 1 , j ) ) `2 = ( G * ( 1 , j ) ) `2 & ( G * ( 1 , j ) ) `2 <= ( G * ( 1 , j ) ) `2 ;
synonym m1 c= m2 means for for for for for p , q being set st p in P & q in P holds the non empty set of U = ( the non empty U ) \ ( the carrier of U ) & ( the carrier of U ) \ ( the carrier of U ) c= the carrier of U & ( the carrier of U ) \ ( the carrier of U ) c= the carrier of U ;
consider a being Element of B ( ) such that x = F ( ) and a in { G ( ) where b is Element of B ( ) : P [ b ] } and a in A ( ) and P [ a ] ;
synonym the multiplicative loop loop structure means : where s being multiplicative non empty multMagma over R means : Def1 : for a being Element of R holds it . a = [ the carrier of it , the carrier of it ] & it . a = [ the carrier of it ] ;
\HM { a , b , c , d + 1 } + \mathop { \rm _ _ { c , d } } = b + \mathop { \rm _ _ { a , b } , \mathop { \rm _ { d } } , \mathop { \rm _ { c , d } } .= \mathop { \rm _ _ { a , b } , \mathop { \rm _ { d } } } ;
cluster strict non empty for Subset of INT , i , i2 , j2 being Element of INT holds ( i , i2 ) = ( i , i1 ) --> ( i2 , j2 ) & ( i , i1 ) --> ( i2 , j2 ) = ( i , i1 ) --> ( i2 , j2 )
( - ( 2 * p1 ) + ( 2 * p2 ) ) * ( p1 + p2 ) = ( - ( 2 * p1 ) + ( 2 * p2 ) ) * ( p1 + p2 ) ;
eval ( ( a | ( n , L ) ) *' p , x ) = eval ( a | ( n , L ) ) * eval ( p , x ) .= a * eval ( p , x ) .= a * eval ( p , x ) .= a * eval ( p , x ) .= a * eval ( p , x ) ;
assume that the TopStruct of S = the TopStruct of T and for D being non empty Subset of S st D = the carrier of S holds sup D in ( the carrier of S ) and for V being Subset of S st V in V holds V in V holds V meets ( the carrier of T ) and V meets ( the carrier of S ) ;
assume that 1 <= k and k <= len w + 1 and TU . ( k + 1 ) = ( TU . k ) . ( k + 1 ) and TU . ( k + 1 ) = ( TU . k ) . ( k + 1 ) and U . ( k + 1 ) = ( TU . k ) . ( k + 1 ) ;
2 * a |^ ( n + 1 ) + ( 2 * a ) * ( a |^ ( n + 1 ) ) >= ( a |^ ( n + 1 ) + a |^ ( n + 1 ) ;
M , v / ( x. 3 , x. 0 ) / ( x. 0 , x. 4 ) / ( x. 0 , x. 0 ) / ( x. 4 , x. 0 ) / ( x. 0 , x. 4 ) / ( x. 0 , x. 4 ) / ( x. 0 , x. 4 ) / ( x. 0 , x. 4 ) / ( x. 0 , x. 4 ) ) / ( x. 0 , x. 4 ) / ( x. 4 , x. 0 ) ) |= M ;
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f . x0 or for x0 st x0 in l holds f . x0 < f . x0 or f . x0 < 0 ;
for G1 being _Graph , W being Walk of G1 , e being Vertex of G2 , e being set st e in W holds e in W iff e in W & W is Walk & e in W
not c22 is not empty iff not ( not q1 is not empty & not q1 is not empty & q1 is not empty & not q2 is not empty & q1 is not empty & q1 is not empty & q2 is not empty & not q2 is not empty & q1 is not empty & q2 is not empty & q1 is not empty & q2 is not empty & q2 is not empty & q2 is not empty & q2 is not empty & q2 is not empty & q2 is not empty & q2 is not empty & q2 is not empty & q2 is not empty & q2 is not empty & not q2 is not empty & q1 is not empty & q2 is not empty & not q2 is not empty & not q2 is not empty & not empty & not q2 is not empty & not empty & not empty &
Indices GoB f = [: dom GoB f , Seg width GoB f :] & dom f = [: Seg ( len GoB f ) , Seg ( len GoB f ) :] & dom f = [: Seg ( len GoB f ) , Seg ( len GoB f ) :] & dom f = Seg ( len GoB f ) & rng f c= Seg ( len GoB f ) ;
for G1 , G2 , H being Group , I , J being strict Subgroup of O st G1 is stable & G2 is stable & I is stable holds ( G1 , G2 ) -N is stable & ( G1 , G2 ) -N is stable
UsedIntLoc ( ( intloc 0 ) .--> 1 ) = { intloc 0 , ( intloc 0 ) .--> 1 , ( intloc 0 ) .--> 1 , ( intloc 0 ) .--> 1 ) , ( intloc 0 ) .--> 1 , ( intloc 0 ) .--> 1 ) , 1 = ( intloc 0 ) .--> 1 , 1 = ( intloc 0 ) .--> 1 , 1 = ( intloc 0 ) .--> 1 , 2 = ( intloc 0 ) .--> 1 , 1 = ( intloc 0 ) .--> 1 , 2 = ( intloc 0 ) .--> 1 , 2 = ( intloc 0 ) .--> 1 , 2 = ( intloc 0 ) , 2 = ( intloc 0 ) .--> 1 , 2 = ( intloc 0 ) , 2 = ( intloc 0 , 2 = ( intloc 0 ) , 2 = ( intloc 0 )
for f1 , f2 being FinSequence of F st f1 ^ f2 is p -element & ( for i st i < len f1 holds Q [ i , f2 . i ] ) & Q [ i , f2 . ( i + 1 ) ] holds Q [ i , f2 . ( i + 1 ) ]
sqrt ( ( p `1 / |. p .| - cn ) / ( 1 + cn ) ) ^2 = ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ;
for x1 , x2 , x3 , x4 , x4 , x5 , x5 , x1 , x2 , x3 , x4 , x4 , x1 , x2 , x4 , x2 , x4 , x2 , x4 , x1 , x2 , x3 , x4 , x4 , x2 , x4 , x4 , x1 , x2 , x3 , x4 , x4 , x2 , x4 , x4 , x1 , x2 , x3 , x4 , x4 , x1 , x2 , x3 , x4 , x4 , x2 , x1 , x2 , x2 , x3 , x4 , x4 , x4 , x1 , x2 , x4 , x4 , x2 , x4 , x2 , x4 , x4 , x2 , x4 , x4 , x4 , x2 , x4 , x4 , x1 , x2 , x4 , x4 , x4 , x2 , x4 , x4 , x4
for x st x in Z holds ( ( ( - ( exp_R | A ) ) `| Z ) . x = - ( ( - ( exp_R | A ) ) `| Z ) . x ) / ( - ( ( - ( exp_R | A ) ) . x ) ^2 )
for T being non empty TopSpace , P , Q being Subset of T , x being Point of T st P c= the topology of T & Q c= the topology of T holds P is Basis of T & Q is Basis of T
( a 'or' b ) . x = 'not' ( ( a 'or' b ) . x ) 'or' ( b . x ) .= 'not' ( a . x ) 'or' 'not' ( b . x ) .= 'not' ( a . x ) 'or' 'not' ( b . x ) .= TRUE .= TRUE ;
for e being set st e in A8 ex X1 being Subset of Y st e = X1 & ex Y1 being Subset of Y st Y1 = Y1 & Y1 = Y1 & Y1 = Y1 & Y1 = Y2 & Y1 = Y2 & Y1 = Y2 & Y1 = Y2 & Y1 = Y2 & Y1 = Y2 & Y1 = Y2 & Y1 = Y2 & Y1 = Y2 & Y1 = Y2 & Y1 = Y2 & Y2 = Y2 & Y1 = Y2 & Y1 = Y2 & Y1 = Y2 & Y2 = Y2 & Y2 = Y2 & Y2 = Y2 & Y1 = Y2 & Y1 = { Y1 , Y2 , Y2 , Y1 = { Y1 , Y2 , Y2 , Y1 } ;
for i be set st i in the carrier of S for f being Function of [: S1 , S2 :] , S2 st f = H . i & f . i = f . i holds F ( f , i ) = f | [: S1 , S2 :]
for v , w st for y st x <> y holds w . y = v . y holds Valid ( VERUM ( Al , A ) , J ) . ( v . y ) = Valid ( VERUM ( Al , A ) , J ) . ( v . y )
card D = card ( D1 + D2 ) - card ( { i , j } ) .= card ( D1 + D2 ) - card ( { i , j } ) .= 2 * ( { i , j } + 1 ) - card ( { i , j } ) .= 2 * ( { i , j } + 1 ) - ( 2 * ( { i , j } ) ) .= 2 * ( ( { i , j } + 1 ) + 1 ) .= 2 * ( ( { i , j } + 1 ) + 1 .= 2 * ( ( { i , j } + 1 ) + 1 .= 2 * ( ( { i , j } + 1 ) + 1 ) + 1 ) + 1 .= 2 * ( ( ( {
IC Exec ( i , s ) = ( s +* ( 0 .--> 1 ) ) . IC SCM+FSA .= ( s +* ( 0 .--> 1 ) ) . IC SCM+FSA .= ( s . IC SCM+FSA ) .= ( s . IC SCM+FSA ) .= ( s . IC SCM+FSA ) . IC SCM+FSA .= ( s . IC SCM+FSA ) .= ( s . IC SCM+FSA ) .= ( s . IC SCM+FSA ) .= ( s . IC SCM+FSA ) ;
len f /. ( \downharpoonright i1 -' 1 ) + 1 + 1 = len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) ;
for a , b , c being Element of NAT st 1 <= a & a < b & b < d holds a + b <= a + b or a = b + c or a = d + b
for f being FinSequence of TOP-REAL 2 , p being Point of TOP-REAL 2 st p in LSeg ( f , p ) & p in LSeg ( f , p ) holds Index ( p , f ) <= len f & Index ( p , f ) <= len f
( curry ( P , k + 1 ) # x ) # x = lim ( ( curry ( P , k + 1 ) # x ) ) + lim ( ( curry ( P , k + 1 ) # x ) # x ) ;
z2 = g /. ( \downharpoonright n1 + 1 ) .= g /. ( i -' n1 + 1 ) .= g /. ( i -' n1 + 1 ) .= g /. ( i -' n1 + 1 ) .= g /. ( i -' n1 + 1 ) .= g /. ( i -' n1 + 1 ) .= g /. ( i -' n1 + 1 ) .= g /. ( i -' n1 + 1 ) ;
[ f . 0 , f . 3 ] in id ( the carrier of G ) or [ f . 0 , f . 3 ] in id ( the carrier of G ) or [ f . 0 , f . 3 ] in the InternalRel of G or [ f . 3 , f . 3 ] in the InternalRel of G ;
for G being Subset-Family of B st G = { R where R is Subset of A ( ) , R is Subset of A ( ) , A ( ) : R in F ( ) & R in G ( ) & R in F ( ) & for X being Subset of A ( ) st X in F ( ) holds X in G ( ) holds X in F ( )
CurInstr ( P1 , Comput ( P1 , s1 , m1 + 1 ) ) = CurInstr ( P1 , Comput ( P2 , s2 , m1 + 1 ) ) .= CurInstr ( P1 , Comput ( P2 , s2 , m1 ) ) .= CurInstr ( P1 , Comput ( P2 , s2 , m1 ) ) .= halt SCMPDS .= halt SCMPDS ;
assume that a on M and b on M and c on N and d on N and p on M and a on N and p on M and d on N and p on M and a on M and a on N and p on M and d on N and a on M and a on N and p on M and a on N and p on N and a on M and a on N and p on N and a on M and d on N and a on N and d on N and a on N and a on N and a on N and d on N and d on N and d on N and d on N and d on N and a on M and a on M and a on M and a on N and a on N and a on M and d on N and d on N and d on N and a on M and d on N and d on N and d on N and d on N
assume that T is \hbox { T _ 4 } and for F being Subset-Family of T holds F is closed iff ex F being Subset-Family of T st F is closed & F is finite-ind & ind F <= 0 & ind F <= 0 & ind F <= 0 & ind F <= 0 & ind F <= 0 & ind F <= 0 ;
for g1 , g2 st g1 in ]. r1 , r2 .[ & g2 in ]. r1 , r2 .[ holds |. ( f . g1 ) - g2 .| <= ( |. g1 . g2 .| - ( f . g2 ) ) / 2
( ( \HM { the } \HM { function } \HM { exp ( 2 , 1 ) ) * ( ( \HM { the } \HM { function } \HM { exp ( 2 , 1 ) ) * ( ( \HM { the } \HM { function } \HM { sin } ) + ( \HM { the } \HM { function } \HM { sin } ) ) * ( ( \HM { the } \HM { function } \HM { sin } ) ) ) = ( \HM { the } \HM { function } ) * ( ( \HM { the } \HM { function } ) + ( \HM { the } \HM { function } ) ) * ( ( \HM { the } \HM { function } ) + ( \HM { the } \HM { function } ) ) * ( ( \HM { the } ) ) ;
F . i = F /. i .= 0. R .= ( b |^ n ) * ( a |^ ( n + 1 ) ) .= <* ( n + 1 ) * a |^ ( n + 1 ) *> .= <* ( n + 1 ) * a |^ ( n + 1 ) *> .= <* ( n + 1 ) * a |^ ( n + 1 ) ;
ex y being set , f being Function st y = f . n & dom f = NAT & f . 0 = [: A , B :] & for n being Nat holds f . ( n + 1 ) = [: A , B :] . ( n + 1 ) & f . ( n + 1 ) = [: A , B :] . ( n + 1 ) ;
func f (#) F -> FinSequence of V means : Def1 : len it = len F & for i being Nat st i in dom it holds it . i = F . i * ( f . i ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , M } = { x1 , x2 , x3 , x4 , x4 , x5 , x5 , x5 , M } \/ { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 7 } \/ { x1 , x2 , x3 , x4 , 7 } \/ { x2 , x4 , 7 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 7 , 8 , 8 , 7 , 7 } \/ { x1 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8
for n being Nat , x being set st x = h . n holds h . ( n + 1 ) = o . ( x , n ) & h . ( n + 1 ) in InnerVertices S ( x , n ) & h . ( n + 1 ) in InnerVertices S ( x , n ) ;
ex S1 be Element of CQC-WFF ( Al , e , l ) st ( SubP ( P , l , e ) ) . ( S1 , e ) = S1 & ( for i st i in dom S1 holds ( for i st i in dom S1 holds S1 . i = ( S2 . i ) `1 ) `1 ) & ( for i st i in dom S1 holds S1 . i = ( S2 . i ) `1 ) `1 ) & ( S1 . i = ( S2 . i ) `1 ) `1 ) `1 ;
consider P being FinSequence of ( the carrier of G ) such that p9 = Product P and for i being Element of NAT st i in dom P ex t1 being Element of the carrier of G st P . i = ( the multF of G ) . i & t1 = ( the multF of G ) . i ;
for T1 , T2 being strict non empty TopSpace , T1 , T2 being Basis of T1 , T2 being Basis of T2 st the topology of T1 = the topology of T2 & the topology of T2 = the topology of T2 & the topology of T2 = the topology of T2 & the topology of T2 = the topology of T2 & the topology of T2 = the topology of T2 & the topology of T2 = the topology of T2 & the topology of T2 = the topology of T2 holds the topology of T1 = the topology of T2 & the topology of T2 = the topology of T2 & the topology of T2 = the topology of T2 & the topology of T2 & the topology of T1 & the topology of T2 = the topology of T2 & the topology of T2 & the topology of T2 = the topology of T2 & the topology of T2 = the topology of T2 = the topology of T2 &
assume that f is_partial differentiable on is_\cal GoB f and r (#) pdiff1 ( f , 3 ) is_differentiable on GoB f and ex u st r (#) pdiff1 ( f , 3 ) + r (#) pdiff1 ( f , 3 ) = r * pdiff1 ( f , 3 ) + r * pdiff1 ( f , 3 ) and ex r st r (#) pdiff1 ( f , 3 ) = r * pdiff1 ( f , 3 ) ;
defpred P [ Nat ] means for F , G being FinSequence of bool ( the carrier of V ) st len F = $1 & len G = $1 & for s being FinSequence of bool ( the carrier of V ) st s = F . ( len F ) holds not s in rng F & not s in rng G ;
ex j st 1 <= j & j < width GoB f & ( GoB f ) * ( 1 , j ) `2 <= ( ( GoB f ) * ( 1 , j + 1 ) ) `2 & ( GoB f ) * ( 1 , j + 1 ) `2 <= ( ( GoB f ) * ( 1 , j + 1 ) ) `2 ;
defpred U [ set , set ] means ex FBi1 being Subset-Family of T st ( $1 = F . $1 & ( for n being Nat st n in dom F holds F . n = ( F . n ) . ( n + 1 ) ) & ( union F ) . ( n + 1 ) is open & ( union F ) . ( n + 1 ) is open & union F is open & union F is open & union F is O & union F is O & union F is O & union F is O & union F is O & union F is O & union F is O & union F is O & union F is O & union F is O & union F is O & union F is O & union ( F . ( n + 1 ) is O & union ( F . ( n + 1 ) is O & union ( F is O & union ( F . ( n + 1 ) is O & union (
for p4 being Point of TOP-REAL 2 st LE p4 , p4 , P , p1 , p2 & LE p4 , p4 , P , p1 , p2 holds LE p4 , p4 , P , p2
f in St ( E , H ) & for y st y <> f . y holds x in St ( E , H ) implies for y st y in St ( E , H ) holds y in St ( E , H ) & f . y = ( the Sorts of H ) . y
ex p0 being Point of TOP-REAL 2 st x = p2 & for q being Point of TOP-REAL 2 st q = p2 & |. q .| >= sn & |. q .| >= sn & |. p2 .| >= sn & |. p2 .| >= sn & |. p2 .| >= sn & |. p2 .| >= sn & |. p2 .| >= sn & p2 .| >= sn & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & |. p2 <> 0. TOP-REAL 2 & |. p2 <> 0. TOP-REAL 2 & |. p2 <> 0. TOP-REAL 2 & |. p2 <> 0.
assume for d7 being Element of NAT st d7 <= |. |. 7 - 2 .| holds |. ( |. 7 - 2 .| ) . K - ( |. 7 - 2 .| ) . K = |. 7 - 2 .| & |. 7 - 2 .| = |. 7 - 2 .| ;
assume that s <> t and s is Point of Closed-Interval-TSpace ( x , r ) and not ex e being Point of Closed-Interval-TSpace ( x , r ) st e in Ball ( x , r ) & not e in Ball ( x , r ) & not e in Ball ( x , r ) ;
given r such that 0 < r and for s st 0 < s ex x1 , x2 being Point of Cst x1 in dom f & ||. x1 - x2 .|| < s & ||. x1 - x2 .|| < s & ||. x1 - x2 .|| < s & ||. x1 - x2 .|| < s ;
( p | 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 | ( 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 | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | (
assume that x , h + x in dom sec and ( \HM { the } \HM { function } \HM { cos } ) . x = ( 4 * ( sin ( x ) ) + ( 2 * ( sin ( x ) ) ^2 ) ) and for x st x in Z holds ( ( 2 * x ) ^2 + ( 2 * ( sin ( x ) ) ^2 ) ) ^2 = ( 4 * ( sin ( x ) ) ^2 + ( 2 * ( sin ( x ) ) ^2 ) ;
assume that i in dom A and i > 1 and len A > 1 and ( the _ of A ) * ( i , j ) = ( the _ of K ) * ( i , j ) and ( the _ of K ) * ( i , j ) = ( the multF of K ) * ( i , j ) and ( the multF of K ) * ( i , j ) = ( the multF of K ) * ( i , j ) ;
for i being non zero Element of NAT st i in Seg n holds ( i divides n implies i divides n implies h . i = ( 1_ F_Complex ) . i ) & ( h . i = ( 1_ F_Complex ) . i ) & ( h . i = ( 1_ F_Complex ) . i ) & ( h . i = ( 1_ F_Complex ) . i ) & h . i = ( 1_ F_Complex ) . i ) & h . i = ( 1_ F_Complex ) . i ;
( ( ( ( b1 '&' b2 ) '&' ( b2 '&' b3 ) ) '&' ( ( b1 '&' c1 ) '&' ( ( b1 '&' c2 ) '&' ( ( b1 '&' c1 ) '&' ( b1 '&' c2 ) ) '&' ( ( b1 '&' c1 ) '&' ( ( b1 '&' c2 ) '&' ( ( b1 '&' c2 ) '&' ( ( b1 '&' c2 ) '&' ( b1 '&' c2 ) '&' ( ( b1 '&' c2 ) '&' ( b1 '&' c2 ) ) ) ) ) ) '&' ( ( b1 '&' b2 ) '&' ( ( b1 '&' c1 ) '&' ( ( b1 '&' c2 ) '&' ( a1 '&' c2 ) ) '&' ( a1 '&' c2 ) ) '&' ( a1 '&' c2 ) '&' ( a1 ) '&' ( a1 '&' c2 ) '&' ( a1 '&' c2 ) '&' ( a1 '&' c2 ) ) '&' ( a1 '&' c2 ) '&' ( a1 '&' c2 ) '&' ( a1 '&' c2 ) ) '&' ( a1 '&' c2 ) ) '&' ( a1 '&' c2 ) ) ) '&'
assume that for x holds f . x = ( ( ( - cot ) * ( sin - cot ) ) `| Z ) and for x st x in Z holds ( ( ( - cot ) * ( ( sin - cot ) * ( ( sin - cot ) * ( ( sin - cot ) * ( ( sin - cot ) * ( ( sin - cot ) * ( ( sin - cot ) * ( ( sin - cot ) * ( ( sin - cot ) * ( ( sin - cot ) * ( ( sin - cot ) * ( ( sin - cot ) * ( ( sin - cot ) ) `| Z ) ) ) `| Z ) ) ) ) . x ) ) = ( ( sin - cot ) `| Z ) . x ) and for x ) `| Z ) . x ) and for x st x in Z ) holds ( ( A ) . x ) and for x st x in Z holds ( ( x ) `| Z ) .
consider R8 , R-8 being Real such that R8 = Integral ( M , F . n ) and Rl = Integral ( M , F . n ) and I = Integral ( M , F . n ) and for i be Nat holds I . i = Integral ( M , F . i ) and I . i = Integral ( M , F . i ) ;
ex k be Element of NAT st ' = k & 0 < d & for q be Element of product G st q in X & q in X holds ||. ( f , q ) . ( k + 1 ) - ( f , q ) . ( k + 1 ) .|| < r ;
x in { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x6 , x6 , x6 , x6 , x6 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7 } iff x in { x1 , x2 , x3 , x4 , x4 , 8 , 7 , 8 , 8 , 7 }
( G * ( j , i ) ) `2 = ( G * ( 1 , i ) ) `2 .= ( G * ( 1 , i ) ) `2 .= ( G * ( 1 , i ) ) `2 .= ( G * ( 1 , i ) ) `2 .= ( G * ( 1 , i ) ) `2 .= ( G * ( 1 , i ) ) `2 .= ( G * ( 1 , i ) ) `2 .= ( G * ( 1 , i ) ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= G * ( 1 , i
f1 * p = p .= ( ( the Arity of S1 ) +* ( the Arity of S2 ) ) . o .= ( the Arity of S2 ) . o .= ( the Arity of S2 ) . o .= ( the Arity of S2 ) . o .= ( the Arity of S2 ) . o ;
func tree ( T , P , T ) -> Tree means : Def1 : for q st q in it holds q in T or q in T or q in T or q in T or q in T or q in T or q in T & q in T or q in T ;
F /. ( k + 1 ) = F . ( p . ( k + 1 ) ) .= FD . ( p . ( k + 1 ) ) .= FD . ( p . ( k + 1 ) ) .= FD . ( p . ( k + 1 ) ) .= FD . ( p . ( k + 1 ) ) .= FD . ( p . ( k + 1 ) ) .= FD . ( p . ( k + 1 ) .= FD . ( p . ( k + 1 ) .= FD . ( p . ( k + 1 ) .= FD . ( p . ( k + 1 ) .= FD . ( p . ( k + 1 ) .= FD . ( p . ( k + 1 ) .= FD . ( p . k ) .= FD . ( p . ( k + 1 ) .= FD . ( p . ( p . ( k + 1 ) .= FD
for A , B , C , D st len B = len C & len C = width A & len B = width C & len A > 0 & len B > 0 & len A > 0 & len B > 0 & len A > 0 & len B > 0 & len B > 0 & len B > 0 & len A > 0 & len B > 0 & len B > 0 & len B > 0 holds B = C * B
seq . ( k + 1 ) = 0. ( ( seq . k ) + seq . ( k + 1 ) ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . k ;
assume that x in ( the carrier of CP ) and y in ( the carrier of CP ) and z = ( the carrier of CP ) and x = ( the carrier of CP ) and y = ( the carrier of CP ) and z = ( the carrier of CP ) and x = ( the carrier of P ) and y = ( the carrier of P ) . z ;
defpred P [ Element of NAT ] means for f st len f = $1 holds ( ( VAL ( g ) ) . ( k + 1 ) ) . ( f . ( k + 1 ) ) = ( ( ( VAL ( g ) ) . ( k + 1 ) ) . ( f . ( k + 1 ) ) ) '&' ( ( ( VAL ( g ) ) . ( k + 1 ) ) . ( f . ( k + 1 ) ) ) ;
assume that 1 <= k and k + 1 <= len f and f /. k = G * ( i , j ) and [ i + 1 , j ] in Indices G and [ i + 1 , j ] in Indices G and [ i + 1 , j ] in Indices G and G * ( i + 1 , j ) = G * ( i + 1 , j ) and G * ( i + 1 , j ) = G * ( i + 1 , j ) and G * ( i + 1 , j ) and G * ( i + 1 ) = G * ( i + 1 , j ) and G * ( i + 1 , j ) and G * ( i + 1 , j ) ;
assume that -4 < 1 and ( q `1 / |. q .| - cn ) / ( 1 + cn ) >= 0 and ( q `1 / |. q .| - cn ) / ( 1 + cn ) >= 0 and ( q `1 / |. q .| - cn ) / ( 1 + cn ) >= 0 and ( q `1 / |. q .| - cn ) / ( 1 + cn ) >= 0 and q `1 / |. q .| >= 0 and q `1 / ( 1 + cn ) and q `1 / ( 1 + cn ) = 0 and q `1 / ( 1 + cn ) and q `1 / ( 1 + cn ) and q `1 / ( 1 + cn ) = 0 and q `2 / ( 1 + cn ) and q `2 / ( 1 + cn ) = 0 and q `1 / ( 1 + cn ) = 0 and q `1 / ( 1 + cn and q `1 / ( 1 + cn and q `2 / ( 1 + cn ) = 0 and
for M being non empty MetrStruct , x being Point of M , f being Function of M , M st x = x holds ex f being Function of M , M st f is continuous & f is continuous & f . 0 = x & f . 1 = f . 1 & f . 1 = f . 1
defpred P [ Element of omega ] means ( f1 - f2 ) . $1 * ( f1 - f2 ) . $1 * ( f1 . $1 ) * ( f2 . $1 ) * ( f2 . $1 ) * ( f2 . $1 ) * ( f2 . $1 ) * ( f2 . $1 ) * ( f2 . $1 ) * ( f2 . $1 ) * ( f2 . $1 ) * ( f2 . $1 ) * ( f2 . $1 ) ) * ( f2 . $1 ) ) * ( f2 . $1 ) * ( f2 . $1 ) * ( f2 . $1 ) * ( f2 . $1 ) * ( f2 . $1 ) * ( f2 . $1 ) * ( f2 . $1 ) * ( f2 . $1 ) * ( f2 . $1 ) * ( f2 . $1 ) * ( f2 . $1 ) * ( f2 . $1 ) = ( f1 . $1 ) + ( f2 . $1 ) * ( f1 . $1 ) * ( f2 . $1 ) * ( f2 . $1
defpred P1 [ Nat , Point of Cmin ( C , $1 ) - ( f /. ( $1 + 1 ) ) ) < r & ||. f /. ( $1 + 1 ) - f /. ( $1 + 1 ) .|| < r ;
( f ^ mid ( g , 2 , len g -' 1 ) ) . i = ( mid ( g , 2 , len g -' 1 ) ) . ( i + 1 ) .= g . ( i + 1 ) .= g . ( i + 1 ) .= g . ( i + 1 ) .= g . ( i + 1 ) .= g . ( i + 1 ) .= g . ( i + 1 ) ;
sqrt ( 1 - ( 2 * n + 1 ) * ( 2 * n + 1 ) ) = ( ( 1 - ( 2 * n + 1 ) * ( 2 * n + 1 ) ) ) * ( 2 * n + 1 ) .= ( 1 - ( 2 * n + 1 ) * ( 2 * n + 1 ) ) * ( 2 * n + 1 ) .= ( 1 - ( 2 * n + 1 ) * ( 2 * n + 1 ) ) * ( 2 * n + 1 ) * ( 2 * n + 1 ) * ( 2 * n + 1 ) * ( 2 * n + 1 ) * ( 2 * n ) * ( 2 * n + 1 ) * ( 2 * n + 1 ) * ( 2 * n ) * ( 2 * n ) .= ( 1 * n ) * ( 2 * n ) .= ( 1 / ( 2 * n + 1 / ( 2 * n + 1 ) * ( 2 * n ) ) * ( 2 * n ) .= ( 1 / (
defpred P [ Nat ] means for G being finite non empty RelStr st G is finite non empty & G is finite holds card ( the carrier of G ) = card ( the carrier of G ) & card ( the carrier of G ) = card ( the carrier of G ) & card ( the carrier of G ) = card ( the carrier of G ) ;
assume that f /. 1 in Ball ( u , r ) and 1 <= m and m <= len f and for i st 1 <= i & i <= len f & i <= len f holds not LSeg ( f /. i , f /. ( i + 1 ) ) /\ LSeg ( f /. i , f /. ( i + 1 ) ) <> {} and LSeg ( f , i ) /\ LSeg ( f , i ) <> {} and LSeg ( f , i ) /\ LSeg ( f , i ) /\ LSeg ( f , i ) <> {} , f /. ( i + 1 ) <> {} , f /. ( i + 1 ) /\ LSeg ( f , i ) <> {} , f /. ( i + 1 ) /\ LSeg ( f , i ) <> {} , f /. ( i + 1 ) /\ LSeg ( f /. ( i + 1 ) /\ LSeg ( f /. ( i + 1 ) /\ LSeg ( f /. ( i + 1 ) /\ LSeg ( f , i ) /\ LSeg ( f /. ( i + 1 ) /\ LSeg (
defpred P [ Element of NAT ] means ( Partial_Sums ( cos * ( $1 + 1 ) ) ) . ( 2 * ( ( cos * ( $1 + 1 ) ) ) / ( 2 * ( ( ( cos * ( $1 + 1 ) ) / ( 2 * ( ( ( ( cos * ( $1 + 1 ) ) ^2 ) ) ) ) ) . ( 2 * ( ( ( ( ( 1 / 2 ) * ( ( 1 / 2 ) ) ^2 ) ) ) ) ) ;
for x being Element of product F holds x is FinSequence & x in dom ( the Sorts of F ) & ( for i being set st i in dom x holds x . i in dom ( the Sorts of F ) & ( for i being set st i in dom x holds x . i = ( the Sorts of F ) . i ) . x ) & ( the Sorts of F ) . i = ( the Sorts of F ) . i
( ( x " ) |^ ( n + 1 ) ) * ( x " ) = ( ( x " ) |^ ( n + 1 ) ) * ( x " ) .= ( ( x " ) |^ ( n + 1 ) ) * ( x " ) .= ( x " ) * ( x " ) .= ( x " ) * ( x " ) .= ( x " ) * ( x " ) .= x " ;
DataPart Comput ( P +* I , Initialized s ) = DataPart Comput ( P +* I , ( LifeSpan ( P +* I ) + 2 ) ) .= DataPart Comput ( P +* I , ( LifeSpan ( P +* I , s1 ) + 2 ) ) .= DataPart Comput ( P +* I , ( LifeSpan ( P +* I , s1 ) ) + 2 ) ;
given r such that 0 < r and for g st g in dom ( f1 + f2 ) & g in dom ( f1 + f2 ) holds ( f1 + f2 ) . g <= ( f1 + f2 ) . g + ( f2 + g2 ) . g ;
assume that X c= dom f1 /\ dom f2 and ( f1 + f2 ) | X is continuous and for r st r < 0 ex g st r < g & g in X & |. f1 - f2 .| < g & |. f1 + f2 .| < g & g in X ;
for L being continuous complete LATTICE for l being Element of L for X being Subset of L st l = sup ( { x } ) for x being Element of L st x in X holds x is Element of L & x is ` holds x is `
Support ( A *' ( A *' p ) ) = { m *' ( p *' q ) where m is Element of NAT : m in dom ( A *' p ) & ex i being Element of NAT st i in dom ( A *' p ) & ex m being Element of NAT st ( A *' ( p *' q ) ) . i = ( m *' p ) . i & ( m *' q ) . i = ( m *' q ) . i ;
( f1 - f2 ) /* ( seq + k ) = lim ( f1 /* ( seq + k ) ) - f2 . ( seq . ( seq + k ) ) .= lim ( f1 /* ( seq + k ) ) - f2 . ( seq . ( seq + k ) ) .= lim ( f1 /* ( seq + k ) ) ;
ex p1 being Element of CQC-WFF ( Al ) st F ( p1 ) = g . p1 & for g being Function of Al ( ) , D ( ) st g . ( p1 ) = f . ( g . ( p1 ) ) & for g being Function st g . ( p1 ) = f . ( g . ( p1 ) ) holds g . ( p1 , g . ( p1 ) ) = f . ( p1 , p2 ) ) ;
( mid ( f , i , len f -' 1 ) ) /. ( j + 1 ) = ( mid ( f , i , len f -' 1 ) ) /. ( j + 1 ) .= ( mid ( f , i , len f -' 1 ) ) . ( j + 1 ) .= ( mid ( f , i , len f -' 1 ) ) . ( j + 1 ) .= ( mid ( f , i , len f -' 1 ) ) . ( j + 1 ) ;
( ( p ^ q ) . k ) . ( len p + k ) = ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( p . k ) . ( len p + k ) .= ( p . k ) . ( len p + k ) .= ( p . k ) . ( len p + k ) .= ( p . k ) . ( k + k ) . ( len p + k ) . ( k + k ) . ( k + k ) . ( k + k ) . ( k + k ) . ( k + k ) . ( k + k ) . ( k + k ) . ( k + k ) .= ( p . k ) .= ( p . k ) .= ( p . k ) . ( len p + k ) .= ( p . k ) . ( len p + k ) . ( k + k ) .= ( p . k ) . ( k + k ) .= ( p . k ) . ( k + k
len mid ( D2 , indx ( D2 , D1 , j1 ) + 1 , indx ( D2 , D1 , j1 ) -' 1 ) + 1 = indx ( D2 , D1 , j1 ) + ( indx ( D2 , D1 , j1 ) -' 1 ) ;
x * y * z = ( ( x * y ) * z ) * ( ( y * z ) * z ) .= ( ( x * y ) * z ) * ( ( y * z ) * z ) .= ( x * y ) * ( z * z ) .= ( x * y ) * ( z * z ) .= x * ( y * z ) .= x * ( y * z ) ;
v . <* x , y *> = ( <* x0 , y *> ) * ( <* x0 , y *> ) + ( ( ( ( ( y - x0 ) * ( y - x0 ) ) * ( y - x0 ) ) + ( ( ( y - x0 ) * ( y - x0 ) ) ) * ( ( y - x0 ) * ( y - x0 ) ) ) ;
i * i = <* 0 , 1 , 0 *> .= <* 0 , 1 *> .= <* 0 , 1 *> .= <* 1 *> .= <* 1 *> .= <* 1 *> .= <* 1 *> .= <* 1 *> .= <* 1 *> ;
Sum ( L (#) F ) = Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( Sum ( F1 ^ F2 ) ) .= Sum ( L (#) ( F1 ^ F2 ) + Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( Sum ( L (#) ( F1 ^ F2 ) + Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( L (#) ( Sum ( F1 ^ F2 ) ) .= Sum ( F1 ^ F2 ) + Sum ( F1 ^ F2 ) + Sum ( F1 ^ F2 ) + Sum ( F1 ^ F2 ) + Sum ( F1 ^ F2 ) + Sum ( F1 ^ F2 ) ) .= Sum ( Sum ( F1 ^ F2 ) + Sum ( F1 ^ F2 ) + Sum ( F1 ^ F2 )
ex r be Real st for e be Real st 0 < e ex Y being finite Subset of X st Y is non empty & for n being Nat st n >= m holds |. ( f . n ) - f . n .| < r & r <= 1 holds |. ( f . n ) - f . n .| < r
( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) & ( GoB f ) * ( i + 1 , j ) = f /. ( k + 1 ) or ( GoB f ) * ( i + 1 , j ) = f /. ( k + 1 ) or ( GoB f ) * ( i + 1 , j ) = f /. ( k + 1 ) ;
( ( ( - 1 ) / ( r - 1 ) ) / ( r - 1 ) ) / ( r - 1 ) = ( ( ( r - 1 ) / ( r - 1 ) ) / ( r - 1 ) ) / ( r - 1 ) .= ( ( r - 1 ) / ( r - 1 ) ) / ( r - 1 ) .= ( r - 1 ) / ( r - 1 ) ;
- sqrt ( - ( - ( a , b ) * ( - ( a , c ) * ( - ( b , c ) * ( - ( a , c ) * ( - ( b , c ) * ( - ( a , c ) * ( - ( b , c ) * ( - ( a , c ) * ( - ( b , c ) * ( - ( a , c ) ) * ( - ( b , c ) ) ) ) ) ) ) ) > 0 ) & - ( - ( a , c ) ) / ( - ( b , c ) ) ) > 0 ;
assume that ex_inf_of ( ( ( ( ( for x being Element of L ) ) /\ C ) , L ) , L and for x being Element of L st x in C holds x in ( ( ( the carrier of L ) /\ C ) /\ C ) and x in ( ( the carrier of L ) /\ C ) and x in ( the carrier of L ) /\ C ;
( ( ( ( the Sorts of B ) . i ) --> ( ( the Sorts of B ) . i ) ) . ( ( the Sorts of B ) . i ) ) = ( ( ( the Sorts of A ) . i ) --> ( ( the Sorts of B ) . i ) ) . ( ( the Sorts of B ) . i ) .= ( ( the Sorts of A ) . i ) . ( ( the Sorts of B ) . i ) ;