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thesis ;
contradiction ;
contradiction ;
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thesis ;
contradiction ;
contradiction ;
contradiction ;
contradiction ;
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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 ;
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contradiction ;
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contradiction ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
B in B ;
a <> c
T c= S
D c= B
c in X ;
b in X ;
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 ;
let k be Nat ;
K ` is being_line ;
assume n >= N ;
assume n >= N ;
assume X is 1 ;
assume x in I ;
q is \cup by A1 ;
assume c in x ;
p > 0 ;
assume x in Z ;
assume x in Z ;
1 <= kLet ;
assume m <= i ;
assume G is prime ;
assume a divides b ;
assume P is closed ;
I > 0 ;
assume q in A ;
W is not bounded ;
f is Assume f is one-to-one ;
assume A is boundary ;
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 `1 <= c `1 ;
A meets W ;
i `1 <= j `1 ;
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 <= 3-2 ;
let X be set ;
let X be set ;
let y be element ;
let x be element ;
P [ 0 ]
let E be set , A be Subset of E ;
let C be Category ;
let x be element ;
let k be Nat ;
let x be element ;
let x be element ;
let e be element ;
let x be element ;
P [ 0 ]
let c be element ;
let y be element ;
let x be element ;
let a be Real ;
let x be element ;
let X be element ;
P [ 0 ]
let x be element ;
let x be element ;
let y be element ;
r in REAL ;
let e be element ;
n1 is \rangle ;
Q halts_on s ;
x in \in \in such that x in \in \in \in \in $ ;
M < m + 1 ;
T2 is open ;
z in b set ;
R2 is well-ordering ;
1 <= k + 1 ;
i > n + 1 ;
q1 is one-to-one ;
let x be trivial set ;
PM 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 ;
y in C1 ;
k + 1 <= n ;
let a be Real , x be Point of X ;
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 \mathord 4 , o ;
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 complex let X , Y be Subset of V ;
not s in Y |^ 0 ;
rng f <= w
b "/\" e = b ;
m = m3 ;
t in h . D ;
P [ 0 ] ;
assume z = x * y ;
S . n is bounded ;
let V be RealUnitarySpace , A be Subset of V ;
P [ 1 ] ;
P [ {} ] ;
C1 is L~ of f ;
H = G . i ;
1 <= i `1 + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
aZ <= being Real ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , A be Subset of V ;
s is trivial & s is non trivial ;
dom c = Q
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be complete LATTICE , f be Function of T , S ;
the ObjectMap of F is one-to-one
sgn x = 1 ;
k in support a ;
1 in Seg 1 ;
rng f = X ;
len T in X ;
vbeing < n ;
S`| X is bounded ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let U2 ;
pp `1 = c ;
j in dom h ;
let k ;
f | Z is continuous ;
k in dom G ;
UBD C = B ;
1 <= len M ;
p in \mathbin { x } ;
1 <= jj ;
set A = Cl A ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H is_structure or H is \rrangle ;
assume x0 <= m ;
T is increasing ;
e2 <> e2 & e2 <> e2 ;
Z c= dom g ;
dom p = X ;
H is proper implies 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 ( ) ;
C c= C-26 \/ C-26 ;
mm <> {} ;
let x be Element of Y ;
let f be ] [#] of G , x be set ;
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 , x be set ;
P0 = 1 ;
assume r in F . k ;
assume f is simple function ;
let A be \lbrack countable set ;
rng f c= NAT * ;
assume P [ k ] ;
f6 <> {} ;
let o be Ordinal ;
assume x is sum of squares ;
assume not v in { 1 } ;
let I1 , I2 ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
d1 in C ;
assume t . 1 in A ;
let Y be non empty TopStruct , 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 [#] ( f | A ) ;
[ a , c ] in X ;
mm . m <> {} ;
M + N c= M + M ;
assume M is \llangle hhL , M \rrangle ;
assume f is cmarr-r) ;
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 & j1 <= j2 ;
f | A is continuous ;
f . x Let x be Real ;
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 ;
CX in S ;
q2 c= C1 & q2 c= C1 ;
a2 < c2 & a2 < c2 ;
s2 is 0 -started ;
IC s = 0 ;
s4 = s4 & s4 = 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 MSAlgebra over L ;
y " <> 0 ;
y " <> 0 ;
0. V = uw ;
y2 , y , v , w as u1 <> u2 ;
R8 in X ;
let a , b be Real , f be PartFunc of REAL , 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 .] ;
let i , j be Nat ;
let s be State of A , a be element ;
s4 . n = N ;
set y = ( x `1 ) * ( x `2 ) ;
[: i , j :] 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 Nf1 ;
reconsider i = i as Ordinal ;
r * v = 0. X ;
rng f c= INT & rng f c= INT ;
G = 0 .--> goto 0 ;
let A be Subset of X ;
assume X0 is dense & 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 ;
x9 c= Z1 & x9 c= Z1 ;
dom f = C1 & dom g = C1 ;
assume [ a , y ] in X ;
Re ( seq ) is convergent ;
assume that a1 = b1 and b1 = b2 ;
A = ( sInt A ) . x ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be Instruction of S , s be State of S ;
assume that r2 > x0 and x0 in dom f ;
let Y be non empty set , f be Function of Y , BOOLEAN ;
2 * x in dom W ;
m in dom ( g2 | n ) ;
n in dom ( g1 | X ) ;
k + 1 in dom f ;
not the still of { s } is finite ;
assume that x1 <> x2 and x2 <> x3 ;
v1 in V1 & v2 in V1 & v1 in V1 ;
not [ b `1 , b ] in T ;
i9 + 1 = i ;
T c= \rangle ( T ) ;
( l `1 ) ^2 = 0 ;
let n be Nat ;
( t `2 ) ^2 = r ;
A\llangle f , A ] is_integrable_on M & A\llangle f , A ] is_integrable_on M ;
set t = Top t ;
let A , B be real-membered set ;
k <= len G + 1 ;
[: C , D :] misses [: C , D :] ;
product seq is non empty ;
e <= f or f <= e ;
cluster non empty ordinal for set ;
assume c2 = b2 & c1 = b2 ;
assume h in [. q , p .] ;
1 + 1 <= len C ;
not c in B . m1 ;
cluster R .: X -> empty ;
p . n = H . n ;
assume that v9 is Cauchy and vseq is Cauchy ;
IC s3 = 0 & IC s2 = 0 ;
k in N or k in K ;
F1 \/ F2 c= F \/ G ;
Int ( G1 \/ G2 ) <> {} ;
( z `2 ) ^2 = 0 ;
p10 `2 <> p1 `2 & p2 `2 <> p1 `2 ;
assume z in { y , w } ;
MaxADSet ( a ) c= F ;
ex_sup_of downarrow s , S ;
f . x <= f . y ;
let T be up-complete reflexive non empty reflexive RelStr ;
q |^ m >= 1 ;
a is_>=_than X & b >= Y ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one implies G is one-to-one ;
A \/ { a } c= B ;
0. V = 0. Y & 0. V = 0. Y ;
let I be the \cal Instruction Instruction of S , s be State of S ;
f-24 . x = 1 ;
assume z \ x = 0. X ;
C4 = 2 to_power n ;
let B be sequence of Sigma ;
assume X1 = p .: D ;
n + l2 in NAT & n + l2 in NAT ;
f " P is compact ;
assume x1 in REAL + ( - 1 ) ;
p1 = K1 & p2 = K1 or p2 = K1 ;
M . k = <*> REAL ;
phi . 0 in rng phi ;
OSML is closed ;
assume z0 <> 0. L & 0. L in W ;
n < N . k ;
0 <= seq . 0 & 0 <= seq . 0 ;
- q + p = v ;
{ v } is Subset of B ;
set g = f /. 1 ;
[: R , S :] is stable Subset of R ;
set [: R , S :] = Vertices R ;
pp c= P4 & P4 c= P4 ;
x in [. 0 , 1 .] ;
f . y in dom F ;
let T be Scott Scott Scott Scott Scott Scott Scott Scott Scott Scott TopAugmentation of S ;
ex_inf_of the carrier of S , S ;
downarrow a = downarrow 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 <> {} ;
a ~ , b ~ ] ;
assume a in A ( ) ;
k in dom ( q | k ) ;
p is $ y is $ y & p is one-to-one ;
i -' 1 = i-1 - 1 ;
f | A is one-to-one ;
assume x in f .: [: X , Y :] ;
i2 - i1 = 0 & i2 - i1 = 0 ;
j2 + 1 <= i2 & j2 + 1 <= j2 ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster -> of of \HM { \rm \cal `2 } -> strict for strict \rm Real ;
|. q .| ^2 > 0 ;
|. p4 .| = |. p .| ;
s2 - s1 > 0 & s1 - s2 > 0 ;
assume x in { Gik } ;
W-min C in C & W-min 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 + j ;
dom S = dom F & dom S = dom F ;
let s be Element of NAT , n be Nat ;
let R be ManySortedSet of A ;
let n be Element of NAT ;
let S be non empty non void non void ManySortedSign ;
let f be ManySortedFunction of I , P ;
let z be Element of F_Complex , x be set ;
u in { \hbox { \boldmath $ g $ } } ;
2 * n < 2 * ( 2 * n ) ;
let x , y be set ;
B-11 c= V1 & B-15 c= V1 ;
assume I is_halting_on s , P ;
U2 = U2 \/ ( U2 \/ C ) ;
M /. 1 = z /. 1 ;
x9 = x9 & x9 = y9 or x9 = y9 & y9 = x9 ;
i + 1 < n + 1 + 1 ;
x in { {} , <* 0 *> } ;
f7 <= f6 & f7 <= len f6 ;
let l be Element of L ;
x in dom ( F . -17 ) ;
let i be Element of NAT , a be Element of REAL ;
r8 is ( COMPLEX * -valued FinSequence of COMPLEX ;
assume <* o2 , o *> <> {} ;
s . x |^ 0 = 1 ;
card ( K1 | K1 ) in M & card ( K1 | K1 ) in M ;
assume X in U & Y in U ;
let D be \rangle of Omega ;
set r = q q | { 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 \oplus A is interval ;
|. <*> A .| . a = 0 ;
cluster strict Sublattice for SubLattice of L ;
a1 in B . s1 & a2 in B . s1 ;
let V be finite VectSp of F , A be Subset of V ;
A * B on B & A on B ;
f-3 = [: NAT , { 0 } :] --> 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 ;
z in dom ( A --> y ) ;
P [ y , h . y ] ;
{ x0 } c= dom f /\ dom g ;
let B be non-empty ManySortedSet of I , A be ManySortedSet of I ;
( PI / 2 ) < Arg z ;
reconsider z9 = 0 as Nat of 0 -tuples_on REAL ;
LIN a , d , c ;
[ y , x ] in IX ;
( Q ) * ( 1 , 3 ) = 0 ;
set j = x0 div m , m = x0 div m ;
assume a in { x , y , c } ;
j2 - ( j - 1 ) > 0 ;
I \! \mathop { N } = 1 ;
[ y , d ] in F-8 ;
let f be Function of X , Y ;
set A2 = ( B \/ C ) \ { {} } ;
s1 , s2 , s1 , s2 , s1 , s2 , s2 , s1 , s2 ;
j1 -' 1 = 0 & j1 -' 1 = 0 ;
set m2 = 2 * n + j ;
reconsider t = t as bag of n ;
I2 . j = m . j ;
i |^ s , n are_congruent_mod m ;
set g = f | D-21 ;
assume X is lower bounded & 0 <= r ;
( ( p1 `1 ) / |. p1 .| ) ^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 <= i2 ;
1 <= i1 -' 1 & i1 -' 1 <= i2 ;
i + i2 <= len h ;
x = W-min ( P ) & x in P ;
[ x , z ] in X ~ ;
assume y in [. x0 , x .] ;
assume p = <* 1 , 2 , 3 *> ;
len <* A1 *> = 1 & len <* A1 *> = 1 ;
set H = h . -3 , I = h . -3 , H = h . -3 ;
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 ** h1 ;
assume x in ( ( X1 union X2 ) union ( X2 union 4 ) ) ;
||. h .|| < dx0 & 0 < dx0 ;
not x in the carrier of f & x in the carrier of f ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k - l = kl2 - k ;
<* p , q *> /. 2 = q ;
let S be Subset of the carrier of Y ;
let P , Q be c9 of s ;
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 transitive RelStr , f be Function of L , L ;
S-20 is x -\leq i -f1 ;
let r be non positive Real ;
M , v |= x 'in' y ;
v + w = 0. ( Z , n ) ;
P [ len F ] implies P [ F ( ) ] ;
assume InsCode ( i9 ) = 8 & InsCode ( i ) = 8 ;
the zero of M = 0 & the carrier of M = 0 ;
cluster z * seq -> summable for Real_Sequence ;
let O be Subset of the carrier of C ;
||. f .|| | X is continuous ;
x2 = g . ( j + 1 ) ;
cluster -> the string of S for string of S ;
reconsider l1 = l-1 as Nat ;
v4 is Vertex of r2 & v4 is \overline { v } ;
T1 is SubSpace of T2 & T2 is SubSpace of T2 ;
Q1 /\ Q1 <> {} & Q1 /\ Q1 <> {} ;
let k be Nat ;
q " is Element of X & q " in X ;
F . t is set of of non zero & F . t = M ;
assume n <> 0 & n <> 1 ;
set en = EmptyBag n , e = EmptyBag n , u = EmptyBag n ;
let b be Element of Bags n ;
assume for i holds b . i is commutative ;
x is root implies ( p `1 ) ^2 = ( p `2 ) ^2
not r in ]. p , q .] ;
let R be FinSequence of REAL , x be set ;
S7 does not destroy b1 & not a1 in S1 & a2 in S2 ;
IC SCM R ( ) <> a & IC SCM R ( ) <> b ;
|. - |[ x , y ]| .| >= r ;
1 * seq = seq & 1 * seq = seq ;
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 ;
H + G = F- ( GG ) ;
CS1 . x = x2 & CS2 . x = y2 ;
f1 = f .= f . 1 .= f2 . 1 .= f /. 1 .= f /. 1 ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u + W ;
{ a1 } = { a2 } & { a2 } = { a2 } ;
a1 , b1 _|_ b , a ;
d1 , o _|_ o , a3 ;
Ix is_reflexive implies C is reflexive & C is reflexive
IX is_antisymmetric implies C is antisymmetric & C is antisymmetric
upper_bound rng ( H1 | i ) = e & upper_bound rng ( H1 | i ) = e ;
x = ( a * a9 ) * ( a * b ) ;
|. p1 .| ^2 >= 1 ^2 ;
assume j2 -' 1 < 1 & j2 -' 1 < len f ;
rng s c= dom f1 /\ dom f2 ;
assume that support a misses support b and not a in support b ;
let L be associative commutative non empty doubleLoopStr , p be Polynomial of L ;
s " + 0 < n + 1 ;
p . c = ( f " ) . 1 .= p . 1 ;
R . n <= R . ( n + 1 ) ;
Directed I1 = I1 & card I1 = card I1 ;
set f = + ( x , y , r ) ;
cluster Ball ( x , r ) -> bounded ;
consider r being Real such that r in A ;
cluster -> non empty for Function ;
let X be non empty directed Subset of S ;
let S be non empty full SubRelStr of L ;
cluster <* *> . <* N . N *> -> complete continuous for non trivial TopSpace ;
( 1 - a ) " = a ;
( q . {} ) `1 = o ;
n - ( i -' 1 ) > 0 ;
assume ( 1 - 2 ) / t `1 <= 1 ;
card B = k + 1 - 1 ;
x in union rng ( f | n ) ;
assume x in the carrier of R & y in the carrier of R ;
d in D ;
f . 1 = L . ( F . 1 ) ;
the vertices of G = { v } ;
let G be : for w be : w in G holds w = 0 ;
let e , v9 be set ;
c . i9 in rng c & c . i9 in rng c ;
f2 /* q is divergent_to+infty & f2 /* ( f1 /* s ) is divergent_to+infty ;
set z1 = - z2 , z2 = - z2 , z2 = - z2 , z2 = - z2 ;
assume w is \mathop { l\mathop ( S , G ) } ;
set f = p \! \mathop { t } , g = p \! \mathop { t } , h = p \! \mathop { t } , n = p \! \mathop { t } , m = p \! \mathop
let c be Object of C ;
assume ex a st P [ a ] ;
let x be Element of REAL m , y be Point of REAL-NS m ;
let I1 be Subset-Family of X , G 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 , I be Program of SCM+FSA ;
p is FinSequence of NAT & q is FinSequence of NAT implies p is FinSequence of NAT
stop I c= P_(#) ( 1 , 1 ) ;
set ci = f// f11 /. i , f21 /. i ;
w ^ t connected w ^ <* s *> ;
W1 /\ W = W1 /\ W ` .= ( W1 + W2 ) /\ W ;
f . j is Element of J . j ;
let x , y be Element of T2 , a , b be Real ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 & c <> 0 ;
ord x = 1 & x is \sum p implies x is \sum p
set g2 = lim ( seq , n ) , g1 = lim ( seq , n ) ;
2 * x >= 2 * ( 1 / 2 ) ;
assume ( a 'or' c ) . z <> TRUE ;
f (*) g in Hom ( c , c ) ;
Hom ( c , c + d ) <> {} ;
assume 2 * Partial_Sums ( q | m ) > m ;
L1 . F1 = 0 & L1 . F1 = 1 ;
/ ( X \/ R1 ) = / ( X \/ R1 ) ;
( ( - sin ) `| Z ) . x <> 0 ;
( ( exp_R * exp_R ) `| Z ) . x > 0 ;
o1 in [: X , Y :] /\ [: X , Y :] ;
let e , v9 be set ;
r3 > ( 1 - 2 ) * 0 ;
x in P .: ( F -ideal ) ;
let J be closed Subset of R , I be Ideal of R ;
h . p1 = f2 . O & h . O = g2 . I ;
Index ( p , f ) + 1 <= j ;
len ( q * M ) = width M & len ( q * M ) = width M ;
the carrier of LK c= A & the carrier of LK c= A ;
dom f c= union rng ( F | n ) ;
k + 1 in support ( support ( support ( n ) ) ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y `2 ] in ( \mathclose R ) \/ ( ( R ~ ) * ( R ~ ) ) ;
i = D1 or i = D2 or i = D1 ;
assume a mod n = b mod n ;
h . x2 = g . x1 & h . x2 = g . x2 ;
F c= 2 |^ ( the carrier of X ) ;
reconsider w = |. s1 .| as Real_Sequence of TOP-REAL 2 ;
( 1 / m * m + r ) < p ;
dom f = dom I-4 & dom I = dom I-4 ;
[#] P-17 = [#] ( ( TOP-REAL 2 ) | K1 ) .= K1 ;
cluster - x -> ExtReal -> ExtReal for ExtReal ;
then { d1 } c= A & A is closed ;
cluster TOP-REAL n -> finite-ind for Subset of TOP-REAL n ;
let w1 be Element of M ;
let x be Element of dyadic ( n ) ;
u in W1 & v in W2 implies u in W2
reconsider y = y as Element of L2 ( ) ;
N is full SubRelStr of [: T , T :] , S be full SubRelStr of T , T ;
sup { x , y } = c "\/" c ;
g . n = n / 1 .= n ;
h . J = EqClass ( u , J ) ;
let seq be summable sequence of X , n be Nat ;
dist ( x , y ) < ( r / 2 ) / 2 ;
reconsider mm = m - n as Element of NAT ;
x- x0 < r1 - x0 & x0 < r2 - x0 ;
reconsider P = P ` as strict Subgroup of N ;
set g1 = p * idseq ( q `1 ) , g2 = idseq ( q `2 ) , g1 = idseq ( q `2 ) ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is lower ;
D2 . I2 in { x } & D2 . I2 in { x } ;
cluster subcondensed closed -> subcondensed for Subset of T ;
let P be compact connected non empty Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
Gik in LSeg ( Gik , Gij ) \/ LSeg ( Gik , Gij ) ;
let n be Element of NAT , x be set ;
reconsider Ss = S as Subset of T | A ;
dom ( i .--> X ` ) = { i } ;
let X be non-empty ManySortedSet of S ;
let X be non-empty ManySortedSet of S ;
op ( 1 ) c= { [ {} , {} ] } ;
reconsider m = mm - 1 as Element of NAT ;
reconsider d = x as Element of C ( ) ;
let s be 0 -started State of SCMPDS , k be Nat ;
let t be 0 -started State of SCMPDS , Q be Subset of SCMPDS ;
b , b , b , x , y , z is_collinear ;
assume that i = n \/ { n } and j = k \/ { n } ;
let f be PartFunc of X , Y ;
N1 >= ( sqrt c / sqrt 2 ) ^2 + ( sqrt c / sqrt 2 ) ^2 ;
reconsider t7 = T" as Point of Euclid n ;
set q = h * p ^ <* d *> ;
z2 in U . ( y2 ) /\ Q2 . ( z2 . z2 ) ;
A |^ 0 = { <* \rangle *> } \/ { <* E *> *> } ;
len W2 = len W + 2 & len W2 = len W + 2 ;
len ( h2 | 1 ) in dom ( h2 | 1 ) ;
i + 1 in Seg ( len s2 ) & i + 1 in Seg ( len s2 ) ;
z in dom g1 /\ dom f & z in dom f1 /\ dom f2 ;
assume that p2 = W-min ( K ) and p1 in K and p2 in K ;
len G + 1 <= i1 + 1 & 1 <= i2 ;
f1 (#) f2 is convergent & lim ( f1 (#) f2 ) = x0 ;
cluster ( s1 + s2 ) ^\ k -> summable for Real_Sequence ;
assume that j in dom ( M1 * ( i , j ) ) ;
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 ;
<* xy *> ^ <* y *> \subseteq x ;
a , b in { a , b } ;
len p2 is Element of NAT & len p2 = len p1 + 1 ;
ex x being element st x in dom R & R . x = x ;
len q = len ( K * G ) ;
s1 = Initialize Initialized s , P1 = P +* I , P2 = P +* stop I , s2 = LifeSpan ( P2 , s2 ) , P2 = P +* stop I , P2 = P +* stop I ;
consider w being Nat such that q = z + w ;
x ` ` is Element of L ` & x ` is Element of L ` ;
k = 0 & n <> k or k > n & n > k ;
then X is discrete for Subset of X ;
for x st x in L holds x is FinSequence of L
||. f /. c - f /. c .|| <= r1 ;
c in ]. p , q .[ & not c in { p } ;
reconsider V = V as Subset of the topology of TOP-REAL n ;
let N , M be \hbox { S } , L ;
then z is_>=_than waybelow x & z is_>=_than compactbelow y ;
M | [. f , g .] = f & M | [. g , g .] = g ;
( ( L~ z ) /. 1 ) `1 = TRUE ;
dom g = dom ( f / X ) .= dom f ;
mode \cal \cal of G is \cal \cal \cal p\rangle ;
[ i , j ] in Indices M & [ i , j ] in Indices M ;
reconsider s = x " as Element of H ( ) ;
let f be Element of dom ( Subformulae p ) , x be set ;
F1 . ( a1 , - 1 ) = G1 . ( a1 , - 1 ) ;
Observe that \HM { the } \HM { carrier } \HM { of } \HM { , } b , r ) is compact ;
let a , b , c , d , e , f , g , h be Real ;
rng s c= dom ( 1 / 2 ) & rng s c= dom ( 1 / 2 ) ;
curry ( ( F . -19 ) , k ) is additive & ( F . -19 ) . k is additive ;
set k2 = card dom B - 1 , k1 = card dom B - 1 , k2 = card dom B - 1 , k2 = card dom B - 1 ;
set G = coprod X , A = ( the carrier of S ) --> NAT ;
reconsider a = [ x , s ] as Object of G ;
let a , b be Element of Mr , M be Matrix of REAL ;
reconsider s1 = s as Element of ( S | 0 ) | ( the carrier of S ) ;
rng p c= the carrier of L & p in the carrier of L ;
let d be Subset of the Sorts of A ;
( x .|. x ) = 0 iff x = 0. W & x = 0. W
I-21 in dom stop I & y in dom stop I ;
g be continuous Function of X | B , Y ;
reconsider D = Y as Subset of Euclid n , G = { p } ;
reconsider i0 = len p1 - 1 as Integer ;
dom f = the carrier of S & dom g = the carrier of S ;
rng h c= union ( the carrier of J ) .= ( the carrier of J ) ;
cluster All ( x , H ) -> reconsider reconsider H1 = H as " ;
d * N1 ^2 > N1 * 1 / ( d * 1 ) ;
]. a , b .[ c= [. a , b .] ;
set g = f " D1 | D1 , p = f | D1 , q = f | D2 ;
dom ( p | [: m , m :] ) = [: m , m :] ;
3 + - 2 <= k + - 2 ;
tan is_differentiable_in ( ( tan * arccot ) `| Z ) . x ;
x in rng ( f /^ ( p -' 1 ) ) ;
let f , g be FinSequence of D ;
[: p , q :] in the carrier of S1 & [: p , q :] in the carrier of S2 ;
rng ( f " ) = dom f & rng ( f " ) = dom f ;
( the Source of G ) . e = v ;
width G -' 1 < width G -' 1 & 1 < width G ;
assume v in rng ( S | ( E . 1 ) ) ;
assume x is root & x is root implies x is root & x is root ;
assume 0 in rng ( ( g2 | A ) | A ) ;
let q be Point of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
let p be Point of TOP-REAL 2 , x be Point of TOP-REAL 2 ;
dist ( O , u ) <= |. p2 .| + 1 ;
assume dist ( x , b ) < dist ( a , b ) ;
<* S *> is in the carrier of C-20 & <* S *> is in the carrier of C-20 ;
i <= len G -' 1 & 1 <= i & i + 1 <= width G ;
let p be Point of TOP-REAL 2 , x be Point of TOP-REAL 2 ;
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 = Sp2 " ( Q \ { x } ) .= Sp2 " ( Q \ { x } ) ;
( ( 1 / 2 ) (#) ( 1 / 2 ) ) (#) ( 1 / 2 ) is summable ;
- p + I c= - p + A + A ;
n < LifeSpan ( P1 , s1 ) + 1 & n < len s1 ;
CurInstr ( p1 , s1 ) = i & CurInstr ( p2 , s2 ) = i ;
A /\ Cl { x } \ { x } <> {} ;
rng f c= ]. r , r + 1 .[ ;
g be Function of S , V ;
let f be Function of L1 , L2 , g be Function of L1 , L2 ;
reconsider z = z as Element of CompactSublatt ( L ) ;
let f be Function of S , T ;
reconsider g = g as Morphism of c opp , b ;
[ s , I ] in S ~ ( A , I ) ;
len ( the connectives of C ) = 4 & len ( the connectives of C ) = 5 ;
let C1 , C2 be SubFunctor of C , C2 , a be object of C ;
reconsider V1 = V as Subset of ( X | B ) | B ;
attr p is valid means : Def21 : All ( x , p ) is valid ;
assume that X c= dom f and f .: X c= dom g and g .: X c= dom g ;
H |^ ( a " ) is Subgroup of H |^ ( a " ) ;
let A1 be Let be Let : A1 on A1 & A2 on A1 ;
p2 , r3 , q1 , q2 is_collinear & q2 , q1 , q2 is_collinear ;
consider x being element such that x in v ^ K ;
not x in { 0. TOP-REAL 2 } \/ { 0. TOP-REAL 2 } ;
p in [#] ( I[01] | B0 ) & p in [#] ( I[01] | B0 ) ;
0 . 0 < M . ( Ed . n ) ;
^ ( c / 2 ) / 2 = c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . s2 ) & a2 in dom ( F . s2 ) ;
cluster *> -> \rbrace -\cal \mathclose \mathclose { \rm \hbox { - } \rm \hbox { - } \rm _ 1 L ;
set i1 = the Nat of G , i2 = the Element of G , n = the Element of NAT ;
let s be 0 -started State of SCM+FSA , k be Nat ;
assume y in ( f1 \/ f2 ) .: A ;
f . ( len f ) = f /. len f .= f /. 1 ;
x , f . x '||' f . x , f . y ;
attr X c= Y means : Def6 : cos ( X ) c= cos ( Y ) ;
let y be upper Subset of Y , x be Element of Y ;
cluster ( x `1 ) ^2 -> transitive for Relation ;
set S = <* Bags n , i9 *> , i = <* <* <* i *> *> , j = <* i *> *> ;
set T = [. 0 , 1 / 2 .] , S = [. 0 , 1 .] ;
1 in dom mid ( f , 1 , 1 ) ;
sqrt ( 4 * PI ) < ( 2 * PI ) / 2 ;
x2 in dom ( f1 + f2 ) /\ dom ( f1 + f2 ) ;
O c= dom I & { {} } = { {} } ;
( the Source of G ) . x = v ;
{ HT ( f , T ) } 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 , u , v , w , u1 , v1 , v2 , u1 , v1 , v2 , u1 , v1 , v2 , v1 , v2 , u1 , v1 , v2 , v1 , v2 , u1 , v1 , v2 , u1 ,
h19 . i = f . ( h . i ) ;
( p `1 ) ^2 = ( p1 `1 ) ^2 + ( p1 `2 ) ^2 ;
i + 1 <= len Cage ( C , n ) ;
len <* P *> / 2 = len P & len <* P *> = len P ;
set N-26 = the \cup of N , NN = the carrier of N ;
len g: g: + ( x , x + 1 ) - 1 <= x ;
a on B & b on B & a on B ;
reconsider rv = r * I . v as FinSequence of REAL ;
consider d such that x = d and a \in d ;
given u such that u in W and x = v + u ;
len ( f /^ n ) = len f -' n ;
set q2 = N-min L~ Cage ( C , n ) , q2 = q2 ;
set S =
MaxADSet ( b ) c= MaxADSet ( P /\ Q ) ;
Cl ( G . q1 ) c= F . r2 ;
f " D /\ h " D meets h " V & f " D /\ h " V c= h " V ;
reconsider D = E as non empty directed Subset of L1 ;
H = ( the_left_argument_of H ) '&' ( the_right_argument_of H ) ;
assume t is Element of ( S ( ) ) . ( X , Y ) ;
rng f c= the carrier of S2 & rng f c= the carrier of S2 ;
consider y being Element of X such that x = { y } ;
f1 . ( a1 , b1 ) = b1 . ( b2 , b1 ) ;
the carrier' of G ` = E \/ { E } ;
reconsider m = len such that len such that k = len such that p = k - 1 and m <= n ;
set S1 = LSeg ( n , UMP C ) , S2 = LSeg ( n , UMP C ) ;
[ i , j ] in Indices ( M1 + M2 ) ;
assume that P c= Seg m and M is \HM { \mathbb N } and P is of M ;
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 * L /. 1 .= p * L /. 1 ;
pp . i = p1 . i .= p1 . i .= p2 . i ;
let PA , PA , G be a_partition of Y ;
pred 0 < r & r < 1 & 1 < ( 1 - r ) / 2 ;
rng \mathop { \rm AffineMap ( a , X ) = [#] ( 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 ) \/ the topology of X ) ;
dom ( f0 * u ) c= dom ( ( u - v ) * ( v - u ) ) ;
pred n divides m & m divides n & n = m ;
reconsider x = x as Point of [: I[01] , I[01] :] | K1 ;
a in ;
not y0 in the carrier of f & not ( ex g st g in f & g in f . x ) ;
Hom ( ( a \times b ) , c ) <> {} ;
consider k1 such that p " < k1 and p . k1 = 0 ;
consider c , d such that dom f = c \ d ;
[ x , y ] in dom g & [ x , y ] in dom g ;
set S1 = Let Let S1 , S2 = \rm many ( x , y , z ) ;
l1 = m2 & l1 = i2 & l1 = j2 & l2 = j2 ;
x0 in dom ( ( u - x0 ) * ( y - x0 ) ) /\ dom ( ( u - x0 ) * ( y - x0 ) ) ;
reconsider p = x as Point of TOP-REAL 2 , q = x `2 / |. q .| as Point of TOP-REAL 2 ;
[: I[01] , I[01] :] = [: the carrier of I[01] , the carrier of I[01] :] .= [: the carrier of I[01] , the carrier of I[01] :] ;
f . p4 `2 <= f . p1 `2 & f . p1 `2 <= f . p2 `2 ;
( ( F . n ) `1 ) ^2 <= ( ( F . n ) `1 ) ^2 ;
( x `2 ) ^2 = ( ( W . ( n + 1 ) ) `1 ) ^2 .= ( W . ( n + 1 ) ) ^2 ;
for n being Element of NAT holds P [ n ] ;
let J , K be non empty Subset of I ;
assume that 1 <= i and i <= len <* a " *> and <* a *> . i = a ;
0 |-> a = <*> ( the carrier of K ) .= ( the carrier of K ) --> a ;
X . i in 2 -tuples_on A . i \ B . i ;
<* 0 *> in dom ( e --> [ 1 , 0 ] ) ;
then P [ a ] & P [ succ a ] ;
reconsider sbeing p2 = s\rbrack as cluster cluster ( the ObjectMap of D ) | ( the carrier of D ) ;
- ( i -' 1 ) <= len such that ( - j ) <= len such that ( - j ) = - j ;
[#] S c= [#] ( T | ( the carrier of S ) ) & [#] S = [#] ( T | S ) ;
for V being strict RealUnitarySpace holds V in ex W being Subspace of V st W in \mathbb V & W is Subspace of V
assume k in dom mid ( f , i , j ) ;
let P be non empty Subset of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ;
let A , B be square Matrix of n , K , n , m be Nat ;
- a * ( - b ) = a * b ;
for A being Subset of A9 holds A // A & A // A implies A = B
( id o2 ) in <* o2 , o1 , o2 *> & ( id o2 ) in { o1 , o2 } ;
then ||. x - y .|| = 0 & x = 0. ( X , Y ) ;
let N1 , N2 be strict normal Subgroup of G , x be Element of G ;
j >= len ( upper_volume ( g , D1 ) | j1 ) & j <= len ( upper_volume ( g , D1 ) | j1 ) ;
b = Q . ( len Q - 1 ) .= Q . ( len Q - 1 ) ;
f2 * f1 /* s is divergent_to+infty & lim ( f2 * f1 ) = x0 ;
reconsider h = f * g as Function of [: N , G :] , 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 ( T | E ) | n ;
{} = the carrier of L1 + L2 & the carrier of L1 = the carrier of L2 + L2 ;
Directed I is_closed_on Initialized s , P & Directed I is_halting_on Initialized s , P ;
Initialized p = Initialize ( ( p +* q ) +* ( i , 1 ) ) ;
reconsider N2 = N1 as strict net of R2 , a be Element of R2 ;
reconsider Y = Y as Element of \langle Ids L , \subseteq \rangle ;
"/\" ( uparrow p \ { p } , L ) <> p ;
consider j being Nat such that i2 = i1 + j and j < len f ;
not [ s , 0 ] in the carrier of S2 & not [ s , 0 ] in the carrier of S2 ;
mm in ( B '&' C ) \/ D \ { {} } ;
n <= len ( P6 + len ( P6 + 1 ) ) + len ( P6 + 1 ) ;
( x1 - x2 ) `1 = ( x2 - x1 ) `1 .= ( x2 - x1 ) `1 ;
InputVertices S = { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 } ;
let x , y be Element of ( F_ 1 ) . n ;
p = |[ p `1 , p `2 ]| .= |[ p `1 , p `2 ]| ;
g * 1_ G = h " * g * g * h .= g " * h ;
let p , q be Element of PFuncs ( V , C ) ;
x0 in dom ( x1 - x2 ) /\ dom ( x1 - x2 ) ;
( R qua Function ) " = R " ( dom R ) .= R " ( rng R ) ;
n in Seg len ( f /^ ( len p -' 1 ) ) ;
for s being Real st s in R holds s <= s2 & s2 <= s1 ;
rng s c= dom ( f2 * f1 ) /\ dom ( f2 * f1 ) ;
synonym ex X being Subset of ex Y being Subset of \mathop { \rm ex X st X in \mathop { \rm ex Y st Y in X & X c= Y
( 1_ K ) * ( 1_ K ) = 1_ K & ( 1_ K ) * ( 1_ K ) = 1_ K ;
set S = Segm ( A , P1 , Q1 ) , Q1 = Segm ( A , B , Q1 ) ;
ex w st e = ( w - f ) . ( w - f ) & w in F ;
curry ' ( P+* ( k , X ) ) # x is convergent ;
cluster -> open for Subset of [: T , T :] | [: S , T :] ;
len f1 = 1 .= len ( f1 | 1 ) .= len ( f1 | 1 ) .= 1 ;
( i * p ) / p < ( 2 * p ) / p ;
let x , y be Element of \mathop { \rm OSSub ( U0 ) ;
b1 , c1 // b9 , c9 & c1 , c1 // b9 , c9 ;
consider p being element such that c1 . j = { p } ;
assume that f " { 0 } = {} and f is total and f is total ;
assume that IC Comput ( F , s , k ) = n and IC Comput ( F , s , k ) = n ;
Reloc ( J , card I + 1 ) is not empty ;
\frac { ( card I + 1 ) * 1 } not f in { c } ;
set m3 = LifeSpan ( p3 , s3 ) , P4 = LifeSpan ( p3 , s3 ) , P4 = LifeSpan ( p3 , s3 ) , P4 = LifeSpan ( p3 , s3 ) , P4 = LifeSpan ( p3 , s3 ) , P4 = LifeSpan ( p3 , s3 )
IC Comput ( P , s , k ) in dom Initialize ( ( intloc 0 ) .--> 1 ) ;
dom t = the carrier of ( SCM R ) | ( the carrier of R ) .= the carrier of ( R ) | ( the carrier of R ) ;
( ( N-min L~ f ) .. f ) .. f = 1 & ( N-min L~ f ) .. f = 1 ;
let a , b be Element of PFuncs ( V , C ) ;
Cl ( union F ) c= Cl ( Cl union F ) ;
the carrier of X1 union X2 misses ( A1 \/ A2 ) \/ ( A1 \/ A2 ) ;
assume not LIN a , f . ( a , f . a ) , g ;
consider i being Element of M such that i = d6 and i in A ;
then Y c= { x } or Y = {} or Y = { x } ;
M , v |= H1 / ( ( y , v ) . x ) ;
consider m being element such that m in Intersect ( F9 . 0 ) and x = ( Intersect ( F9 . 0 ) ) . m ;
reconsider A1 = support u1 as Subset of X | ( { x } ) ;
card ( A \/ B ) = k-1 + ( 2 * 1 ) ;
assume that a1 <> a3 and a2 <> a4 and a3 <> a4 and a4 <> a5 ;
cluster s -\mathop { V } -> .| for string of S ;
LG2 /. ( n2 + 1 ) = LG2 . ( n2 + 1 ) .= LG2 /. ( n2 + 1 ) ;
let P be compact connected non empty Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
assume that r-7 in LSeg ( p1 , p2 ) and rp2 in LSeg ( p1 , p2 ) ;
let A be non empty compact Subset of TOP-REAL n , p be Point of TOP-REAL n ;
assume that [ k , m ] in Indices ( D1 | j1 ) and m in dom D1 ;
0 <= ( ( 1 / 2 ) to_power p ) . ( p / 2 ) ;
( F . N ) | E8 . x = +infty ;
pred X c= Y & Z c= V & X \ V c= Y \ Z ;
( y `2 ) * ( z `2 ) <> 0. I & ( y `2 ) * ( z `1 ) <> 0. I ;
1 + card ( X \/ Y ) <= card u + card ( X \/ Y ) ;
set g = z \circlearrowleft ( ( N-min L~ z ) .. z ) ;
then k = 1 & p . k = <* x , y *> . k ;
cluster -> total for Element of C -\hbox { - } \rm \hbox { - } \rm \hbox { - } \rm such that for Function of C , D ;
reconsider B = A as non empty Subset of TOP-REAL n , p be Point of TOP-REAL n ;
let a , b , c be Function of Y , BOOLEAN ;
L1 . i = ( i .--> g ) . i .= g . i .= f . i .= g . i ;
Plane ( x1 , x2 , x3 , x4 , x5 , v2 , P ) c= P ;
n <= indx ( D2 , D1 , j1 ) + 1 & n <= len D1 ;
( ( ( g2 ) . O ) `1 ) ^2 = - 1 ;
j + p .. f -' len f <= len f - 1 + 1 ;
set W = W-bound C , S = S-bound C , E = E-bound C , N = E-bound C , S = E-bound C , N = E-bound C , S = E-bound C , S = E-bound C , N = E-bound C , S = E-bound C , N = E-bound
S1 . ( a ` , e ) = a + e .= a ` ;
1 in Seg width ( M * ( ColVec2Mx p ) ) ;
dom ( i (#) Im ( f ) ) = dom ( Im ( f ) ) .= dom ( Im ( f ) ) ;
Diff . ( x `1 , x `2 ) = W . ( a , p `2 ) ;
set Q = contradiction \ contradiction ( g , f , h ) ;
cluster -> many sorted for Relation of U1 , U2 ;
attr F = { A } means : Def6 : F is discrete ;
reconsider z9 = *> . y as Element of product ( G . i ) ;
rng f c= rng ( f1 + f2 ) \/ rng ( f1 + f2 ) ;
consider x such that x in f .: A and x in f .: C ;
f = <*> ( the carrier of F_Complex ) & ( the carrier of F_Complex ) c= ( the carrier of F_Complex ) ;
E , j |= All ( x1 , x2 , x3 , x4 ) ;
reconsider n1 = n as Morphism of o1 , o2 ;
assume that P is idempotent and R is idempotent and P ** R = R ** P ;
card ( B2 \/ { x } ) = k-1 + 1 ;
card ( ( x \ B1 ) /\ ( B \ B1 ) ) = 0 ;
g + R in { s : g-r - s < s & s < g + r } ;
set q00 = ( q , <* s *> ) -\hbox { - } ;
for x being element st x in X holds x in rng f1 & x in X
h0 /. ( i + 1 ) = h0 . ( i + 1 ) ;
set mw = max ( B , min ( B , R ) ) ;
t in Seg width ( I ^ ( n , n ) ) & t in Seg n ;
reconsider X = dom f /\ C as Element of Fin NAT ;
IncAddr ( i , k ) = <% x %> + k .= x ;
( ( GoB f ) * ( 1 , 1 ) ) `2 <= ( ( GoB f ) * ( 1 , 1 ) ) `2 ;
attr R is condensed means : Def6 : ( Cl R is condensed & Cl R is condensed & Cl R is condensed ) ;
pred 0 <= a & a <= 1 & b <= 1 & a * b <= 1 ;
u in ( ( c /\ ( ( d /\ b ) /\ e ) ) /\ f ) /\ j ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) /\ f ) /\ j ) /\ j ;
len C + - 2 >= 9 + ( - 3 ) + ( - 3 ) ;
x , z , y is_collinear & x , z , y is_collinear implies x = y
a |^ ( n1 + 1 ) = a |^ ( n1 + 1 ) * a ;
<* \underbrace ( 0 , \dots , 0 } , x ) in Line ( x , a * x ) ;
set y1 = <* y , c *> ;
FG2 /. 1 in rng Line ( D , 1 ) & FG2 /. 1 in rng Line ( D , 1 ) ;
p . m > r /. m & r /. ( m + 1 ) in rng r ;
( p `2 ) ^2 = ( f /. ( i1 + 1 ) ) `2 .= ( f /. ( i1 + 1 ) ) `2 ;
( W-bound X \/ Y ) = W-bound X + ( W-bound Y ) / 2 & ( W-bound X + E-bound Y ) = W-bound Y ;
0 + ( p `2 ) ^2 <= 2 * r + ( p `2 ) ^2 ;
x in dom g & not x in g " { 0 } implies x in dom g & g . x in dom g
f1 /* ( seq ^\ k ) is divergent_to+infty & f2 /* ( seq ^\ k ) is divergent_to+infty ;
reconsider u2 = u as VECTOR of Pmin ( X , Y ) , f be VECTOR of X ;
p \! \mathop { Product ( X , Y ) } = 0 & p in ( X * Y ) " { 0 } ;
len <* x *> < i + 1 & i + 1 <= len c + 1 ;
assume that I is non empty and { x } /\ { y } = { 0. I } ;
set ii = ( card I + 4 ) .--> ( 0 + 4 ) , i2 = goto 0 ;
x in { x , y } & h . x = {} T & y in T ;
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 ) ) ;
reconsider m = M , i = I , n = N as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . j ;
set NN = : \leq |. GN - GN .| ;
rng F c= the carrier of gr { a } & rng F c= the carrier of gr { a } ;
( \vert p2 \vert ) . ( K . ( K , n , r ) , n ) is min ;
f . k , f . ( Let ( n ) , n ) in rng f ;
h " P /\ [#] T1 = f " P & h " P = f " P ;
g in dom ( f2 \ f2 " { 0 } ) \ f2 " { 0 } ;
gN /\ dom f1 = g1 " ( { g } ) .= g1 " ( { g } ) ;
consider n being element such that n in NAT and Z = G . n ;
set d1 = c9 ( x1 , y1 , y2 ) , d2 = c9 ( y1 , y2 , z2 ) , d1 = h . ( y1 , y2 ) , d2 = h . ( y1 , y2 ) , d1 = h . ( y1 , y2 ) , d2 = h . ( y1 ,
b `1 + ( 1 - sqrt 2 ) < ( 1 - sqrt 2 ) ^2 + ( 1 - sqrt 2 ) ^2 ;
reconsider f1 = f as VECTOR of the carrier of X , g be Function of X , Y ;
pred i <> 0 means : Def7 : i ^2 mod ( i + 1 ) = 1 ;
j2 in Seg len ( ( g2 . i2 ) * ( ( g2 . i2 ) * ( 1 , j ) ) ) ;
dom ( i - 1 ) = dom ( i - 1 ) .= Seg ( i - 1 ) .= Seg ( i + 1 ) ;
cluster sec | ]. PI / 2 , PI / 2 .[ -> one-to-one ;
Ball ( u , e ) = Ball ( f . p , e ) ;
reconsider x1 = x0 as Function of S , IF , I be Function of S , IF ;
reconsider R1 = x , R2 = y , R2 = x as Relation of L , L ;
consider a , b being Subset of A such that x = [ a , b ] ;
( <* 1 *> ^ p ) ^ <* n *> in RL & <* n *> in RL ;
S1 +* S2 = S2 +* S1 & S2 +* S2 = S2 +* S2 & S1 +* S2 = S2 +* S2 ;
( ( ( id Z ) (#) ( ( id Z ) ^ ) ) `| Z ) = f ;
cluster -> continuous for Function of C , REAL n , x be element ;
set C7 = 1GateCircStr ( <* z , x *> , f3 ) , C7 = 1GateCircStr ( <* z , x *> , f3 ) ;
E8 . e2 = E8 . ( e2 . e2 ) .= ( 2 to_power ( e2 + 1 ) ) * T ;
( ( arctan * arctan ) `| Z ) . x = ( ( arctan * arccot ) `| Z ) . x .= ( ( arctan * arccot ) `| Z ) . x ;
upper_bound A = ( cos * 3 ) / 2 & lower_bound A = 0 & lower_bound A = 0 ;
F . ( dom f , - F ) is_transformable_to F . ( cod f , - F ) ;
reconsider pp = q8 as Point of Euclid 2 , q = q `2 / |. q .| as Point of TOP-REAL 2 ;
g . W in [#] ( Y | X0 ) & [#] ( Y | X0 ) c= [#] ( Y | X0 ) ;
let C be compact connected non vertical non vertical non horizontal Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) \/ LSeg ( g , k ) ;
rng s c= dom f /\ ]. x0 - r , x0 .[ & rng s c= dom f /\ ]. x0 - r , x0 + r .[ ;
assume x in { idseq ( 2 ) , Rev ( idseq 2 ) } ;
reconsider n2 = n , m2 = m , n1 = n as Element of NAT ;
for y being ExtReal st y in rng seq holds g . y <= g . x
for k st P [ k ] holds P [ k + 1 ] ;
m = m1 + m2 .= m1 + m2 .= m1 + m2 .= m1 + m2 + m2 .= m1 + m2 + m2 ;
assume for n holds H1 . n = G . n -H . n ;
set Bx = f .: ( the carrier of X1 ) , Bx = f .: ( the carrier of X2 ) ;
ex d being Element of L st d in D & x << d ;
assume that R ~ c= R ~ and R ~ c= R ~ and R ~ c= R ~ and R ~ c= R ~ ;
t in ]. r , s .] or t = r or t = s & s = s ;
z + v2 in W & x = u + ( z + ( z + v2 ) ) ;
x2 |-- y2 iff P [ x2 , y2 , y2 , z2 , z2 , z2 , z2 , z2 , z2 , z2 , z2 , z2 , z2 , z2 , z2 , z2 , z2 , z2 , z2 , z2 , z2 ;
pred x1 <> x2 means : Def21 : |. x1 - x2 .| > 0 & x1 <> x2 ;
assume that p2 - p1 , p3 - p1 , p2 - p1 , p3 - p1 is_collinear and p2 - p1 , p3 - p1 , p3 - p1 , p2 - p1 is_collinear ;
set q = ( Ant f ) ^ <* 'not' 'not' A *> ;
let f be PartFunc of \langle REAL-NS 1 , \Vert * \Vert \rangle , REAL-NS 1 , REAL-NS 1 , REAL-NS 1 ;
( n mod ( 2 * k ) ) + 1 = n mod k ;
dom ( T * ( \mathop { \rm \lbrace t } ) ) = dom ( \mathop { \rm <* t *> *> ) ;
consider x being element such that x in wX and x in c and x in c ;
assume ( F * G ) . ( v . x3 ) = v . x3 ;
assume that the carrier of D1 c= the carrier of D2 and the carrier of D1 = the carrier of D2 and the carrier of D2 = the carrier of D2 ;
reconsider A1 = [. a , b .] as Subset of R^1 | [. a , b .] ;
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 = W-min L~ Cage ( C , n ) , r = W-min L~ Cage ( C , n ) ;
n1 -' len f + 1 - 1 <= len - 1 + 1 - 1 ;
ConsecutiveSet ( q , O1 ) = [ u , v , v , a , b , c , d ] ;
set C-2 = ( ( \mathclose { \mathclose { i } ) \/ G ) . ( k + 1 ) ;
Sum ( L * p ) = 0. R * Sum p .= 0. V .= 0. V ;
consider i being element such that i in dom p and t = p . i ;
defpred Q [ Nat ] means 0 = Q ( $1 ) & $1 <= len Q & Q [ $1 ] ;
set s3 = Comput ( P1 , s1 , k ) , P3 = P1 +* I , s4 = P2 +* I , P4 = P2 +* I , s4 = Comput ( P2 , s2 , k ) , P4 = P2 +* I , P4 = P3 ;
let l be variable of k , A , B be Subset of V ;
reconsider U2 = union G-24 as Subset-Family of [: T , T :] | A , G be Subset-Family of [: T , T :] ;
consider r such that r > 0 and Ball ( p `1 , r ) c= Q ` ;
( h | ( n + 2 ) ) /. ( i + 1 ) = p9 /. ( n + 1 ) ;
reconsider B = the carrier of X1 , C = the carrier of X2 as Subset of X ;
psqrt 5 = <* - ( c * c9 ) , - ( c * c9 ) *> .= <* - ( c * c9 ) *> ;
synonym f is complex-valued for rng f c= NAT & rng f c= NAT & rng f c= { f . n } ;
consider b being element such that b in dom F and a = F . b ;
x9 < card ( X0 \/ ( X0 \/ Y ) ) & card ( Y \/ ( X0 \/ Y ) ) < card ( X0 \/ Y ) ;
attr X c= B1 means : Defp) for X st X c= succ B1 holds X in succ B1 ;
then w in Ball ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( \HM { the } \HM { function } , 0 , \HM { the } \HM { function } )
pred 1 <= len s means : Def8 : len ( s * ( 0 , 0 ) ) = s & for i being Nat st i in dom s holds s . i = 0 ;
fReconsider 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 : Def: q '&' p in TAUT ( A ) & q '&' p in TAUT ( A ) ;
- ( t `1 ) < ( ( t `2 ) - ( t `1 ) ) / ( 1 - t `1 ) ;
U2 . 1 = U2 /. 1 .= ( U2 /. 1 ) .= ( ( B * A ) + ( B * A ) ) . 1 .= ( B * A ) . 1 .= ( B * A ) . 1 .= ( B * A ) . 1 ;
f .: ( the carrier of x ) = the carrier of x & f .: ( the carrier of x ) = the carrier of x ;
Indices ( ( - O ) * ( i , j ) ) = [: Seg n , Seg n :] ;
for n being Element of NAT holds G . n c= G . ( n + 1 ) ;
then V in M ^ \square & ex x being Element of M st V = { x } ;
ex f being Element of F-9 st f is \cup ( A * B ) & ( f is unital implies f is unital ) ;
[ h . 0 , h . 3 ] in the InternalRel of G & [ h . 2 , h . 3 ] in the InternalRel of G ;
s +* Initialize ( ( intloc 0 ) .--> 1 ) = s3 +* Initialize ( ( intloc 0 ) .--> 1 ) ;
|[ w1 , v1 ]| - |[ w1 , v1 ]| <> 0. TOP-REAL 2 & |[ w1 , v1 ]| - |[ w1 , v1 ]| = 0. TOP-REAL 2 ;
reconsider t = t as Element of ( Z , X ) * ;
C \/ P c= [#] ( ( G | [#] ( ( ( G | A ) \ A ) ) \ A ) ;
f " V in ( for X being Subset of [ X , the carrier of S ] ) /\ D ( ) ;
x in [#] ( ( the carrier of X ) /\ A ) /\ ( the carrier of ( X | A ) ) ;
g . x <= h1 . x & h . x <= h1 . x & h . x <= h1 . x ;
InputVertices S = { xy , y , z } \/ { xy , z } ;
for n being Nat st P [ n ] holds P [ n + 1 ] ;
set R = Line ( M , i , a * 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 M1 is being_line and M2 is being_line ;
reconsider a = f0 . i0 -' 1 as Element of K ( ) ;
len B2 = Sum Len ( ( F1 ^ F2 ) ^ <* F1 *> ) .= len F1 + len F2 .= len F2 + len <* F2 *> ;
len ( ( the _ of K ) * ( i , j ) ) = n & ( ( the _ of K ) * ( i , j ) ) = n ;
dom max ( - ( f + g ) , - ( f + g ) ) = dom ( f + g ) ;
( the superior of seq ) . n = upper_bound Y1 & ( the \upharpoonright of seq ) . n = upper_bound Y1 + upper_bound Y2 ;
dom ( p1 ^ p2 ) = dom ( ( f ^ ) | ( dom p2 + 1 ) ) .= dom ( f ^ ) .= dom ( f ^ ) ;
M . [ 1 , y ] = 1 / ( 1 - M ) * v1 .= ( 1 - M ) * v1 .= ( 1 - M ) * v1 ;
assume that W is non trivial and W .vertices() c= the \frac of G2 , { v } } and W c= the \frac { v } , G ;
C6 * ( i1 , i2 ) = G1 * ( i1 , i2 ) & C6 * ( i1 , i2 ) = G2 * ( i1 , i2 ) ;
C8 |- 'not' Ex ( x , p ) 'or' p . ( x , y ) ;
for b st b in rng g holds lower_bound rng f"\/" f"\/" b <= b - 1
- ( ( q `1 ) / |. q .| - sn ) = 1 - ( ( q `1 ) / |. q .| ) ;
( LSeg ( c , m ) \/ [: l , k :] ) \/ LSeg ( l , k ) c= R ;
consider p being element such that p in LSeg ( x , p ) and p in L~ f and x = f /. p ;
Indices ( X @ ) = [: Seg n , Seg n :] & dom ( X @ ) = Seg n ;
cluster s => ( q => p ) => ( q => ( s => p ) ) -> valid ;
Im ( ( Partial_Sums F ) . m ) is_measurable_on E & ( Partial_Sums F ) . m is_measurable_on E ;
cluster f . ( x1 , x2 ) -> Element of D ( ) ;
consider g being Function such that g = F . t and Q [ t , g . t ] ;
p in LSeg ( ( N-min L~ Cage ( C , n ) ) , ( NW-corner L~ Cage ( C , n ) ) ) ;
set R8 = R ^ 1 \ ]. b , +infty .[ ;
IncAddr ( I , k ) = SubFrom ( da , db ) .= I . ( da + k ) ;
seq . m <= ( ( the Sorts of seq ) . m ) . ( ( seq ^\ k ) . n ) ;
a + b = ( a ` *' ) *' ( b ` *' ) .= ( a ` *' ) *' ( b ` *' ) .= ( a ` *' ) *' ( b ` *' ) ;
id ( X /\ Y ) = id ( X /\ Y ) /\ id ( X /\ Y ) ;
for x being element st x in dom h holds h . x = f . x ;
reconsider H = U1 \/ U2 as non empty Subset of U0 ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) /\ f ) /\ j ) /\ m /\ m ;
consider y being element such that y in Y and P [ y , lower_bound B ] ;
consider A being finite stable set of R such that card A = ( R * A ) and for x being set st x in A holds x in A ;
p2 in rng ( f |-- p1 ) \ rng <* p1 *> & not p2 in rng <* p1 *> ;
len s1 - 1 > 0 & len s2 - 1 > 0 & len s1 - 1 > 0 ;
( ( N-min L~ Cage ( C , n ) ) .. Cage ( C , n ) ) `2 = ( N-min L~ Cage ( C , n ) ) `2 ;
Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) \/ LeftComp Cage ( C , k + 1 ) ;
f . a1 ` ` = f . ( a1 ` ` ` ) .= f . ( a1 ` ` ) .= f . ( a1 ` ) .= ( f | ( a1 ` ) ) ` ;
( seq ^\ k ) . n in ]. x0 - r , x0 + r .[ & ( seq ^\ k ) . n in ]. x0 - r , x0 + r .[ ;
gg . seq = g . ( ( seq . 0 ) | G . ( seq . n ) ) .= g . ( seq . n ) ;
the InternalRel of S is If & ( the InternalRel of S ) \/ ( the InternalRel of S ) is If & the InternalRel of S is thesis
deffunc F ( Ordinal , Ordinal ) = phi . ( $1 , $2 ) ;
F . ( s1 . a1 ) = F . ( s2 . a1 ) .= F . ( s2 . a1 ) ;
x `1 = A . ( o . a ) .= Den ( o , A . a ) ;
Cl ( f " P1 ) c= f " ( ( Cl P1 ) \/ ( Cl P1 ) ) ;
FinMeetCl ( ( the topology of S ) \/ ( the topology of T ) ) c= the topology of T ;
synonym o is " means : Def11 : o <> \ast & o <> * & o <> * ;
assume that X = Y + 1 and card X = card Y and Y <> {} and X <> {} ;
the Lin of s <= 1 + ( the Lin of s ) . ( len s ) & ( the Lin of s ) . ( len s ) = ( the Lin of s ) . ( len s ) ;
LIN a , a1 , d or b , c // b1 , c1 or b , c // b1 , c1 ;
e . 1 = 0 & e . 2 = 1 & e . 3 = 0 & e . 3 = 0 ;
EE in S1 & not EE in SE & not EE in { NE } ;
set J = ( l , u ) \mathop { I } ;
set A1 = qua non empty set , p = [ <* A1 , cin *> , '&' ] , A2 = [ <* A1 , cin *> , '&' ] ;
set c9 = [ <* c9 , cin *> , '&' ] , A2 = [ <* c , d *> , '&' = [ <* d , c *> , '&' ] , \lbrack c , d *> , '&' = [ <* c , d *> , '&' ] ;
x * z * x * x " in x * ( z * N ) * ( x * N ) " ;
for x being element st x in dom f holds f . x = f3 . x & f . x = g2 . x ;
Int cell ( f , 1 , G ) c= RightComp f \/ RightComp f \/ L~ f \/ L~ f \/ L~ f ;
U2 is_an_arc_of W-min ( C ) , W-min ( C ) , W-min ( C ) , W-min ( C ) , W-min ( C ) ;
set f-17 = f @ "/\" @ g @ ;
attr S1 is convergent & S2 is convergent means : Def21 : for n holds ( S1 - S2 ) . n = ( S1 - S2 ) . n ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a + ( 0 qua Ordinal ) .= a ;
cluster be such that for { reflexive , transitive } -symmetric RelStr holds the InternalRel of ( the InternalRel of C ) -symmetric is reflexive
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 ) \/ dom Start-At ( ( card I + 2 ) , SCMPDS ) ;
( z + a ) + x = z + ( a + y ) .= z + ( a + y ) ;
len ( l \lbrack ( a , A ) |^ 0 ) = len l .= len l ;
t4 \/ {} is ( {} \/ rng t4 ) -valued FinSequence of ( {} \/ rng t4 ) * * , ( {} , rng t4 ) * ) -valued FinSequence ;
t = <* F . t *> ^ ( C . p ) ^ ( C . q ) .= ( C . p ) ^ ( C . q ) ;
set pp = W-min L~ Cage ( C , n ) , p = W-min L~ Cage ( C , n ) , q = W-min L~ Cage ( C , n ) , r = W-min L~ Cage ( C , n ) ;
( k -' ( i + 1 ) ) = ( k - i ) + ( k - 1 ) ;
consider u being Element of L such that u = u ` ` and u in D and u in D ` ;
len ( ( width ( ( ( ( ( B ) ) |-> a ) ) ^ <* a *> ) ^ <* b *> ) ) = width ( ( ( A * B ) ^ <* a *> ) ^ <* b *> ) ;
FM . x in dom ( ( G * the_arity_of o ) . x ) ;
set H2 = the carrier of H2 , H1 = the carrier of H2 , H2 = the carrier of H2 ;
set H1 = the carrier of H1 , H2 = the carrier of H2 , H2 = the carrier of H2 ;
( Comput ( P , s , 6 ) ) . intpos m = s . intpos m .= s . intpos m ;
IC Comput ( Q2 , t , k ) = ( l + 1 ) + 1 .= ( l + 1 ) ;
dom ( ( - 1 ) (#) ( ( id Z ) * ( ( id Z ) * ( ( id Z ) * ( id Z ) * ( id Z ) ) ) ) = REAL ;
cluster <* l *> ^ phi -> ( 1 + 1 ) -element for string of S ;
set b5 = [ <* A1 , cin *> , '&' ] , b5 = [ <* A1 , cin *> , '&' ] , b5 = [ <* A1 , cin *> , '&' ] ;
Line ( Segm ( M , P , Q ) , x ) = L * Sgm Q .= ( Sgm Q ) . x ;
n in dom ( ( the Sorts of A ) * ( the_arity_of o ) ) & ( ( the Sorts of A ) * ( the_arity_of o ) ) . n = ( the Sorts of A ) . ( the_arity_of o ) ;
cluster f1 + f2 -> continuous for PartFunc of REAL , REAL n , x be Point of S ;
consider y being Point of X such that a = y and ||. y - x .|| <= r ;
set x3 = t2 . DataLoc ( s2 . SBP , 2 ) , x2 = s2 . DataLoc ( s2 . SBP , 2 ) , x3 = s2 . DataLoc ( s2 . SBP , 2 ) , x4 = s2 . DataLoc ( s2 . SBP , 2 ) ;
set pp = stop I , p1 = P +* stop I , p2 = Comput ( P2 , s2 , 1 ) , p2 = P +* stop I ;
consider a being Point of D2 such that a in W1 and b = g . a and a in W2 and b in W2 ;
{ A , B , C , D } = { A , B , C , D , E , F , J , M , N , N , F , M , N , N , F , M , N , N , F , M , N , N , F , M , N , N , F , M ;
let A , B , C , D , E , F , J , M , N , N , F , J , M , N , N , F , M , N , N , N , F , J , M , N , N , N , F , M , N , N , F , M , N , N ,
|. p2 .| ^2 - ( ( p2 `2 ) ) ^2 >= 0 & ( ( p2 `2 ) ) ^2 / ( 1 + ( p2 `1 ) ) ^2 >= 0 ;
l - 1 + 1 = n-1 * ( l1 + 1 ) + ( mm - 1 ) ;
x = v + ( a * w1 + b * w2 ) + ( c * w2 ) .= c * ( ( c * w2 ) + ( c * w2 ) ) ;
the TopStruct of L = TopSpaceMetr ( the Scott Scott functor of L ) & the TopStruct of L = TopSpaceMetr ( the topology of L ) ;
consider y being element such that y in dom H1 and x = H1 . y and y in H1 . x ;
fg \ { n } = ( Free All ( v1 , H ) ) \/ ( Free ( H1 , H ) ) ;
for Y being Subset of X st Y is summable & Y is summable holds Y is \overline Y is \overline of Y
2 * n in { N : 2 * Partial_Sums ( p | N ) = N & N > 0 } ;
for s being FinSequence holds len ( the _ of K ) = len s & len ( the _ of K ) = len s & len ( the _ of K ) = len s
for x st x in Z holds exp_R * f is_differentiable_in x & exp_R * f is_differentiable_in x & f . x > 0
rng ( ( h2 * f2 ) | X ) c= the carrier of ( ( h1 * f2 ) | X ) .= the carrier of ( ( h1 * f2 ) | X ) ;
j + ( - len f ) <= len f + ( len f - 2 ) - ( len f - 2 ) ;
reconsider R1 = R * I as PartFunc of REAL , \langle REAL-NS n *> , REAL-NS n , REAL-NS n ;
C8 . x = s1 . ( ( a + b ) / 2 ) .= s1 . ( ( a + b ) / 2 ) .= s1 . ( ( a + b ) / 2 ) .= s1 . x ;
( power F_Complex ) . ( z , n ) = 1 .= ( x |^ n ) * ( x |^ n ) .= ( x |^ n ) * ( x |^ n ) ;
t at ( C , s ) = f . ( ( the connectives of S ) . o ) .= f . ( ( the connectives of S ) . o ) ;
( support f + g ) c= ( support f \/ { C } ) \/ ( { C } \/ { D } ) ;
ex N st N = j1 & 2 * Sum ( ( r | N ) | N ) > N & N > 0 ;
for y , p st P [ p ] holds P [ All ( y , p ) ]
{ [ x1 , x2 ] where x1 , x2 is Point of [: X , Y :] : x1 in X & x2 in Y } is Subset of [: X , Y :] ;
h = ( i , j ) |-- ( id B ) . i .= H . i .= H . i ;
ex x1 being Element of G st x1 = x & x1 * N c= A & x1 in A & x1 in A ;
set X = ( ( |. d ( q , O1 ) ) . ( 1 + 4 ) ) `1 , Y = ( ( d ( q , O1 ) ) . 2 ) `1 , Z = { [ q , p ] } ;
b . n in { g1 : x0 < g1 & g1 < x0 & g1 < x0 + r } ;
f /* s1 is convergent & f /. x1 = lim ( f /* s1 ) & lim s1 = lim ( f /* s1 ) ;
the lattice of T = the lattice of the lattice of Y & the carrier of T = the carrier of X & the carrier of T = the carrier of X ;
'not' ( a . x ) '&' b . x 'or' a . x '&' 'not' ( b . x ) = TRUE ;
2 = len ( ( q ^ <* r1 *> ) + ( len <* r1 *> ) ) + len ( <* r1 *> ) .= len ( ( q ^ <* r1 *> ) ^ <* r1 *> ) ;
( 1 / a ) * ( sec * f1 ) - id Z is_differentiable_on Z ;
set K1 = upper ( ( lim ( H , A ) || H ) , ( lim ( H , A ) `| H ) ) ;
assume that e in { ( ( w1 + w2 ) - ( w1 + w2 ) ) / 2 : w1 in F & w2 in G & w2 in G } ;
reconsider d7 = dom a `1 , d6 = dom F , d6 = dom F , d6 = dom G as finite set ;
LSeg ( f /^ q , j ) = LSeg ( f , j ) \/ LSeg ( f , j + q .. f -' 1 ) ;
assume X in { T . ( N2 , K ) : h . ( N2 , K ) = N2 & h . ( N2 , K ) = K } ;
assume that Hom ( d , c ) <> {} and <* f , g *> * f1 = <* f , g *> * f1 ;
dom S29 = dom S /\ Seg n .= dom S /\ Seg n .= dom ( L | n ) .= dom ( L | n ) .= dom ( L | n ) .= dom ( L | n ) ;
x in ( H |^ a ) implies ex g st x = g |^ a & g in H |^ b
a * ( ( n , 1 ) --> ( a , 1 ) ) = a ` - ( 0 * n ) .= a ` - ( 0 * n ) .= a ` ;
D2 . j in { r : lower_bound A <= r & r <= D1 . i & D1 . i <= r } ;
ex p being Point of TOP-REAL 2 st p = x & P [ p ] & p <> 0. TOP-REAL 2 ;
for c holds f . c <= g . c implies f @ c ^ @ g ;
dom ( f1 (#) f2 ) /\ X c= dom ( f1 (#) f2 ) /\ X .= dom ( f1 (#) f2 ) /\ X .= dom ( f1 (#) f2 ) /\ X .= dom ( f1 (#) f2 ) /\ X .= dom ( f1 (#) f2 ) /\ X .= dom ( f1 (#) f2 ) /\ X .= dom ( f1 (#) f2 ) /\ X ;
1 = ( p * p ) * p .= p * ( p * ( 1 , 1 ) ) .= p * 1 .= p * 1 ;
len g = len f + len <* x + y *> .= len f + len <* y *> .= len f + 1 .= len f + 1 ;
dom ( F | [: N1 , S :] ) = dom ( F | [: N1 , S :] ) .= dom ( F | [: N1 , S :] ) .= dom ( F | [: N1 , S :] ) ;
dom ( f . t * I . t ) = dom ( ( f . t ) * g . t ) ;
assume a in ( "\/" ( ( T |^ \alpha ) , F ) ) .: D ) .: D & ( the carrier of S ) .: D in the carrier of S ;
assume that g is one-to-one and ( the carrier' of S ) /\ rng g c= dom g and g is one-to-one ;
( ( x \ y ) \ z ) \ ( ( x \ z ) \ ( y \ z ) ) = 0. X ;
consider f such that f * f `1 = id b & f * f = id b and f is one-to-one ;
( ( cos * cos ) | [. 2 * PI , 0 + 1 .] ) | [. 2 , 0 + 1 .] is increasing ;
Index ( p , co ) <= len LS - Index ( Gij , LS ) + 1 - Index ( Gij , LS ) - 1 ;
t1 , t2 , t2 , t1 , t2 , t2 , t1 , t2 , t2 , t1 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 ;
( ( Frege ( H ) ) . h ) . h <= ( ( Frege ( H ) ) . h ) . ( j + 1 ) .= ( ( Frege ( H ) ) . j ) . ( j + 1 ) ;
then P [ f . i0 , f . i0 ] means F ( f . i0 , f . i0 ) < j & F ( f . i0 , f . i0 ) < j ;
Q [ ( [ D . ( [ D . x , 1 ] ) , F . [ D . ( D . x , 1 ] ) ] ) ;
consider x being element such that x in dom ( F . s ) and y = ( F . s ) . x ;
l . i < r . i & [ l . i , r . i ] is for of G . i , G . i ;
the Sorts of A2 = ( the carrier of S2 ) --> TRUE .= ( the carrier of S2 ) --> TRUE .= ( the carrier of S2 ) --> TRUE .= ( the carrier of S2 ) --> TRUE ;
consider s being Function such that s is one-to-one & dom s = NAT & rng s c= F and rng s c= F ;
dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( a , b ) + dist ( a , b ) ;
( Lower_Seq ( C , n ) ) /. len ( Cage ( C , n ) ) = W /. len ( Cage ( C , n ) ) ;
q `2 <= ( ( UMP L~ Cage ( C , 1 ) ) * ( ( UMP L~ Cage ( C , 1 ) ) `1 ) ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} ;
given a being ExtReal such that a <= Ir and A = ]. a , I1 .[ and a in A and b in B ;
consider a , b being complex number such that z = a & y = b & z + y = a + b ;
set X = { b |^ n where b is Element of NAT : b in n & b in n } , Y = { b } ;
( ( x * y * z \ x ) \ z ) \ ( x * y * z \ x ) = 0. X ;
set xy = [ <* xy , \mathopen { x , y } , f1 ] , yz = [ <* y , z *> , f2 ] , zx = [ <* z , x *> , f3 = [ <* y , z *> , f3 ] , f4 = [ <* z , x *> , f3 = [ <* y , z *> , f3 ] ;
lk /. len ( lk ) = ( lk /. len ( lk ) ) .= ( ( k + 1 ) + 1 ) * ( ( k + 1 ) + 1 ) ;
( ( q `2 ) / |. q .| - sn ) / ( 1 + sn ) = 1 ;
( ( p `2 ) - ( |. p .| ) ) ^2 - ( ( p `2 ) ) ^2 < 1 - ( ( p `2 ) ) ^2 ;
( ( ( S \/ Y ) \/ { x } ) \/ { x } ) `2 = ( ( S \/ Y ) \/ { x } ) `2 ;
( s1 - s1 ) . k = s1 . k - s1 . k .= s1 . k - s1 . k .= s1 . k - s1 . k .= ( s1 - s1 ) . k ;
rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , u0 ) /\ dom SVF1 ( 1 , f , u0 ) ;
the carrier of X = the carrier of ( the carrier of X ) & the carrier of X = the carrier of ( X ) ;
ex p3 st p3 = p4 & |. p3 - |[ a , b ]| .| = r & |. p3 - |[ a , b ]| .| = r ;
set [: h , g :] = [: A , B :] , [: A , B :] , [: A , B :] :] ;
R / ( 0 * n ) = \mathop { I*> ( X , X ) .= R / ( n * n ) .= R / ( n * n ) ;
( Partial_Sums ( ( curry ( F ) ) . n ) ) . ( ( Partial_Sums ( F ) ) . n ) is nonnegative & ( Partial_Sums ( ( F ) ) . n ) . ( ( Partial_Sums ( F ) ) . m ) is nonnegative ;
f2 = C7 . ( ( E8 . ( V , K , len V ) ) * ( V , len 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 , p11 ) \/ LSeg ( p1 , p10 ) /\ LSeg ( p1 , p10 ) ;
dom ( f . t ) = Seg n & dom ( I . t ) = Seg n & dom ( I . t ) = Seg n ;
assume o = ( the connectives of S ) . 11 & 11 in ( the carrier' of S ) . 12 ;
set phi = ( l1 , l2 ) \mathop { l1 } , phi = ( l1 , l2 ) \mathop { l1 } ;
synonym p is invertible for p is invertible for ( p , T ) * ( p , T ) = 1 / p * ( p , T ) ;
( Y1 `2 ) ^2 = - 1 & ( 0. TOP-REAL 2 ) = ( - 1 ) * ( ( - 1 ) * ( 1 + ( sn - sn ) ) ) & ( - 1 ) * ( 1 + sn ) = ( - 1 ) * ( ( - 1 ) * ( 1 + sn ) ) ;
defpred X [ Nat , set , set , set , set , set , set , set , set , set , $2 = $2 $2 $2 $2 = $2 & $2 = $2 ;
consider k being Nat such that for n being Nat st k <= n holds s . n < x0 + g and g in dom f ;
Det ( I ^ ( m -' n ) , ( m -' n ) * ( m -' n ) ) = ( 1_ K ) * ( m -' n ) .= ( 1_ K ) * ( m -' n ) ;
( - b - sqrt ( b ^2 - 4 * a * c ) ) / 2 * a < 0 ;
CC . d = CC . ( dC . d ) mod CC . ( dC . d ) .= CC . ( CC . d ) mod C . ( CC . d ) ;
attr X1 is dense dense means : DefLet : X1 is dense dense dense & X2 is dense dense dense SubSpace of X ;
deffunc F6 ( Element of E , Element of I , Element of E ) = $1 * $2 + ( $1 * $2 ) ;
t ^ <* n *> in { t ^ <* i *> : Q [ i , T . ( i + 1 ) ] } ;
( x \ y ) \ x = ( x \ x ) \ y .= y \ 0. X .= 0. X ;
for X being non empty set for X being Subset-Family of [: X , product <* Y *> :] holds X is Basis of [: X , product <* Y *> :]
synonym A , B , C , A , B , C , D , E , F , J , M , N , N , F , M , N , N , N , F , M , N , N , F , J , M , N , N , M , N , N , F , M , N , N , F , J , M , N , N , N , M ;
len ( M @ ) = len p & width ( M @ ) = width p & width ( M @ ) = width p & width ( M @ ) = width p ;
J = { v where v is Element of K : 0 < v & v < 1 } ;
( Sgm ( Seg m ) ) . d - ( Sgm ( Seg m ) ) . e <> 0 ;
lower_bound divset ( D2 , k + 1 ) = D2 . ( k + 1 - 1 ) .= D2 . ( k + 1 - 1 ) .= D2 . ( k + 1 - 1 ) ;
g . r1 = ( 2 * r1 ) * r1 + 1 & dom h = [. 0 , 1 .] & dom h = [. 0 , 1 .] ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `2 & g . x = ( h . x ) `2 ;
ex w st w in dom B1 & <* 1 *> ^ s = <* 1 *> ^ w & <* 1 *> ^ w = <* 1 *> ^ w ;
[ 1 , {} , <* d1 *> ] in ( { [ 0 , {} , {} ] } \/ ( { [ 0 , {} , {} ] } \/ { [ 1 , {} , {} ] } ) ;
IC Exec ( i , s1 ) + n = IC Exec ( i , s2 ) + n .= IC Exec ( i , s2 ) + n .= ( n + 1 ) ;
IC Comput ( P , s , 1 ) = IC Comput ( P , s , 1 ) .= ( 5 + 1 ) .= 5 + 1 .= 5 ;
( IExec ( W6 , Q , t ) ) . intpos ( 8 + 1 ) = t . intpos ( 8 + 1 ) ;
LSeg ( f /^ ( q -' i + 1 ) , i ) misses LSeg ( f /^ ( q -' i + 1 ) , j ) ;
assume for x , y being Element of L st x in C & y in C holds x <= y or x <= y ;
\displaystyle { \int \limits ( C , f ) . x } = f . ( upper_bound C ) - f . ( lower_bound C ) ;
for F , G being one-to-one FinSequence st rng F misses rng G & rng F misses rng G holds F ^ G is one-to-one
||. R /. ( L . h ) - R /. ( L . h ) .|| < e1 * ( K + 1 ) * ( K + 1 ) ;
assume a in { q where q is Element of M : dist ( z , q ) <= r & q <= p & q <= p } ;
set p4 = [ 2 , 1 ] .--> [ 2 , 0 , 1 ] ;
consider x , y being Subset of X such that [ x , y ] in F and x c= d and y c= d ;
for y , x being Element of REAL st y ` in Y & x in X & y in Y holds y ` <= x ` ;
func |. p : p ^ <* p *> -> variable of A , p ^ <* p *> ) = ( p ^ <* NBI *> ) . 1 ;
consider t being Element of S such that x ` , y ` , z , t is_collinear and x , y , t is_collinear and x , y , t is_collinear ;
dom x1 = Seg len ( x1 ^ y1 ) & len y1 = len ( x1 ^ y1 ) & len x1 = len ( x1 ^ y1 ) + len ( y1 ^ y2 ) ;
consider y2 being Real such that x2 = y2 and 0 <= y2 and y2 <= 1 and y2 <= 1 and y2 <= 1 and y2 <= 1 ;
||. f | X /* s1 .|| = ||. f | X .|| | ( X /\ dom s1 ) .= ||. f /. s1 .|| ;
( the InternalRel of A ) ~ \ ( x ` \ x ` ) = {} \/ {} .= {} \/ {} .= {} \/ {} .= {} ;
assume that i in dom p and for j being Nat st j in dom q holds P [ i , j ] and for i st i in dom p holds P [ i , j ] ;
reconsider h = f | [: X , Y :] as Function of [: X , Y :] , [: X , Y :] ;
u1 in the carrier of W1 & u2 in the carrier of W2 implies ( ( the carrier of W1 ) + ( the carrier of W2 ) ) = the carrier of W2
defpred P [ Element of L ] means M <= f . $1 & f . $1 <= f . $1 & f . $1 <= f . $1 ;
l . ( u , a , v ) = s * x + ( - ( s * x ) + y ) .= b * x + y .= b ;
- ( y - x ) = - x + ( - y ) .= - x + ( - y ) .= - x + ( - y ) .= - x + y ;
given a being Point of Gs such that for x being Point of Gs holds a , x , a , x , y is_collinear ;
fY. = [ [ dom @ ( @ ( @ ( @ f2 ) ) , cod ( @ f2 ) ) , cod ( @ f2 ) ] , fY. = [ @ ( @ f2 ) , @ ( @ f2 ) ] ;
for k , n being Nat st k <> 0 & k < n & k < n holds ( k |^ n ) divides ( k |^ n )
for x being element holds x in A |^ d iff x in ( ( A ` ) |^ d ) ` & x in ( ( A ` ) |^ d ) `
consider u , v being Element of R , a being Element of A such that l /. i = u * a * v ;
- ( ( p `1 / |. p .| - cn ) / ( 1 + cn ) ) ^2 > 0 ;
L-13 . k = Lt . ( F . k ) & F . k in dom ( Lt . k ) & F . k in dom ( Lt . k ) ;
set i2 = AddTo ( a , i , - n ) , n = - ( n + 1 ) ;
attr B is max means : Def5 : for S being Subsqrt of ( B , S9 ) holds S is ( B , S9 ) `1 ;
a9 "/\" D = { a "/\" d where d is Element of N : d in D & d in D & a in D } ;
|( \square , exp_R . ( q9 - q ) * |( \square , q )| >= |( \square , q )| * |( \square , q )| ;
( - f ) . ( upper_bound A ) = ( ( - f ) | A ) . ( upper_bound A ) .= ( - f ) . ( upper_bound A ) ;
( G * ( len G , k ) ) `1 = ( G * ( len G , k ) ) `1 .= ( G * ( len G , k ) ) `1 .= ( G * ( len G , k ) ) `1 ;
( Proj ( i , n ) ) . ( Lx ) = <* ( proj ( i , n ) ) . ( Lx ) *> .= <* ( proj ( i , n ) ) . ( Lx ) *> ;
f1 + f2 * reproj ( i , x ) is_differentiable_in ( ( reproj ( i , x ) ) . x0 ) .= ( ( reproj ( i , x ) ) . x0 ) * diff ( f1 , x ) ;
pred ( ( - tan ) `| Z ) . x <> 0 & ( ( tan ) `| Z ) . x = ( tan . x ) ^2 ;
ex t being SortSymbol of S st t = s & h1 . t = ( h2 . t ) . x & t . x = ( h2 . t ) . x ;
defpred C [ Nat ] means P8 . $1 is empty & P8 is as as S -connected Subset of ( TOP-REAL 2 ) | A ;
consider y being element such that y in dom p9 and q9 . i = p9 . y and y in dom p9 and p = p9 . i ;
reconsider L = product ( { x1 } +* ( index ( B ) , l ) ) as Subset of product A ;
for c being Element of C ex d being Element of D st T . ( id c ) = id d & ( id c ) . ( id c ) = id d
Comput ( f , n , p ) = ( f | n ) ^ <* p *> .= f ^ <* p *> .= f ^ <* p *> ;
( 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 ` - { p } = ( f | ( n , L ) ) *' - ( f | ( n , L ) ) .= ( f - ( - ( - ( g | ( n , L ) ) ) ) ) *' - ( ( - ( g | ( n , L ) ) ) ;
consider r being Real such that r in rng ( f | divset ( D , j ) ) and r < m + s and m < m + s ;
f1 . ( |[ ( 8 - r ) , ( 8 - r ) ]| ) in f1 .: ( W1 /\ W2 ) & f2 . ( |[ 8 - r , ( 8 - r ) ]| ) in f1 .: ( W2 /\ W3 ) ;
eval ( a | ( n , L ) , x ) = eval ( a | ( n , L ) ) .= a * ( x | ( n , L ) ) .= a * ( x | ( n , L ) ) .= a * ( x | ( n , L ) ) ;
z = DigA ( tk , x9 ) .= DigA ( tk , x9 ) .= DigA ( tk , x9 ) .= DigA ( tk , x9 ) .= DigA ( tk , x9 ) .= DigA ( tk , x9 ) ;
set H = { Intersect S ( S ) where S is Subset-Family of X : S c= G ( S ) & S c= G ( S ) & S c= G ( S ) } ;
consider S19 being Element of D * , d being Element of D * such that S `1 = S29 ^ <* d *> and S `2 = d ^ <* e *> ;
assume that x1 in dom f and x2 in dom f and f . x1 = f . x2 and f . x2 = f . x2 ;
- 1 <= ( ( q `2 ) / |. q .| - sn ) / ( 1 + sn ) ;
( 0. ( V , m ) ) is Linear_Combination of A & Sum ( ( 0. ( V , m ) ) * ( 0. ( V , m ) ) = 0. ( V , m ) ) ;
let k1 , k2 , k1 , k2 , k2 , x4 , 6 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 8 be Element of NAT ;
consider j being element such that j in dom a and j in g " { k } and x = a . j and x = a . j ;
H1 . x1 c= H1 . x2 or H1 . x2 c= H1 . x2 or H1 . x1 c= H1 . x2 & H1 . x2 c= H1 . x2 ;
consider a being Real such that p = 1- p1 * p1 + ( a * p2 ) and 0 <= a and a <= 1 and a <= 1 and a <= 1 ;
assume that a <= c and c <= d and [ a , b ] c= dom f and f . [ a , b ] c= dom g ;
cell ( Gauge ( C , m ) , len Gauge ( C , m ) -' 1 , 0 ) is non empty ;
A-1 in { ( S . i ) `1 where i is Element of NAT : not contradiction } ;
( T * b1 ) . y = L * ( b2 /. y ) .= L * ( ( F * b1 ) . y ) .= ( F * b2 ) . y .= ( ( F * b2 ) . y ) * ( ( F * b2 ) . y ) ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x - ( s . y ) .| ;
( log ( 2 , k + 1 ) ) / 2 >= ( log ( 2 , k + 1 ) ) / 2 ;
then that not p => q in S and not x in the still of p and not x in S and not p in S ;
dom ( the \subseteq of ( the consider of r\in ) ) misses dom ( the \subseteq of ( the consider of r\in ) ) & dom ( the --> ( r\in ) ) = dom ( the --> ( r-10 ) ) ;
synonym f is extended real means : Def21 : for x being set st x in rng f holds x is ExtReal ;
assume for a being Element of D holds f . { a } = a & for X being Subset-Family of D holds f .: ( f .: X ) = f . union X ;
i = len p1 .= len p1 + len <* x *> .= len p1 + len <* x *> .= len p1 + len <* x *> .= len <* x *> + len <* x *> .= len <* x *> + len <* x *> .= len <* x *> + len <* x *> .= len <* x *> + len <* x *> ;
( l , 3 ) `1 = ( g . ( 1 , 3 ) ) `1 + ( g . ( 1 , 3 ) ) `1 .= ( g . ( 1 , 3 ) ) `1 + ( g . ( 1 , 3 ) ) `1 ;
CurInstr ( P2 , Comput ( P2 , s2 , l ) ) = ( halt SCM+FSA ) . IC SCM+FSA .= ( halt SCM+FSA ) . IC SCM+FSA .= ( halt SCM+FSA ) . IC SCM+FSA .= ( IC SCM+FSA ) .= ( IC SCM+FSA ) ;
assume for n be Nat holds ||. seq . n - seq . n .|| <= ( ||. seq . n - seq . n .|| ) & ( for n be Nat holds ||. seq . n - seq . n .|| < p ) ;
sin . st = sin . r2 * cos ( - 1 ) * cos ( - 1 ) .= sin ( ( - 1 ) * cos ( - 1 ) ) .= 0 ;
set q = |[ g1 . t1 , g2 . t2 ]| , g1 = |[ g2 . t2 , g2 = |[ g2 . t2 , g2 . t2 ]| , g2 = |[ g2 . t2 , g2 . t2 ]| , g1 = |[ g2 . t2 , g2 = |[ g2 . t2 ]| , g2 = |[ g2 . t2 , g2 = |[ g2 . t2 , g2 ]| ;
consider G being sequence of S such that for n being Element of NAT holds G . n in G\overline ( F . n ) and G is : contradiction ;
consider G such that F = G and ex G1 , G2 st G1 in SX & G2 in SX & G = ( X \/ Y ) & G2 in SX ;
the root of [ x , s ] in ( ( the Sorts of Free ( C , X ) ) * ( the Arity of S ) ) . s & ( ( the Sorts of Free ( C , X ) ) . ( o , s ) ) . s = ( ( the Sorts of Free ( C , X ) ) . s ;
Z c= dom ( exp_R * ( exp_R + ( exp_R + f ) ) ) /\ dom ( exp_R * ( exp_R + f ) ) ;
for k being Element of NAT holds rV . k = ( ( Im ( f , S ) ) . k ) * ( ( Im ( f , S ) ) . k )
assume that - 1 < n and ( q `2 / |. q .| - sn ) < 0 and ( q `2 / |. q .| - sn ) < 0 and ( q `2 / |. q .| - sn ) < 0 ;
assume that f is continuous and a < b and a < b and c < d and f . a = c and f . b = d and f . c = d ;
consider r being Element of NAT such that s1 = Comput ( P1 , s1 , 1 ) and r <= q and r <= q and q <= r ;
LE f /. ( i + 1 ) , f /. j , L~ f & LE f /. ( j + 1 ) , f /. ( j + 1 ) , f /. ( j + 1 ) , f /. ( j + 1 ) ;
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 x in { y } ;
assume that f +* ( i1 , \xi ) . ( i1 , j1 ) in ( proj ( F , i2 ) ) . ( i1 , j1 ) and f . ( i1 , j1 ) in ( proj ( F , i2 ) ) . ( i1 , j1 ) ;
rng ( ( ( Flow M ) | ( the carrier of M ) ) | ( the carrier of M ) ) c= the carrier of M | ( the carrier of M ) ;
assume z in { ( the carrier of G ) \times { t } where t is Element of T : t in X & t in Y } ;
consider l be Nat such that for m be Nat st l <= m holds ||. s1 . m - g .|| < g / 2 ;
consider t being VECTOR of product G such that mt = ||. Dt . t .|| and ||. t . t - x .|| <= 1 / 2 ;
assume that the carrier of degree v = 2 and v ^ <* 0 *> in dom p and v ^ <* 1 *> in dom p and p ^ <* 1 *> in dom p ;
consider a being Element of the topology of [: X1 , X2 :] , A being Element of the topology of X such that a on A and A c= { a } and a on A ;
( - x ) |^ ( k + 1 ) * ( ( - x ) |^ ( k + 1 ) ) " = 1 / ( ( - x ) |^ k ) " ;
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 , element ] means ex x , y st [ x , y ] = $1 & $2 = [ x , y ] ;
L~ f2 = union { LSeg ( p10 , p1 ) , LSeg ( p10 , p2 ) } .= LSeg ( p1 , p2 ) \/ LSeg ( p10 , p2 ) ;
i -' len h11 + 2 -' 1 + 2 - 1 < i -' len h11 + 2 - 1 + 1 + 1 - 1 + 2 - 1 + 1 - 1 + 1 ;
for n being Element of NAT st n in dom F holds F . n = |. ( ns1 . n ) - ( 1 - n ) .| ;
for r , s1 , s2 , s1 , s2 , s3 , s3 , s3 , s1 , s2 , s3 , s3 , s1 , s2 , s3 , s3 , s1 , s2 , s3 , s3 , s3 , s1 , s2 , s3 , s3 , s3 , s1 , s3 , s3 , s3 , s1 , s2 , s3 , s3 , s3 , s3 , s3 , s3 , s3 , s3 , s3 , s3 , s3 , s3 , s1 , s3 , s3 , s3 , s1
assume v in { G where G is Subset of T2 : G in ( B \/ C ) & G in ( B \/ C ) & G in ( B \/ C ) & G in ( B \/ C ) & G in ( B \/ C ) & G in ( B \/ C ) ;
let g be Element of A , X be Element of INT , f | X , g | X , g | X , f | X be Function of X , REAL n , g | X , 0 ;
min ( g . [ x , y ] , k . [ y , z ] ) = ( min ( g , k ) ) . ( y , z ) ;
consider q1 being sequence of CP such that for n holds P [ n , q1 . n , q1 . n ] and for n holds q1 . n = q2 . n ;
consider f being Function such that dom f = NAT & for n being Element of NAT holds f . n = F ( n ) and for n being Element of NAT holds f . n = F ( n ) ;
reconsider B-6 = B /\ B , O = O /\ ( A /\ B ) , Z = O /\ ( A /\ B ) as Subset of T ;
consider j being Element of NAT such that x = the the H of n and j in dom ( the _ of n ) and 1 <= j and j <= n and n <= len ( ( the _ of n ) * ( j , n ) ) ;
consider x such that z = x and card ( x . O ) in card ( x . O ) and x in L1 . O and x in L1 . O ;
( C * _ T4 ( k , n2 ) ) . 0 = C . ( ( _ T4 ( k , n2 ) ) . 0 ) .= C . ( ( _ T4 ( k , n2 ) ) . 0 ) ;
dom ( X --> rng f ) = X & dom ( ( X --> f ) . x ) = dom ( X --> f ) ;
( ( N-min L~ Cage ( C , n ) ) `2 <= ( ( N-min L~ Cage ( C , n ) ) `2 ) / 2 & ( ( N-min L~ Cage ( C , n ) ) `2 <= ( ( N-min L~ Cage ( C , n ) ) `2 ) / 2 ;
synonym x , y , z means : Def1 : x = y or ex l being Nat st { x , y } c= l & ex x being Subset of S st { x , y } c= l ;
consider X be element such that X in dom ( f | ( n + 1 ) ) and ( f | ( n + 1 ) ) . X = Y ;
assume that ( for k , x , y being Element of L st a = x & b = y & x = y holds x << y ) and ( for a , b being Element of L st a = b & b = y holds a << b ) ;
( 1 / 2 * ( ( $1 + 1 ) * ( ( ( $1 + 1 ) * ( ( $1 + 1 ) / 2 ) ) ) ) * ( ( ( $1 + 1 ) * ( ( ( $1 + 1 ) / 2 ) * ( ( $1 + 1 ) / 2 ) ) ) is_differentiable_on REAL ;
defpred P [ Element of omega ] means ( ( the partial of A1 ) * ( $1 , 1 ) ) . ( $1 , 1 ) = A1 . ( $1 , 1 ) ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 2 ) .= IC Comput ( P , s , 1 ) .= 6 + 1 .= 6 + 1 .= 6 ;
f . x = f . g1 * f . g2 .= ( f . g1 ) * ( f . g2 ) .= ( f . g1 ) * ( f . g2 ) .= ( f . g2 ) * ( f . g2 ) .= ( f . g2 ) * ( f . g2 ) .= ( f . g2 ) * ( f . g2 ) ;
( M * ( F . n ) ) . n = M . ( ( F . n ) . x ) .= M . ( ( ( ( ( M * ( F . n ) ) ) . x ) ) .= M . ( ( ( M * ( F . n ) ) . x ) ) ;
the carrier of L1 + L2 c= ( the carrier of L1 ) \/ ( the carrier of L2 ) \/ ( the carrier of L2 ) \/ ( the carrier of L2 ) ;
pred a , b , c , x , y , z , u , v , w , y , z , w , y , z , w , y , z , w , y , w , z , w , y , w , x , y , z , w , w , y , w , w , x , y ;
( the partial of product s ) . n <= ( ( the partial of s ) . n ) * ( ( the partial of s ) . n ) ;
pred - 1 <= r & r <= 1 & ( - 1 ) * ( ( arccot ) / ( 1 + r ) ) = - ( ( - 1 ) / ( 1 + r ) ) ;
s in { p ^ <* n *> where p is Nat : p ^ <* n *> in T1 & p ^ <* n *> in T1 & p ^ <* n *> in T1 } ;
|[ x1 , x2 , x3 , x4 ]| . 2 - |[ x1 , x2 , x3 , x4 ]| . 2 = x2 - x3 - x3 ;
attr for m being Nat holds F . m is nonnegative & ( Partial_Sums F ) . m is nonnegative & ( Partial_Sums F ) . m is nonnegative ;
len ( such that ( for z be Element of X holds \mathopen { \Vert ( G , z ) - ( G . x ) ) .|| = len ( ( G . x ) - ( G . y ) ) + ( ( G . y ) - ( G . x ) ) ;
consider u , v being VECTOR of V such that x = u + v and u in W1 /\ W2 and v in W2 and u in W2 /\ W3 ;
given F being FinSequence of NAT such that F = x & dom F = n & rng F c= { 0 , 1 } and for k being Nat st k in n holds F . k = G ( k ) ;
0 = ( 1 * exp_R ) * uu1 iff 1 = ( ( 1 - ( 1 - ( 1 - ( 1 - ( 1 / exp_R ) ) ) ) * ( 1 - ( 1 - ( 1 / exp_R ) ) ) ) ;
consider n be Nat such that for m be Nat st n <= m holds |. ( f # x ) . m - lim ( f # x ) . n .| < e ;
cluster -> \mathclose Boolean for non empty RelStr , ( ( len ( <* f *> ) ) | ( 2 , 1 ) ) , ( ( <* f *> ) | ( 2 , 1 ) ) | ( 2 , 3 ) ) is Boolean
"/\" ( BB , {} ) = Top ( B ) .= "/\" ( B , {} ) .= "/\" ( [#] S , {} ) .= "/\" ( [#] S , {} ) .= "/\" ( [#] S , {} ) .= "/\" ( [#] S , {} ) .= "/\" ( [#] S , {} ) .= "/\" ( [#] S , {} ) ;
( r / 2 ) ^2 + ( r / 2 ) ^2 <= ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for x being element st x in A /\ dom ( ( f `| X ) `| A ) holds ( ( f `| X ) `| A ) . x >= r2
2 * r1 - c * |[ a , c ]| - ( 2 * |[ b , c ]| - ( 2 * |[ b , c ]| ) ) = 0. TOP-REAL 2 ;
reconsider p = P * ( \square , 1 ) , q = a " * ( ( - 1 ) * ( ( - 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 y = [ x1 , x2 ] ;
for n being Nat st 1 <= n & n <= len q1 holds q1 . n = ( ( upper_volume ( g , M ) ) | ( n + 1 ) ) . ( len q1 + 1 )
consider y , z being element such that y in the carrier of A & 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 & y = H2 & H1 , H2 // H2 , H1 & H2 = H2 and H1 , H2 // H2 .] ;
for S , T being non empty RelStr , d being Function of T , S st T is complete & d is complete holds d is monotone & d is monotone
[ a + 0 , i + b2 , b2 + b1 , b2 + b2 ] in ( the carrier of V ) \/ ( the carrier of V ) & [ a , b2 ] in the carrier of V ;
reconsider mm = max ( len F1 . ( len F1 ) , len ( p . ( n + 1 ) * <* x *> ) ) as Element of NAT ;
I <= width GoB ( ( GoB f ) * ( len GoB f , j ) , ( GoB f ) * ( len GoB f , j ) ) & ( GoB f ) * ( len GoB f , j ) = ( GoB f ) * ( 1 , 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 : Def21 : A1 is linearly-independent & A2 is linearly-independent & ( for x st x in A1 holds not x in A1 holds x in A2 ) & ( not x in A1 & x in A2 ) ;
func A -carrier of C -> set means : Def7 : union { A . s where s is Element of R : s in A & s in C & s in C } ;
dom ( Line ( v , i + 1 ) ^ ^ ( ( Subset ( p , m ) ) (#) ( 1. ( K , m ) ) ) ) = dom ( F ^ ) .= dom ( F ^ ) .= dom ( F ^ ) ;
cluster [ ( x `1 ) , ( x `2 ) ) , ( x `2 ) , ( x `2 ) , ( x `2 ) , ( x `2 ) , ( x `2 ) , ( x `2 ) , ( x `2 ) , ( x `2 ) , ( x `2 ) , ( x `2 ) , ( x `2 ) , ( x `2 ) , ( x `2 ) , ( x `2 ) , ( x `2 ) ) ;
E , All ( x2 , All ( x2 , x2 ) '&' All ( x1 , x2 ) '&' All ( x2 , x3 ) '&' All ( x2 , x3 ) '&' All ( x2 , x3 ) '&' All ( x2 , x3 ) '&' All ( x2 , x3 ) '&' All ( x2 , x3 ) '&' All ( x2 , x3 ) '&' All ( x2 , x3 ) '&' All ( x2 , x3 ) ) = E ;
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 ) ;
R . ( h . m ) = F . ( x0 + h . m ) - ( h . m ) + ( h . m ) - ( h . m ) ;
cell ( G , ( [: X , Y :] -' 1 , t + 1 ) , ( Y + 1 ) \ ( { t } ) meets UBD L~ f ;
IC Comput ( P2 , s2 , 2 ) = IC Comput ( P2 , s2 , 2 ) .= IC Comput ( P2 , s2 , 2 ) .= IC Comput ( P2 , s2 , 2 ) .= ( card I + 2 ) .= ( card I + 2 ) .= card I + 2 .= card I + 2 ;
sqrt ( 1 - ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 ) > 0 ;
consider x0 being element such that x0 in dom a and x0 in dom g and g . x0 = a . ( k + 1 ) and x0 in dom g and g . ( k + 1 ) = a . ( k + 1 ) ;
dom ( r1 (#) ( chi ( A , A ) ) | A ) = dom ( ( r1 (#) ( chi ( A , A ) ) ) | A ) .= dom ( ( r1 (#) ( A . m ) ) | A ) .= dom ( ( r1 (#) ( A . m ) ) | A ) .= dom ( ( r1 (#) ( A . m ) ) ) .= dom ( ( r1 (#) ( A . m ) ) ) ;
d-7 . [ y , z ] = ( ( y , z ) `2 ) * ( ( y , z ) `2 ) - ( ( y , z ) `2 ) * ( ( y , z ) `2 ) ;
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 is continuous and f is continuous & for x st x in dom f holds ||. f /. x - f /. x0 .|| < r and f /. x <> 0 ;
p in Cl A implies for K being Basis of p , Q being Subset of T st Q in K & P = { p } holds A meets Q
for x being Element of REAL n st x in Line ( x1 , x2 ) holds |. y1 - y2 .| <= |. y1 - y2 .|
func Sum ( a ) -> Ordinal means : Def6 : a in it & for a being Ordinal st a in it holds a in it iff a in b & a is Ordinal ;
[ a1 , a2 , a3 ] in ( the carrier of A ) \/ ( the carrier of B ) & [ a1 , a2 , a3 ] in ( the carrier of A ) \/ ( the carrier of B ) ;
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 carrier of S2 ;
||. ( vseq . n ) - ( vseq . m ) - ( vseq . m ) .|| * ||. x - y .|| < ( e / ( ||. x - y .|| ) * ||. x - y .|| ) * ||. x - y .|| ;
then for Z being set st Z in { Y where Y is Element of I7 : F c= Y & Y in Z & z in Z & z in Z } holds z in x ;
sup compactbelow ( [ s , t ] ) = [ sup { [ s , t ] where s , t is Element of L : s in compactbelow ( [ s , t ] ) & s in compactbelow ( [ s , t ] ) } ;
consider i , j being Element of NAT such that i < j and [ y , f . j ] in Ij and [ f . i , f . j ] in Ij and [ f . i , f . j ] in Ij ;
for D being non empty set , p , q being FinSequence of D st p c= q & q in D holds ex p being FinSequence of D st p ^ q = q ^ p & p ^ q in D
consider e1 being Element of the carrier of X such that c9 , a9 // a9 , c9 and not ( a9 , c9 // c9 , a9 & not ( a9 , c9 // c9 , a9 & a9 , c9 // c9 , a9 & not ( a9 , c9 // c9 , a9 ) & ( a9 , c9 // c9 , a9 ) & ( c9 , c9 // a9 , c9 ) ) ;
set U2 = I \! \mathop { \rm \hbox { - } F } , E = I \! \mathop { \rm \hbox { - } F } , N = I \! \mathop { \rm \hbox { - } F } , N = I \! \mathop { \rm \hbox { - } F } , E = I \! \mathop { \rm \hbox { - } F } , N = I \! \mathop { - } N ;
|. q2 .| ^2 = ( ( ( q `2 ) ) ^2 + ( q `2 ) ^2 ) + ( ( q `2 ) ) ^2 + ( ( q `2 ) ) ^2 .= ( |. q .| ) ^2 + ( |. q .| ) ^2 ;
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 "\/" y
dom ( ( the charact of U1 ) * ( the charact of U2 ) ) = dom ( ( the charact of U1 ) * ( the charact of U2 ) ) & dom ( ( the charact of U2 ) * ( the charact of U2 ) ) = dom ( ( the charact of U2 ) * ( the charact of U2 ) ) ;
dom ( h | X ) = dom h /\ X .= dom ( ( h | X ) | X ) .= dom ( ( h | X ) | X ) .= dom ( ( h | X ) | X ) .= dom ( ( h | X ) | X ) .= dom ( ( h | X ) | X ) .= dom ( ( h | X ) | X ) .= dom ( ( h | X ) | X ) .= dom ( ( h | X ) | X ) .= dom ( ( h | X ) | X ) .= dom ( ( h | X ) | X ) .= dom ( ( h | X ) | X )
for N1 , N2 , N1 , N1 , N2 being Element of [: G , F :] holds dom ( h . ( N1 . ( N1 . ( N1 . ( N1 . ( N1 . ( N1 . ( N1 . ( N1 . ( N1 . ( N1 . ( N1 . ( N1 . ( N1 . ( N1 . ( N1 . ( N1 . ( N1 . ( N1 . ( N1 . ( N1 . ( N1 . ( N1 . ( N1 . 1 j . 1 ) ) ) ) ) ) ) ) ) ) ) ) , ( ( N1 . ( N1 . ( N1 . ( N1 . ( N1 . ( N1 . ( N1 . (
( mod ( u , m ) + mod ( v , m ) ) . i = ( mod ( u , m ) ) . i + ( mod ( v , m ) ) . i .= ( mod ( v , m ) ) . i + ( mod ( v , m ) ) . i ;
- ( q `1 ) ^2 < - 1 or ( q `2 ) ^2 / ( |. q .| ) ^2 < - ( q `2 ) ^2 & ( q `2 ) ^2 / ( |. q .| ) ^2 <= - ( q `2 ) ^2 ;
attr r1 = f9 & r2 = f9 & for x , y being Element of REAL st x in dom f9 & y in dom ( f9 * ( x , y ) ) holds r1 * ( x , y ) = r2 * ( x , y ) ;
vseq . m is bounded Function of X , Y & x9 . m = ( vseq . m ) * ( vseq . m ) & x9 . m = ( vseq . m ) * ( vseq . m ) ;
pred a <> b & b <> c & c <> 0 & angle ( b , a , c ) = PI & angle ( b , c , a ) = PI & angle ( b , c , a ) = PI ;
consider i , j being Nat , r being Real such that p1 = [ i , r ] & p2 = [ j , s ] and r < j and s < j and j < i ;
|. p .| ^2 - ( 2 * |( p , q )| ) ^2 + ( 2 * |( p , q )| ) ^2 = |. p .| ^2 + ( |. p - q .| ) ^2 ;
consider p1 , q1 being Element of ( X * ) such that y = p1 ^ q1 and q1 in ( X * ) and ( p1 ^ q1 ) ^ q1 in ( X * q1 ) and ( p1 ^ q1 ) ^ q1 in ( X * q1 ) ;
( the addF of A ) . ( r1 , r2 , s1 , s2 , s1 ) = ( ( s2 - s1 ) * ( s1 , s2 ) ) / ( s1 , s2 ) .= ( s2 - s1 ) / ( s1 , s2 ) ;
( for w being Element of A st w = lower_bound ( proj2 .: ( A /\ holds w in A ) ) & ( proj2 .: ( A /\ ( A /\ { w } ) ) = lower_bound ( proj2 .: ( A /\ { w } ) ) ) ) & ( proj2 .: ( A /\ { w } ) = lower_bound ( proj2 .: ( A /\ { w } ) ) )
s , ( ( H / ( H1 , H2 ) ) |= H1 '&' H2 iff s |= All ( H1 , H2 ) ) & s |= All ( H2 , H1 ) & s |= All ( H2 , H2 )
len ( s + 1 ) = card ( support b1 ) + 1 .= card ( support b1 ) + 1 .= card ( support b2 ) + 1 .= card ( support b1 ) + 1 .= card ( support b2 ) + 1 .= card ( support b1 ) + 1 .= card ( support b2 ) + 1 .= card ( support b1 ) + 1 .= card ( support b2 ) ;
consider z being Element of L1 such that z >= x and z >= y and for z being Element of L1 st z in x & z in { x } holds z >= x ;
LSeg ( UMP D , |[ ( W-bound D ) / 2 , ( W-bound D ) / 2 + ( W-bound D ) / 2 ]| ) /\ D = { UMP D } ;
lim ( ( ( f `| N ) / g ) `| N ) = ( ( f `| N ) / g `| N ) . b .= ( ( f `| N ) / g `| N ) . b .= ( ( f `| N ) / g `| N ) . b ;
P [ i , ( pr1 ( f ) ) . i , ( pr1 ( f ) ) . ( i + 1 ) , ( pr1 ( f ) ) . ( i + 1 ) ] ;
for r be Real st 0 < r ex m be Nat st for k be Nat st m <= k holds ||. ( seq . k ) - ( R /. m ) .|| < r
for X being set , P being a_partition of X , x , y , z being set st x in X & y in P & z in P & x in P & y in P & z in P & x = y holds x = z
Z c= dom ( ( ( id Z ) ^ ) (#) ( ( id Z ) ^ ) \ ( ( id Z ) ^ ) " { 0 } ) \ dom ( ( id Z ) ^ ) " { 0 } ;
ex j being Nat st j in dom ( l ^ <* x *> ) & j < i & j < i & i = 1 + j & j = 1 + j & i = 1 + j & j = 1 + j ;
for u , v being VECTOR of V st 0 < r & u < 1 & v in \cal N holds r * u + ( 1 - r ) * v in \cal N
A , Int A , Cl ( A \/ B ) , B , C , D , E , F , J , M , N , N , N , F , M , N , N , F , J , M , N , N , F , M , N , N , F , J , M , N , N , F , M , N , N , F , J , M , N , N , F , M , N , N , F , M , N , N , N , F , M , N , N , N , N , N , N , N , F , N , N , N , N
- Sum <* v , u , w *> = - ( v + u + u ) .= - ( v + u ) + ( u + w ) .= - ( v + u ) + ( u + w ) .= - ( v + u ) + ( w + w ) .= - ( v + u ) ;
( Exec ( a := b , s ) ) . IC SCM R = ( Exec ( a := b , s ) ) . IC SCM R .= ( Exec ( a := b , s ) ) . NAT .= 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 & for x being element st x in I holds h . x in ( the carrier of J ) . x ;
for S1 , S2 being non empty reflexive RelStr , D being non empty Subset of S1 , f being Function of S1 , S2 , g being Function of S2 , S2 st f is monotone & g is monotone holds ( cos * f ) * ( D , f ) is directed
card X = 2 implies ex x , y st x in X & y in X & not ( ex x , y st x in X & y in X & not ( x in X & y in Y & not ( x in Y ) & x in X ) & not ( x in Y ) & ( ex x st x in X & x in Y ) & not ( x in X ) & x in Y )
( E-max L~ Cage ( C , n ) ) in rng ( Cage ( C , n ) \circlearrowleft W-min L~ Cage ( C , n ) ) & ( W-min L~ Cage ( C , n ) ) .. Cage ( C , n ) in rng Cage ( C , n ) ;
for T , T , p , q , r , s being Element of dom T st p in dom T & q in dom T holds ( T -tree ( p , T ) ) . q = T . ( q , s )
[ i2 + 1 , j2 ] in Indices G & [ i2 + 1 , j2 ] in Indices G & f /. k = G * ( i2 + 1 , j2 ) & f /. k = G * ( i2 + 1 , j2 ) ;
cluster -> \llangle ( k -' n ) , ( k -' 4 ) , ( k -' n ) , ( k -' n ) , ( k -' 4 ) , ( k -' n ) + ( k -' 4 ) ) -> set ;
dom F " = the carrier of X2 & rng F = the carrier of X1 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = 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 = n and the carrier of C = A \/ B and the carrier of V = A \/ B and the carrier of V = A \/ B \/ B ;
V is prime implies for X , Y being Element of \langle the topology of T , \subseteq \rangle st X /\ Y c= V & Y c= V holds X c= Y or Y c= V
set X = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] & P [ v1 ] } , Y = { F ( v1 ) : P [ v1 ] } , Z = { F ( v2 ) : P [ v2 ] } , Y = { F ( v2 ) : P [ v2 ] } , Z = { F ( v2 ) : P [ v1 ] } , Z = { F ( v2 ) : P [ v2 ] } ;
angle ( p1 , p3 , p4 ) = 0 .= angle ( p2 , p3 , p2 ) .= angle ( p3 , p2 , p3 ) .= angle ( p2 , p3 , p2 ) .= angle ( p3 , p2 , p3 ) .= angle ( p2 , p3 , p3 ) .= angle ( p3 , p3 , p2 ) ;
- sqrt ( ( - ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ) = - ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 .= - ( ( q `2 / |. q .| - cn ) / ( 1 + cn ) ) ^2 .= - ( - 1 ) ;
ex f being Function of I[01] , TOP-REAL 2 st f is continuous one-to-one & rng f = P & f is continuous one-to-one & f . 0 = p1 & f . 1 = p2 & f . 0 = p2 & f . 1 = p3 & f . 1 = p4 & f . 1 = p4 & f . 0 = p2 & f . 1 = p4 ;
attr f is partial differentiable on // SVF1 ( 2 , pdiff1 ( f , 1 ) , u0 ) means : Def: SVF1 ( 2 , pdiff1 ( f , 1 ) , u0 ) is continuous & SVF1 ( 2 , pdiff1 ( f , 3 ) , u0 ) is continuous ;
ex r , s st x = |[ r , s ]| & ( G * ( len G , 1 ) `1 < r & r < s & s < G * ( 1 , 1 ) `2 & s < G * ( 1 , 1 ) `2 & s < G * ( 1 , 1 ) `2 ;
assume that f is special and 1 <= t and t <= len G and t <= width G and G * ( t , width G ) `2 >= s and s <= G * ( t , 1 ) `2 and G * ( t , 1 ) `2 >= N-bound L~ f ;
pred i in dom G means : Def: r * ( f (#) reproj ( G , i ) ) = r * f * reproj ( G , i ) + r * reproj ( G , i ) ;
consider c1 , c2 being bag of o1 + o2 such that ( decomp c ) /. k = <* c1 , c2 *> and ( decomp c ) /. k = <* c1 , c2 *> and ( decomp c ) /. k = <* c1 , c2 *> ;
u0 in { |[ r1 , s1 ]| : r1 < s1 & s1 < s1 & s1 < s2 & s1 < s2 & s2 < s1 & s2 < s1 & s1 < s1 & s1 < s1 & s1 < s2 & s2 < s2 & t1 < s1 & s1 < s2 & s2 < s2 & t1 in { s1 , s2 } } ;
Cl ( X ^ Y ) = the carrier of X . ( ( k + 1 ) + 1 ) .= ( C . ( k + 1 ) ) .= C . ( ( k + 1 ) + 1 ) .= C . ( ( k + 1 ) + 1 ) .= C . ( ( k + 1 ) + 1 ) .= C . ( ( k + 1 ) + 1 ) .= C . ( ( k + 1 ) + 1 ) ;
attr len M1 = len M2 & width M1 = width M2 & width M1 = width M2 & len M2 = width M2 & width M1 = width M2 & len M1 = width M2 & width M1 = width M2 & len M1 = width M2 ;
consider g2 be Real such that 0 < g2 and { y where y is Point of S : ||. y - x0 .|| < g2 & g2 . ( y - x0 ) < g2 & g2 . ( y - x0 ) < g2 . ( y - x0 ) & g2 is convergent & lim g2 = g2 . ( y - x0 ) ;
assume x < ( - b + sqrt ( a , b , c ) ) / 2 * a or x > ( - b ) / 2 * a + ( - b ) / 2 * a ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ G1 ) . i & ( G1 ^ G2 ) . i = ( <* 3 *> ^ G1 ) . i & ( G1 ^ G2 ) . i = ( <* 3 *> ^ G1 ) . i ;
for i , j st [ i , j ] in Indices ( ( M1 + M2 ) * ( i , j ) ) & ( M1 + M2 ) * ( i , j ) < M2 * ( i , j ) holds ( M1 + M2 ) * ( i , j ) < M2 * ( i , j )
for f being FinSequence of NAT , i being Element of NAT st i in dom f & i in dom f & f . i = f /. i holds i divides f /. ( i + 1 )
assume F = { [ a , b ] where a , b is Subset of X : for c st c in B & c in B & a in B & b in B holds b c= c & c c= a & b c= c ;
b2 * q2 + ( b3 * q3 ) + ( ( - ( b2 * q2 ) ) * ( - ( b2 * q2 ) ) + ( - ( b2 * q2 ) ) * ( - ( b2 * q2 ) ) = 0. TOP-REAL n + ( ( - ( b2 * q2 ) ) * ( - ( b2 * q2 ) ) ;
Cl ( Cl F ) = { D where D is Subset of T : ex B being Subset of T st D = Cl ( B ) & B in F & D c= Cl ( B \/ C ) & D is closed & B c= D & D is closed & B c= D & D is closed ;
attr seq is summable means : Def7 : seq is summable & for n holds seq . n is summable & ( for n holds seq . n = ( seq . n ) * ( seq . n ) ) & ( for n holds seq . n = ( seq . n ) * ( seq . n ) ) ;
dom ( ( cn -FanMorphN ) | D ) = ( the carrier of ( TOP-REAL 2 ) | D ) /\ D .= ( the carrier of ( TOP-REAL 2 ) | D ) /\ D .= the carrier of ( TOP-REAL 2 ) | D .= the carrier of ( TOP-REAL 2 ) | D .= D ;
[ X \to Z ] is full full full full SubRelStr of ( ( [#] Z ) |^ ( \alpha , Y ) ) & [ X \to Z ] is full full SubRelStr of ( ( [#] Z ) |^ ( \alpha , Y ) ) |^ ( \alpha , Y ) ;
( G * ( 1 , j ) ) `2 = ( G * ( i , j ) ) `2 & ( G * ( 1 , j ) ) `2 <= ( G * ( 1 , j ) ) `2 ;
synonym m1 c= m2 & ( for p , q being set st p in P & q in P & not p in P & not q in P & not p in P & not q in P & not p in P & not q in P & not p in P & not q in P & not p in P & p in P & p in P & q in P & p in P & p in P & q in P & p in P & p in P ) ;
consider a being Element of B ( ) such that x = F ( a ) and a in { G ( b ) where b is Element of B ( ) : P [ b ] } and a in A ( ) & P [ b ] ;
attr IT is multiplicative means : Def21 : the carrier of it = [ the carrier of it , the carrier of it , the carrier of it #) = [ the carrier of it , the carrier of it , the carrier of it #) ;
sequence ( a , b , 1 ) + sequence ( b , c , 1 ) = b + c + sequence ( b , c , 1 ) .= b + c + d .= b + ( c + d ) .= b + ( c + d ) .= sequence ( a + c + d ) .= b + ( b + c ) .= sequence ( a + c + d ) ;
cluster ( 1 + 1 ) * -> real for Element of INT , i , j being Element of INT , k being Element of INT holds ( i + 1 ) * ( i , j ) = ( i + 1 ) * ( i , j ) + ( j + 1 ) * ( i , j ) ;
( - ( s2 * p1 + ( s1 * p2 ) ) + ( s1 * p2 ) ) = ( ( - 1 ) * ( ( s1 * p2 ) + ( s1 * p2 ) ) ) + ( ( s1 * p2 ) + ( s1 * p2 ) ) .= ( ( - 1 ) * ( ( s1 * p2 ) + ( s1 * 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 TopSpace st D in the carrier of S & D in the carrier of S holds D meets [#] S and for V being Subset of S st V in the carrier of T holds V meets V and V is open ;
assume that 1 <= k and k <= len w + 1 and T . ( ( q , w ) -to w ) = ( ( q , w ) -\hbox { - } m . ( q , w ) ) . k and T . ( ( q , w ) -\hbox { - } m . k ) = ( ( T , w ) -\hbox { - } m . k } ) ;
2 * ( a |^ ( n + 1 ) + ( 2 * b |^ ( n + 1 ) ) >= ( a |^ n ) + ( b |^ ( n + 1 ) ) + ( b |^ ( n + 1 ) ) ;
M , v |= All ( All ( x , All ( x , H ) ) , All ( x , H ) ) implies M , v / ( All ( x , H ) ) |= All ( x , H ) '&' All ( x , H )
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f . x0 or 0 < f . x0 & for x1 st x1 in l holds f . x1 - f . x0 < f . x1 - f . x0 ;
for G1 being _Graph , W being Walk of G1 , e being Vertex of G2 , x being Vertex of G1 , y being Vertex of G2 st e in W & x in W & y in W holds not W is Walk of G
c01 is not empty iff ( ex y1 st y1 is not empty & q1 is not empty & not ( ex x1 st y1 is not empty & not ( q1 is not empty & q1 is not empty & not q1 is not empty & not q1 is not empty & not q1 is not empty ) & not ( q1 is not empty & not q1 is not empty & q1 is not empty & not q1 is not empty & not q1 is not empty & not q2 is not empty ) & not ( q1 is not empty & not q2 is not empty & not q2 is not empty & not q2 is not empty & not q2 is not empty & not q2 is not empty & not q2 is not empty & not q2 is not empty & not q2 is not empty & not q2 is not empty & not q2
Indices GoB ( f ) = dom ( GoB f ) & ( for i st 1 <= i & i < len GoB f holds ( GoB f ) * ( i , 1 ) = ( GoB f ) * ( i , 1 ) & ( GoB f ) * ( i + 1 , 1 ) = ( GoB f ) * ( i + 1 , 1 )
for G1 , G2 , G2 , G1 , G2 being Subgroup of O st G1 is_stable _Subgroup O & G2 is_stable _Subgroup O & G1 is_stable _Subgroup O & G2 is_stable O & G2 is stable & G1 is stable & G2 is stable holds G1 is stable
UsedIntLoc ( int ( f , n ) ) = { ( ( 0 qua Nat ) - 1 ) * ( ( 1 - 1 ) * ( 1 - 1 ) ) + ( ( 1 - 1 ) * ( 1 - 1 ) ) * ( 1 - 1 ) ;
for f1 , f2 being FinSequence of F st f1 ^ <* p *> is p -element & f2 is p -element & p in rng f1 & <* p *> in F & <* p *> in F holds Q [ f1 ^ <* p *> ]
( ( p `1 ) ^2 + ( p `2 ) ^2 ) = ( ( q `1 ) ^2 + ( p `2 ) ^2 ) * ( ( q `2 ) ^2 ) .= ( ( q `1 ) ^2 + ( q `2 ) ^2 ) * ( ( q `2 ) ^2 ) ;
for x1 , x2 , x3 , x4 , x5 , x2 , x3 , x4 , x5 , x5 , x5 , x2 , x4 , x5 , x5 , x2 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , M being Real st x1 , x2 , x2 , x3 , x4 , N , N , N , N being Real holds |( ( x1 , x2 , N , N , N , x1 , N , x1 , x2 , x2 , x4 , x5 , x5 , N , x1 , x1 , x2 , x4 , x5 , x2 , x4 , x5 , x5
for x st x in dom ( ( - 1 ) (#) ( ( - 1 ) (#) ( - 1 ) ) `| Z ) holds ( ( - 1 ) (#) ( - 1 ) (#) ( - 1 ) ) `| Z ) . x = - ( ( - 1 ) (#) ( - 1 ) (#) ( - 1 ) ) . x
for T being non empty TopSpace , P being Subset of T st P c= the topology of T & P is Basis of T & for x being Point of T st x in P ex B being Basis of x st B c= P & B c= P & P is Basis of x
( a 'or' b 'imp' c ) . x = 'not' ( ( a 'or' b ) . x ) 'or' c . x .= 'not' ( ( a 'or' b ) . x ) 'or' c . x .= TRUE 'or' TRUE .= TRUE 'or' TRUE .= TRUE .= TRUE 'or' TRUE .= TRUE .= TRUE 'or' TRUE .= TRUE 'or' TRUE .= TRUE 'or' TRUE .= TRUE 'or' TRUE .= TRUE 'or' TRUE .= TRUE 'or' TRUE .= TRUE 'or' TRUE .= TRUE 'or' TRUE .= TRUE 'or' TRUE .= TRUE ;
for e being set st e in Ad ex X1 being Subset of [: X , Y :] st e = [: X1 , Y1 :] & [: X1 , Y1 :] is open & [: X1 , Y1 :] is open & [: X1 , Y1 :] is open & [: X1 , Y1 :] is open & [: X1 , Y1 :] is open & [: Y1 , Y1 :] is open & [: Y1 , Y1 :] is open & [: Y1 , Y1 :] is open
for i being set st i in the carrier of S for f being Function of [: S , T :] , S , g being Function of S , T st f = H . i & g is continuous & f is continuous holds g is continuous & g is continuous
for v , w st for x , y st x <> y & w . y = v . y holds J . ( x , y ) = J . ( y , x ) & J . ( y , w ) = Valid ( x , y , w )
card D = card D1 + card D2 - card D1 .= card D1 + card D2 - card D2 .= card D1 + card D2 - 1 .= ( D1 + D2 ) - 1 .= ( D1 + D2 ) - ( D1 + D2 ) .= ( D1 + D2 ) - ( D1 + D2 ) .= ( D1 + D2 ) - ( D1 + D2 ) .= ( D1 + D2 ) - ( D1 + D2 ) .= ( D1 + D2 ) - ( D1 + D2 ) .= ( D1 + D2 ) - ( D1 + D2 ) - ( D1 + D2 ) - ( D1 + D2 ) - ( D1 + D2 ) - ( D1 + D2 ) - ( D1 + D2 ) - ( D1 + D2 ) - ( D1 + D2 ) - ( D1
IC Exec ( i , s ) = ( s +* ( 0 .--> ( s . 0 ) ) ) . 0 .= ( s +* ( 0 .--> ( s . 0 ) ) ) . 0 .= ( s +* ( 0 .--> ( s . 0 ) ) ) . 0 .= ( s +* ( 0 .--> ( s . 0 ) ) ) . 0 .= ( s +* ( 0 .--> ( s . 0 ) ) ) . 0 .= ( s +* ( 0 .--> ( s . 0 ) ) ) . 0 .= ( s . 0 ) ) . 0 .= ( s . 0 ) .= ( s . 0 .--> ( s . 0 ) ) . 0 .= ( s . 0 ) .= ( s . 0 ) ) . 0 .= ( s . 0 )
len f /. ( \downharpoonright i1 -' 1 + 1 ) = len f -' ( i1 -' 1 + 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 & 2 <= b & a < b & b < a & c < b holds a + b < a + b or a = b + b + c
for f being FinSequence of TOP-REAL 2 , p being Point of TOP-REAL 2 , x being Point of TOP-REAL 2 st x in LSeg ( f , p ) & p in LSeg ( f , 1 ) & x = Index ( p , f /. 1 ) holds Index ( p , f /. 1 ) = x
( ( curry ( P , k + 1 ) ) # x ) . x = ( ( curry ( P , k ) ) # x ) . x + ( ( curry ( P , k ) ) # x ) . x ;
z2 = g /. ( \downharpoonright n1 -' 1 ) .= g /. ( i -' 1 + ( i -' 1 ) + 1 ) .= g . ( i -' 1 + 1 ) .= g . ( i -' 1 + 1 ) .= g . ( i -' 1 + 1 ) .= g . ( i -' 1 + 1 ) .= ( g /^ 1 ) . ( i -' 1 ) ;
[ f . 0 , f . 3 ] in id ( the carrier of G ) \/ ( the InternalRel of G ) or [ f . 0 , f . 3 ] in the InternalRel of G & [ f . 0 , f . 3 ] in the InternalRel of ( G ) \/ the carrier of G ;
for G being Subset-Family of B st G = { [ R , X ] where R is Subset of A ( ) , X is Subset of A ( ) st R in F & X in G & Y = ( Intersect ( R ) ) . X & X is finite } holds ( Intersect ( R ) ) . X = Intersect ( ( F ) ) . X
CurInstr ( P1 , Comput ( P1 , s1 , m1 + 1 ) ) = CurInstr ( P1 , Comput ( P1 , s1 , m1 + 1 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m1 + 1 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= halt SCMPDS ;
assume that a on M and b on M and c on N and d on N and c on N and p in M and p in M and c in N and p in M and c in N and p in M and c in N and p in M and p in M and c in N and p in M and c in N and p in M and c in N and c in N and p in N and c in N and c in N and c in N and c in N and p in M and p in M and p in M and p in M and c in N and p in N and p in N and p in N and p in M and p in M and p in M and c in M and p in M and M in N and M in N and M in N and M in N and M in N and M in N and M in N and M in N and M in N
assume that T is \hbox { 4 , T , 4 , B is _ of T and ex F being Subset-Family of T st F is closed & for n being Nat st n in dom F ex F be Subset-Family of T st F is finite-ind & ( for n being Nat st n <= 0 holds F . n <= 0 ) & ( for n being Nat st n <= 0 holds F . n <= 0 ) ;
for g1 , g2 st g1 in ]. r - 1 , r + 1 .[ & g2 in ]. r - 1 , r + 1 .[ & |. f . g1 - g1 . g2 .| <= ( ( r1 - 1 ) * ( r - 1 ) ) / ( 1 - r ) & |. f . g2 - g1 . r .| <= ( r1 - 1 ) / ( 1 - r )
( ( exp_R * z ) + ( - 1 ) ) * ( ( z + - 1 ) * ( z + - 1 ) ) = ( ( exp_R * z ) + ( - 1 ) ) * ( ( z + - 1 ) * ( z + - 1 ) ) .= ( ( exp_R * z ) + ( - 1 ) ) * ( ( z + - 1 ) * ( z + - 1 ) ) ;
F . i = F /. i .= 0. R + ( 0 qua Element of R ) .= ( b |^ n ) * ( b |^ n ) .= ( b |^ n ) * ( b |^ n ) .= ( b |^ ( n + 1 ) ) * ( b |^ n ) .= ( b |^ ( n + 1 ) ) * ( b |^ n ) ;
ex y being set , f being Function st y = f . n & dom f = NAT & rng f = NAT & for n being Nat holds f . ( n + 1 ) = R ( n , f . n ) & for n being Nat holds f . ( n + 1 ) = R ( n , f . n ) ;
func f (#) F -> FinSequence of V means : Def6 : len it = len F & for i being Nat st i in dom F holds it . i = F . i * ( F . i ) & for i be Nat st i in dom F holds it . i = f . i * ( F . i ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } = { x1 , x2 , x3 , x5 } ;
for n being Nat , x being set st x = h . n . n & h . ( n + 1 ) = o ( x , n ) & x in InputVertices S ( ) & o . ( n + 1 ) in InnerVertices S ( x , n ) & o . ( n + 1 ) in InnerVertices S ( x , n )
ex S1 being Element of CQC-WFF ( Al ( ) ) st SubP ( P , l , e ) = S1 & ( for e being Element of Al ( ) st e in S ( ) holds ( S . e ) . e = S1 . e ) & ( S . e = e ) & ( S . e = e implies S . e = e ) ;
consider P being FinSequence of GG2 such that p9 = product P and for i st i in dom P ex t being Element of the carrier of the carrier of K st P . i = t & t . i = i & t . i = j ;
for T1 , T2 being strict non empty TopSpace , P being Subset of T1 , Q being Subset of T2 st the topology of T1 = the topology of T2 & P = the topology of T2 & P is Basis of T2 & Q is Basis of T2 holds P is Basis of T2
Suppose f is_partial differentiable on u0 and r (#) pdiff1 ( f , 3 ) is_differentiable_in u0 and r (#) pdiff1 ( f , 3 ) is_differentiable_in u0 & pdiff1 ( r (#) pdiff1 ( f , 3 ) , 3 ) is_differentiable_in u0 & partdiff ( r (#) pdiff1 ( f , 3 ) , 3 ) = r * SVF1 ( f , 3 ) . 3 and r (#) pdiff1 ( f , 3 ) = r * pdiff1 ( f , 3 ) . 3 ;
defpred P [ Nat ] means for F , G being FinSequence of REAL , s being FinSequence of REAL st len F = $1 & for s being Permutation of Seg ( $1 + 1 ) st len s = $1 & len G = $1 & for s being Permutation of Seg ( $1 + 1 ) st s in Seg ( $1 + 1 ) holds s = ( Sum F ) . s ;
ex j st 1 <= j & j < width GoB f & ( GoB f ) * ( 1 , j ) `2 <= ( GoB f ) * ( 1 , j ) `2 & ( GoB f ) * ( 1 , j + 1 ) `2 <= ( GoB f ) * ( 1 , j + 1 ) `2 ;
defpred U [ set , set , set ] means ex F9 being Subset-Family of T st $1 = $2 & $2 = ( union F ) . $1 & ( union F ) is open & ( union F ) is discrete & ( union F ) is discrete & ( union F ) is discrete & ( union F ) is discrete discrete & ( union F is discrete & union F is discrete & union F is discrete & union F is discrete & union F is discrete & union F is discrete ;
for p4 being Point of TOP-REAL 2 st LE p4 , p2 , P & LE p4 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P holds LE p4 , p1 , P
f in Funcs ( E , H ) & for g st g . y <> f . y & x = g . y holds x in the carrier of ( ( the carrier of X ) | E ) & g in the carrier of ( ( the carrier of X ) | E ) | E ) implies f in the carrier of ( ( the carrier of X ) | E ) | E
ex p9 being Point of TOP-REAL 2 st x = p9 & ( ( for p being Point of TOP-REAL 2 st p in P holds ( p `1 / |. p .| - sn ) / ( 1 + sn ) ) ^2 >= 0 & ( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) ^2 >= 0 ;
assume for d7 being Element of NAT st d7 <= 8 & d7 <= 8 holds s1 . ( d7 ) = ( s2 . ( |. 7 - 2 * ( t - 2 ) ) .| ) & s1 . ( |. 7 - 2 * ( t - 2 ) ) = s2 . ( |. 7 - 2 * ( t - 2 ) .| ) ;
assume that s <> t and s is Point of Sphere ( x , r ) and s in Sphere ( x , r ) and ex e being Point of TOP-REAL n st e in { x , y } & e in Ball ( x , r ) /\ Sphere ( x , r ) ;
given r such that 0 < r and for s holds 0 < s or ex x1 , x2 being Point of ( TOP-REAL 2 ) st x1 in dom f & x2 in dom f & ||. x1 - x2 - x0 .|| < s & ||. x1 - x2 .|| < r & ||. x1 - x2 - x0 .|| < r ;
( p | x ) | ( p | ( x | x ) ) = ( ( ( x | x ) | x ) | ( x | x ) ) | ( p | x ) .= ( ( x | x ) | x ) | ( x | x ) ) | p ;
assume that x , x + h in dom sec and ( there exists exists $ h st ( for x st x in dom sec holds ( ( sec * sec ) `| Z ) . x = ( 4 * sec . x ) * ( sin . x ) + ( sin . x ) ^2 ) and ( for x st x in Z holds ( ( sec * sec ) `| Z ) . x = ( 4 * sec ) `| Z ) . x ;
assume that i in dom A and len A > 1 and for i st i in dom B holds B * ( i , j ) = A * ( i , j ) and B * ( i , j ) = A * ( i , j ) and B * ( i , j ) = A * ( i , j ) ;
for i being non zero Element of NAT st i in Seg n holds ( i divides n implies i = <* 1_ F_Complex *> ) & ( i divides n implies h . i = <* 1_ F_Complex *> ) & ( i divides n implies h . i = <* 1_ F_Complex *> ) & ( i divides n implies h . i = <* 1. F_Complex *> )
( ( b1 'imp' b2 ) '&' ( c1 'or' c2 ) ) '&' ( ( b1 'or' c1 ) '&' ( c1 'or' c2 ) ) '&' ( ( b1 'or' c1 ) '&' ( c1 'or' c2 ) ) '&' ( ( c1 'or' c2 ) '&' ( c1 'or' c2 ) ) '&' ( ( c1 'or' c2 ) '&' ( c1 'or' c2 ) ) '&' ( ( c1 'or' c2 ) '&' ( c1 'or' c2 ) ) '&' ( ( c1 'or' c2 ) '&' ( c1 'or' c2 ) ) ) '&' ( c1 'or' c2 ) ) '&' ( c1 'or' c2 ) ) '&' ( c1 'or' c2 ) ) '&' ( c1 'or' c2 ) '&' ( c1 'or' c2 ) '&' ( c1 'or' c2 ) '&' ( c1 'or' c2 ) '&' ( c1 'or' c2 ) '&' ( c1 'or' c2 ) '&' ( c1 'or' c2 ) '&' ( c1 'or' c2 ) '&' ( c1 'or' c2 ) '&' ( c1 'or' c2 ) '&' ( c1 'or' c2 ) '&' ( c1 'or' c2 ) '&' ( c1
assume that for x holds f . x = ( ( - 1 ) (#) ( cot * cot ) ) . x and for x st x in dom ( ( - 1 ) (#) ( cot * cot ) ) holds ( ( - 1 ) (#) ( cot * cot ) ) . x = ( - 1 ) * ( sin . x ) and ( ( - 1 ) (#) ( cot * cot ) ) . x = - 1 ;
consider Rd , Id be Real such that Rd = Integral ( M , ( Re F ) . n ) & Rd = Integral ( M , ( Im F ) . n ) and I = Integral ( M , ( Im F ) . n ) and I = Integral ( M , ( Im F ) . n ) ;
ex k being Element of NAT st ' = k & 0 < d & for q be Element of product G st q in X & 0 < q & ||. q - f /. x - f /. x0 .|| < d holds ||. partdiff ( f , q , k ) - diff ( f , x ) - diff ( f , q ) .|| < r ;
x in { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , 6 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 7 } iff x in { x1 , x2 , x3 , x4 , 7 , 8 } & x in { x1 , x2 , x3 , x4 , 8 , 7 } ;
( G * ( j , jj ) ) `2 = ( G * ( 1 , i ) ) `2 .= ( G * ( 1 , i ) ) `2 .= ( G * ( 1 , j ) ) `2 .= ( G * ( 1 , j ) ) `2 .= ( G * ( 1 , j ) ) `2 .= ( G * ( 1 , j ) ) `2 .= ( G * ( 1 , j ) ) `2 .= ( G * ( 1 , j ) ) `2 .= ( G * ( 1 , j ) `2 .= ( G * ( 1 , j ) `2 .= ( G * ( 1 , j ) `2 .= ( G * ( 1 , j ) `2 .= ( G * ( 1 , j ) `2 .= ( G * ( 1 , j ) `2 .= ( G * ( 1 , j ) `2 .= ( G * ( 1 , j ) `2 .= ( G * ( 1 , j ) `2 .= ( G * ( 1 ,
f1 * p = p .= ( ( the Arity of S1 ) * ( the Arity of S1 ) ) . o .= ( ( the Arity of S1 ) * ( the Arity of S1 ) ) . o .= ( ( the Arity of S1 ) * ( the Arity of S1 ) ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o ;
func tree ( T , P , T1 ) -> Tree means : Def1 : q in it & q in P & p = T ^ <* x *> or p = T ^ <* x *> & q = T ^ <* x *> or p = <* x *> ;
F /. ( k + 1 ) = F . ( k + 1 -' 1 ) .= F . ( p . ( k + 1 -' 1 ) , F /. ( k + 1 -' 1 ) ) .= F . ( p . ( k + 1 -' 1 ) , p . ( k + 1 -' 1 ) ) .= F . ( p . ( k + 1 -' 1 ) , p . ( k + 1 -' 1 ) ) .= 0. ( TOP-REAL n ) ;
for A , B , C , D being Matrix of K st len B = len C & width B = len C & len C = width C & len B = width C & len C = width C & width B = width C & len C = width C & width B = width C & len B = width C & width B = width C & len B = width C & width B = width C holds B * C = C * ( B * C , C * C )
seq . ( k + 1 ) = 0. ( F_Complex ) + 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 + ( Partial_Sums ( seq ) ) . ( k + 1 ) ;
assume that x in ( the carrier of C1 ) and y in ( the carrier of C2 ) and z in ( the carrier of C2 ) and x = ( the carrier of C2 ) \/ ( the carrier of C2 ) and y in ( the carrier of C2 ) ;
defpred P [ Element of NAT ] means for f st len f = $1 & for k st k = $1 holds ( for k st k in dom g holds g . k = ( ( VAL g ) | ( k -' 1 ) ) . ( f /. k ) ) '&' ( ( VAL g ) | ( k -' 1 ) ) . ( f /. k ) ) ;
assume that 1 <= k and k + 1 <= len f and f is special and [ i , j ] in Indices G and [ i + 1 , j ] in Indices G and f /. k = G * ( i + 1 , j ) and f /. ( k + 1 ) = G * ( i + 1 , j ) ;
assume that sn < 1 and ( q `1 / |. q .| - sn ) > 0 and ( q `2 / |. q .| - sn ) >= 0 and ( q `2 / |. q .| - sn ) >= 0 and ( q `2 / |. q .| - sn ) >= 0 and ( q `2 / |. q .| - sn ) < 0 ;
for M being non empty metric , x being Point of M , f being Point of M st x = x & f = x & ex g being Point of M st g in Ball ( x , ( 1 / 2 ) * f ) & for n being Nat st n in dom f holds f . n = Ball ( x , ( 1 / 2 ) * f )
defpred P [ Element of omega ] means ( f1 is differentiable & f2 is differentiable on Z & for x st x in Z holds f1 . x = 1 / ( x - a ) * ( x - a ) ) & ( f1 - f2 ) is_differentiable_on Z & ( f1 - f2 ) `| Z = ( f1 - f2 ) `| Z ) . x = ( f1 - f2 ) . x - ( f1 - f2 ) . x ;
defpred P1 [ Nat , Point of C1 , Point of C ] means ( $1 in Y & $2 in Y & ||. $2 - x0 .|| < r & ||. $2 - x0 .|| < r ) & ||. $2 - x0 .|| < r ;
( f ^ mid ( g , 2 , len g ) ) . i = ( mid ( g , 2 , len g ) ) . i .= ( mid ( g , 2 , len g ) ) . i .= ( ( mid ( g , 2 , len g ) ) . i .= ( g /^ 1 ) . i .= ( g /^ 1 ) . i .= ( g /^ 1 ) . i .= ( g /^ 1 ) /. i .= ( g /^ 1 ) /. i .= ( g /^ 1 ) /. i .= ( g /^ 1 ) . i .= ( g /^ 1 ) . i .= ( g /^ 1 ) . i .= ( g /^ 1 ) . i .= ( g /^ 1 ) . i .= ( g /^ 1 ) . i .= ( g /^ 1 ) . i .= ( g /^ 1 ) . ( ( ( len g ) . ( ( ( len g ) . ( ( len g ) . ( len g ) . ( i -' 1 ) . ( i -' 1 ) .= ( g /^ 1 ) .
( 1 - 2 * ( n + 2 ) ) * ( 2 * ( n + 1 ) + 2 * ( n + 1 ) ) = ( ( 1 - 2 ) * ( n + 1 ) ) * ( 2 * ( n + 1 ) ) .= ( 1 - 2 ) * ( n + 1 ) .= ( 1 - 2 ) * ( n + 1 ) .= ( 1 - 2 ) * ( n + 1 ) ;
defpred P [ Nat ] means for G being non empty RelStr , A being non empty finite RelStr , x being set st G is $1 -symmetric symmetric & x in A & x in A & y in A holds ( the RelStr of G ) . ( x , y ) = ( the RelStr of G ) . ( x , y ) ;
assume that not f /. 1 in Ball ( u , r ) and 1 <= m and m <= len - 1 and for i st 1 <= i & i <= len - 1 & i <= len f holds not LSeg ( f , i ) /\ LSeg ( f , m ) <> {} ;
defpred P [ Element of NAT ] means ( Partial_Sums ( cos ) ) . $1 = ( Partial_Sums ( cos ) ) . ( $1 + 1 ) * ( Partial_Sums ( cos ) . ( $1 + 1 ) ) + ( Partial_Sums ( cos ) . ( $1 + 1 ) * ( Partial_Sums ( cos ) ) . ( $1 + 1 ) * ( Partial_Sums ( cos ) ) . ( $1 + 1 ) ) ;
for x being Element of product F holds x is FinSequence of G & dom x = I & for i being set st i in I holds x . i = ( ( the carrier of F ) | I ) . i & x . i = ( ( the carrier of F ) | I ) . i
( x " ) |^ ( n + 1 ) = ( ( x " ) * x ) |^ n .= ( x " ) * ( x " ) .= ( x " ) * ( x " ) .= ( x " ) * ( x " ) .= ( x " ) * ( x " ) .= ( x " ) * ( x " ) .= ( x " ) * ( x " ) .= ( x " ) * ( x " ) .= ( x " ) * ( x " ) .= ( x " ) * ( x " ) .= ( x " ) * ( x " ) * ( x " ) .= ( x " ) * ( x " ) ;
DataPart Comput ( P +* I , ( LifeSpan ( P +* I , s ) ) , k ) = DataPart Comput ( P +* I , ( LifeSpan ( P +* I , s ) ) + 3 ) .= DataPart Comput ( P +* I , ( LifeSpan ( P +* I , s ) ) + 3 .= DataPart Comput ( P +* I , s ) ;
given r such that 0 < r and ]. x0 - r , x0 + r .[ c= dom ( f1 + f2 ) /\ dom ( f1 + f2 ) and for g st g in ]. x0 - r , x0 + r .[ /\ dom ( f1 + f2 ) holds ( f1 + f2 ) . g <= 0 ;
assume that X c= dom f1 /\ dom f2 and f1 | X is continuous and f2 | X is continuous and ( for x st x in X /\ dom f2 holds f1 . x = f1 . x ) and ( f1 | X is continuous implies f2 | X is continuous ) & ( for x st x in X /\ dom f2 holds f1 . x = f2 . x ) ;
for L being continuous complete LATTICE for l being Element of L ex X being Subset of L st l = sup X & for x being Element of L st x in X holds x in X & x is directed & x is Ideal of L holds x is prime
Support ( e *' A ) = { m *' ( p *' A ) where m is Element of NAT : m in Support ( m *' p ) & p <> 0 & ( p *' A ) . m = ( p *' A ) . m & ( p *' A ) . m = ( p *' A ) . ( m + 1 ) ;
( f1 - f2 ) /. ( lim s1 ) = lim ( ( f1 - f2 ) /* s1 ) .= lim ( ( f1 - f2 ) /* s1 ) .= lim ( ( f1 - f2 ) /* s1 ) .= lim ( ( f1 - f2 ) /* s1 ) .= lim ( ( f1 - f2 ) /* s1 ) .= lim ( ( f1 - f2 ) /* s1 ) .= lim ( ( f1 - f2 ) /* s1 ) ;
ex p1 being Element of CQC-WFF ( Al ) st F . p = g . p & for g being Element of D ( ) st F . p = g . p & for g being Function st g in D ( ) holds P [ g . p , p1 , g . g ] ;
( mid ( f , i , len f -' 1 ) ) /. j = ( mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) /. j .= ( mid ( f , i , len f -' 1 ) /. j ) ;
( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) = ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( ( p ^ q ) . ( len p + k ) ) . ( len p + k ) .= ( ( p ^ q ) . k ) . ( len p + k ) .= ( ( p ^ q ) ^ q ) . ( k + 1 ) .= ( ( p ^ q ) . ( k + 1 ) .= ( p ^ q ) . ( k + 1 ) .= ( p ^ q ) . ( k + 1 ) .= ( p ^ q ) . ( k + 1 ) .= ( p ^ q ) . ( k + 1 ) . ( k + 1 ) .= ( p ^ q ) . ( k + 1 ) .= ( ( p ^ q ) . ( k + 1 ) .= ( ( p + q ) .= ( p ^ q ) . ( k + 1 ) . ( k + 1 ) .= ( p ^ q ) . ( k + 1 ) .= ( p ^ q )
len mid ( upper_volume ( f , D1 , j1 ) , indx ( D2 , D1 , j1 ) + 1 , indx ( D2 , D1 , j1 ) + 1 ) = indx ( D2 , D1 , j1 ) + 1 .= indx ( D2 , D1 , j1 ) + 1 - 1 .= indx ( D2 , D1 , j1 ) + 1 - 1 ;
x * y * z = ( M * ( x * y ) ) * ( x * z ) .= x * ( M * ( y * z ) ) .= x * ( M * ( y * z ) ) .= x * ( M * ( y * z ) ) .= x * ( M * ( y * z ) ) .= x * ( M * ( y * z ) ) ;
v . <* x , y *> - ( <* x0 *> ) . i = ( <* x0 , y0 *> ) . i * ( <* y , y0 *> ) + ( diff ( <* x0 , y0 *> , 1 ) ) . ( ( reproj ( 1 , 1 ) ) . ( ( reproj ( 1 , 1 ) ) . ( y , y0 ) ) + ( diff ( u , 1 ) ) . ( ( reproj ( 1 , 1 ) ) . ( y , y0 ) ) ) ;
i * i = <* 0 * ( 1 - i ) - ( 0 * i ) + ( 0 * i ) - ( 0 * i ) + ( 0 * i ) - ( 0 * i ) .= <* 0 * ( 1 - i ) + 0 * i - ( 0 * i ) + 0 * i - ( 0 * i ) .= 0 * ( 1 - i ) + 0 * i - 0 * i ;
Partial_Sums ( L * F ) = Partial_Sums ( L * ( F1 ^ F2 ) ) + 0. V .= ( L * ( F1 ^ F2 ) ) + 0. V .= ( L * ( F1 ^ F2 ) ) + 0. V .= ( L * ( F1 ^ F2 ) ) + 0. V .= ( L * ( F1 ^ F2 ) ) + 0. V .= ( L * ( F1 ^ F2 ) ) + 0. V .= ( L * ( F1 ^ F2 ) ) + 0. V .= ( L * ( F1 ^ F2 ) .= ( L * ( F1 ^ F2 ) ) + ( L * ( F1 ^ F2 ) .= ( L * ( F1 ^ F2 ) .= ( L * F1 ^ F2 ) + ( L * ( F2 ^ F2 ) .= ( L * ( F1 ^ F2 ) ) + ( L * ( F2 ^ F2 ) + ( L * ( F2 ^ F2 ) + ( L * ( F1 ^ F2 ) ) + ( L * ( F1 ^ F2 ) .= ( L * ( F1 ^ F2 ) .= ( L * ( F2 ^ F2 ) ) + ( L * F2 ) .= ( L * ( F1 ^ F2 )
ex r be Real st for e be Real st 0 < e ex Y be finite Subset of REAL st 0 < Y & Y c= dom ( ||. f .|| ) & for Y1 be finite Subset of X st Y1 in Y & Y c= Y holds |. ( f | Y ) . Y1 - 0 .| < r
( GoB f ) * ( i , j + 1 ) = f /. ( k + 2 ) & ( GoB f ) * ( i + 1 , j ) = f /. ( k + 2 ) or ( GoB f ) * ( i + 1 , j + 1 ) = f /. ( k + 2 ) or ( GoB f ) * ( i + 1 , j + 1 ) = f /. ( k + 2 ) ;
( ( - 1 ) (#) cos ) . x = ( ( r / ( 2 * x ) ) * ( cos . x ) ) ^2 .= ( ( r / ( 2 * x ) ) * ( cos . x ) ) ^2 .= ( ( r / ( 2 * x ) ) * ( cos . x ) ) ^2 .= ( r / ( 2 * x ) ) ^2 .= ( r / ( 2 * x ) ) ^2 ;
( - b + sqrt ( a , b , c ) ) * a + ( - b + sqrt ( a , b , c ) ) * a < 0 & ( - b + sqrt ( a , b , c ) ) * a < 0 & ( - b ) < 0 ;
Suppose ex_inf_of uparrow ( "\/" ( X , L ) /\ C ) and for X st X in L & X in C holds not X in C & not X in C & not Y in C & not Y in C & not Y in C & not Y in C & Y in C & not Y in C & Y in C & Y in C & Y in C & Y in C ) implies X in C & Y in C & X in C & Y in C & X in C & X in C & X in C & Y in C & X in C & Y in C & X in C & Y in C & X in C & Y in C implies "/\" ( ( ( \mathopen ( "/\" ) /\ C ) implies X in C & X in C & Y in C & X in C & Y in C & Y in C & Y in C & Y in C & Y in C & Y in C & Y in C & Y in C & Y in C & Y in C & Y in C & Y in C & Y in C & Y in C & Y in C & Y in C & Y in C &
( for B holds ( ( the Sorts of B ) . ( j , i ) ) . ( j , i ) = ( j |-> id ( the carrier of S ) ) . ( j , i ) ) & ( j = i implies ( j = i implies j = i ) ) implies ( j = i )