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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 ;
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 ;
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 finite ;
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 , c ;
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 L ;
not x in Y ;
z = +infty ;
k be Nat ;
K ` is being_line ;
assume n >= N ;
assume n >= N ;
assume X is let I ;
assume x in I ;
q is as Nat ;
assume c in x ;
p > 0 ;
assume x in Z ;
assume x in Z ;
1 <= k} ;
assume m <= i ;
assume G is finite ;
assume a divides b ;
assume P is closed ;
b-a > 0 ;
assume q in A ;
W is non 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 atomic ;
b `2 <= c `2 ;
A meets W ;
i `2 <= j `2 ;
assume H is universal ;
assume x in X ;
let X be set ;
let T be DecoratedTree ;
let d be element ;
let t be element ;
let x be element ;
let x be element ;
let s be element ;
k <= \rm implies k <= \rm implies k <= 5 ;
let X be set ;
let X be set ;
let y be element ;
let x be element ;
P [ 0 ]
let E be set , F be FinSequence of E ;
let C be category ;
let x be element ;
k be Nat ;
let x be element ;
let x be element ;
let e be element ;
let x be element ;
P [ 0 ]
let c be element ;
let y be element ;
let x be element ;
a be Real ;
let x be element ;
let X be element ;
P [ 0 ]
let x be element ;
let x be element ;
let y be element ;
r in REAL ;
let e be element ;
n1 is , \cdot ;
Q halts_on s ;
x in for for -1 st x in dom } holds x in 3 ;
M < m + 1 ;
T2 is open ;
z in b lim a ;
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 REAL ;
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 , r ;
let E be Ordinal ;
o implies o on o1 ;
O <> O2 ;
let r be Real ;
let f be FinSeq-Location ;
let i be Nat ;
let n be Nat ;
Cl A = A ;
L c= Cl L ;
A /\ M = B ;
let V be complex normed space , v be VECTOR of V ;
not s in Y |^ 0 ;
rng f is_<=_than w
b "/\" e = b ;
m = m3 ;
t in h . D ;
P [ 0 ] ;
assume z = x * y ;
S . n is bounded ;
let V be RealUnitarySpace , W be Subspace of V ;
P [ 1 ] ;
P [ {} ] ;
C1 is component ;
H = G . i ;
1 <= i `2 + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
a` <= non L~ \HM { a } ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , W be Subspace of V ;
s is trivial non empty ;
dom c = Q
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be complete LATTICE , X be Subset of T ;
the Arrows 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 ;
SNAT is bounded ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let UA , A , B ;
pp = 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 & 1 <= j ;
set A = [: NAT , NAT :] ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H is with_no for F is \cdot or F is set
assume n0 <= m ;
T is increasing ;
e2 <> e2 & e2 <> e2
Z c= dom g ;
dom p = X ;
H is proper ;
i + 1 <= n ;
v <> 0. V ;
A c= Affin A ;
S c= dom F ;
m in dom f ;
let X0 be set ;
c = sup N ;
R is_connected implies union M in 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 ;
mm <> {} ;
let x be Element of Y ;
let f be + being ) Chain Chain of P ;
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 card dom that e in v ;
- y in I ;
let A be non empty set , f be Function ;
Px0 = 1 ;
assume r in F . k ;
assume f is simple function of S ;
let A be \cdot \ -1 set ;
rng f c= NAT ;
assume P [ k ] ;
ff <> {} ;
o be Ordinal ;
assume x is sum of non squares ;
assume not v in { 1 } ;
let IB , C ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
d1 in dom f ;
assume t . 1 in A ;
let Y be non empty TopSpace , f be Function of Y , Y ;
assume a in uparrow s ;
let S be non empty Poset ;
a , b // b , a ;
a * b = p * q ;
assume x , y are_the space ;
assume x in Omega ( f ) ;
[ a , c ] in X ;
mm <> {} ;
M + N c= M + M ;
assume M is \mathclose k1 k1 k1 ;
assume f is additive for brbeing set ;
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 = k1 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
n + i >= n + 1 ;
assume Re y = 0 ;
k1 <= j1 & j1 <= len f ;
f | A is non as as as continuous ;
f . x - b <= 0 ;
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 ;
CB in X ;
q2 c= C1 & q2 c= C2 ;
a2 < c2 & c2 < c1 ;
s2 is 0 -started ;
IC s = 0 ;
s4 = s4 , P4 = P3 ;
let V , C ;
let x , y be element ;
let x be Element of T ;
assume a in rng F ;
x in dom T `2 ;
let S be <> <> <> <> the carrier of L ;
y " <> 0 ;
y " <> 0 ;
0. V = u-w ;
y2 , y , w be Element of V ;
R8 ;
let a , b be Real , x be Point of REAL ;
let a be object of C ;
let x be Vertex of G ;
let o be object of C , a be object of A ;
r '&' q = P \lbrack l .] ;
let i , j be Nat ;
let s be State of A , v be Element of S ;
s4 . n = N ;
set y = x `1 ;
NAT in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in CX0 ;
V-19 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 Nffffffffffffffffffffffffffff
reconsider i = i as Ordinal ;
r * v = 0. X ;
rng f c= INT & rng g c= INT ;
G = 0 .--> goto 0 ;
let A be Subset of X ;
assume A0 is dense & A is open ;
|. 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 ;
[: x , Y :] c= Z1 ;
dom f = C1 & dom g = C2 ;
assume [ a , y ] in X ;
Re seq is convergent & Im seq is convergent ;
assume a1 = b1 & a2 = b2 ;
A = sInt A ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be Instruction of S , I be Element of S ;
assume that r2 > x0 and r1 > x0 ;
let Y be non empty set , f be Function of Y , Z ;
2 * x in dom W ;
m in dom g2 & n + 1 in dom g2 ;
n in dom g1 /\ dom g2 ;
k + 1 in dom f ;
the still of { s } is finite ;
assume that x1 <> x2 and y1 <> y2 ;
v3 in V0 & v2 in V0 ;
not [ b `1 , b ] in T ;
i-35 + 1 = i ;
T c= <> * ( X , T ) ;
l `1 = 0 & l `2 = 0 ;
let n be Nat ;
t `2 = r & t `2 = s ;
AcA is_integrable_on M ;
set t = Top t ;
let A , B be real-membered set ;
k <= len G + 1 ;
C ( ) misses V ( ) ;
product ( s ) is non empty ;
e <= f or f <= e ;
cluster non empty normal for Ordinal ;
assume c2 = b2 & c2 = b1 ;
assume h in [. q , p .] ;
1 + 1 <= len C ;
not c in B . m1 ;
cluster R .: X -> empty ;
p . n = H . n ;
assume that vseq is convergent and vseq is convergent ;
IC s3 = 0 & IC s3 = 0 ;
k in N or k in K ;
F1 \/ F2 c= F ;
Int G1 <> {} & Int G2 <> {} ;
z `2 = 0 & z `2 = 0 ;
p11 <> p1 & p11 <> p2 ;
assume z in { y , w } ;
MaxADSet ( a ) c= F ;
ex_sup_of downarrow s , S ;
f . x <= f . y ;
let T be up-complete non empty reflexive antisymmetric RelStr ;
q |^ m >= 1 ;
a is_>=_than X & b is_>=_than Y ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one one-to-one full ;
A \/ { a } \not c= B ;
0. V = 0. Y ;
let I be halting Instruction of S , s be State of S ;
f-24 . x = 1 ;
assume z \ x = 0. X ;
C2 = 2 to_power n ;
let B be SetSequence of Sigma ;
assume X1 = p .: D ;
n + l2 in NAT ;
f " P is compact & f " P is compact ;
assume x1 in REAL & x2 in REAL ;
p1 = K1 & p2 = K1 ;
M . k = <*> REAL ;
phi . 0 in rng phi ;
OSMM^ A is closed
assume z0 <> 0. L ;
n < ( N . k ) ;
0 <= seq . 0 & seq . 0 <= seq . 0 ;
- q + p = v ;
{ v } is Subset of B ;
set g = f `| 1 ;
cR is stable Subset of R ;
set cR = Vertices R ;
pp0 c= P3 & pp0 c= P3 ;
x in [. 0 , 1 .[ ;
f . y in dom F ;
let T be 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 ~ are_isomorphic ;
assume a in A ( ) ;
k in dom ( q | i ) ;
p is \HM { finite } -valued FinSequence of S ;
i -' 1 = i-1 ;
f | A is one-to-one ;
assume x in f .: X ( ) ;
i2 - i1 = 0 ;
j2 + 1 <= i2 ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster strict for for for for let `1 ;
|. q .| ^2 > 0 ;
|. p4 .| = |. p .| ;
s2 - s1 > 0 ;
assume x in { Gij } ;
W-min C in C & W-min C in C ;
assume x in { Gij } ;
assume i + 1 = len G ;
assume i + 1 = len G ;
dom I = Seg n .= dom J ;
assume that k in dom C and k <> i ;
1 + 1-1 <= i + j ;
dom S = dom F /\ dom G ;
let s be Element of NAT ;
let R be ManySortedSet of A ;
let n be Element of NAT ;
let S be non empty non void non void holds S is H -holds S is non void
let f be ManySortedSet of I ;
let z be Element of F_Complex , v be Element of COMPLEX ;
u in { ag } ;
2 * n < 2 * n ;
let x , y be set ;
B-11 c= V-15 \/ { x } ;
assume I is_closed_on s , P ;
UA = [: U1 , U2 :] ;
M /. 1 = z /. 1 ;
x = x22 & x = x22 ;
i + 1 < n + 1 + 1 ;
x in { {} , <* 0 *> } ;
( f | X ) . x <= ( f | X ) . x ;
let l be Element of L ;
x in dom ( F . 0 ) ;
let i be Element of NAT ;
seq1 is COMPLEX -valued & seq2 is COMPLEX -valued ;
assume <* o2 , o *> <> {} ;
s . x |^ 0 = 1 ;
card K1 in M & card K1 in M ;
assume that X in U and Y in U ;
let D be Subset-Family of Omega ;
set r = q - { k + 1 } ;
y = W . ( 2 * an ) ;
assume dom g = cod f & cod g = cod f ;
let X , Y be non empty TopSpace , f be Function of X , Y ;
x ++ A is interval ;
|. <*> A .| . a = 0 ;
cluster strict for SubLattice of L ;
a1 in B . s1 & a2 in B . s2 ;
let V be finite < F , v be Vector of V ;
A * B on B , A ;
fg = NAT --> 0 ;
let A , B be Subset of V ;
z1 = P1 . j & z2 = P1 . j ;
assume f " P is closed ;
reconsider j = i as Element of M ;
let a , b be Element of L ;
assume q in A \/ ( B "\/" C ) ;
dom ( F * C ) = o ;
set S = INT |^ X ;
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 B ;
PI / 2 < Arg z ;
reconsider z9 = 0 , z9 = 1 as Nat ;
LIN a , d , c ;
[ y , x ] in II ;
Q * ( 1 , 3 ) `2 = 0 ;
set j = x0 gcd m , m = x0 gcd m ;
assume a in { x , y , c } ;
j2 - jj > 0 & j2 - jj > 0 ;
I the I the string of phi = 1 ;
[ y , d ] in F-8 ;
let f be Function of X , Y ;
set A2 = ( B - C ) / ( A |^ n ) ;
s1 , s2 are_` & s2 , s1 are_` ;
j1 -' 1 = 0 & j2 -' 1 = 1 ;
set m2 = 2 * n + j ;
reconsider t = t as bag of n ;
I2 . j = m . j ;
i |^ s , n are_relative_prime ;
set g = f | D-21 ;
assume that X is lower bounded and 0 <= r ;
p1 `1 = 1 & p1 `2 = - 1 ;
a < p3 `1 & p3 `1 < b ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= i1 -' 1 & i1 + 1 <= len f ;
1 <= i1 -' 1 & i1 + 1 <= len f ;
i + i2 <= len h ;
x = W-min ( P ) & x = W-min ( P ) ;
[ x , z ] in [: X , Z :] ;
assume y in [. x0 , x .] ;
assume p = <* 1 , 2 , 3 *> ;
len <* A1 *> = 1 ;
set H = h . gg ;
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 (*) h1 ;
assume x in X3 /\ X3 & x in X3 /\ X3 ;
||. h .|| < d1 & ||. h .|| < d1 ;
not x in the carrier of f ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k - l = kbeing - k\leq ;
<* p , q *> /. 2 = q ;
let S be Subset of the carrier of Y ;
let P , Q be \rm \geq s ;
Q /\ M c= union ( F | M )
f = b * canFS ( S ) ;
let a , b be Element of G ;
f .: X is_<=_than f . sup X
let L be non empty transitive RelStr , x , y be Element of L ;
S-20 is x -basis i -basis
let r be non positive Real ;
M , v |= x \hbox \hbox { = } y ;
v + w = 0. ( Z , p ) ;
P [ len F ( ) ] ;
assume that InsCode ( i ) = 8 and InsCode ( i ) = 8 ;
the zero of M = 0 implies 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 -> N \rm implies for Element of S ;
reconsider l1 = l- 1 as Nat ;
( for v being Vertex of r2 holds v . v = r2 . v ) implies v is Vertex of r2
T2 is SubSpace of T2 implies the carrier of T1 = the carrier of T2
Q1 /\ Q19 <> {} & Q29 /\ Q29 <> {} ;
k be Nat ;
q " is Element of X & q " is Element of X ;
F . t is set of of of set , M ;
assume that n <> 0 and n <> 1 ;
set en = EmptyBag n , en = EmptyBag n ;
let b be Element of Bags n ;
assume for i holds b . i is commutative ;
x is root & x `2 = - 1 ;
not r in ]. p , q .[ ;
let R be FinSequence of REAL , a be Real ;
( not 7 does not destroy b1 ) & not b1 on b2 ;
IC SCM R <> a & IC SCM R <> a ;
|. - |[ x , y ]| .| >= r ;
1 * seq = seq & 1 * seq = seq ;
let x be FinSequence of 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 .= IC s ;
H + G = F- ( G-GG ) ;
CC1 . x = x2 & CC2 . x = y2 ;
f1 = f .= f2 .= f2 * f1 ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u + W ;
{ a1 } = { a2 } & { a2 } = { a1 } ;
a1 , b1 _|_ b , a ;
d1 , o _|_ o , a3 ;
IO is reflexive & IO is transitive implies IO is reflexive
IO is antisymmetric implies [: O , O :] is antisymmetric
sup rng H1 = e & sup rng H2 = e ;
x = ( a * a9 ) * ( b * b9 ) ;
|. 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 support b misses support a ;
let L be associative commutative associative non empty doubleLoopStr , p be Polynomial of L ;
s " + 0 < n + 1 ;
p . c = ( f " ) . 1 ;
R . n <= R . ( n + 1 ) ;
Directed ( I1 , I2 ) = I1 +* Directed ( I2 , 1 ) ;
set f = + ( x , y , r ) ;
cluster Ball ( x , r ) -> bounded ;
consider r being Real such that r in A ;
cluster non empty NAT -defined for Function ;
let X be non empty directed Subset of S ;
let S be non empty full SubRelStr of L ;
cluster <* [ ] , ] *> -> complete non trivial ;
( 1 - a ) " = a ;
( q . {} ) `1 = o ;
( n - 1 ) > 0 ;
assume 1 / 2 <= t `1 ;
card B = k + 1-1 ;
x in union rng ( f | ( len f ) ) ;
assume x in the carrier of R & y in the carrier of R ;
d in X ;
f . 1 = L . ( F . 1 ) ;
the vertices of G = { v } & not v in V ;
let G be st G is : Let w be set ;
e , v9 be set , x be element ;
c . ( i9 - 1 ) in rng c ;
f2 /* q is divergent_to-infty & f2 /* q is divergent_to-infty ;
set z1 = - z2 , z2 = - z2 , z2 = - z1 ;
assume w is llas of S , G ;
set f = p |-count ( t ) , g = p |-count ( t ) ;
let c be Object of C ;
assume ex a st P [ a ] ;
let x be Element of REAL m , r be Real ;
let IB be Subset-Family of X , x be Point of X ;
reconsider p = p , q = q 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 SCM+FSA , the carrier of SCM+FSA ;
stop I ( ) c= P-12 ( ) ;
set ci = fbeing /. i ;
w ^ t ^ s ^ w ^ t ^ w ^ t ^ s ^ t ^ t ^ t ^ w ^ t ^ w ^ t ^ w ^ t ^ w ^ t ^ w ^ w ^
W1 /\ W = W1 /\ W ` .= W2 /\ W ;
f . j is Element of J . j ;
let x , y be \rm \cdot of T2 ;
ex d st a , b // b , d ;
a <> 0 & b <> 0 & c <> 0
ord x = 1 & x is dom p implies x is dom p
set g2 = lim ( seq ^\ k ) ;
2 * x >= 2 * ( 1 / 2 ) ;
assume ( a 'or' c ) . z <> TRUE ;
f (*) g in Hom ( c , c ) ;
Hom ( c , c + d ) <> {} ;
assume 2 * Sum ( q | m ) > m ;
L1 . F-21 = 0 & L2 . F-21 = 1 ;
h \/ R1 = h implies R1 = R2
( sin . x ) ^2 <> 0 & ( sin . x ) ^2 <> 0 ;
( #Z n ) . x > 0 & ( #Z n ) . x > 0 ;
o1 in X-5 /\ O2 & o2 in Xor o1 = o2 ;
e , v9 be set , x be element ;
r3 > ( 1 / 2 ) * 0 ;
x in P .: ( F -Ideal ( L ) ) ;
let J be closed Ideal of R ;
h . p1 = f2 . O & h . p2 = g2 . O ;
Index ( p , f ) + 1 <= j ;
len ( q | M ) = width M .= width M ;
the carrier of `1 c= A & the carrier of L = A ;
dom f c= union rng ( F | ( n + 1 ) )
k + 1 in support ( s ) & k + 1 in dom ( s ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y `2 ] in ( \/ \/ R ) ;
i = D1 or i = D2 or i = D1 ;
assume a mod n = b mod n & b mod n = 0 ;
h . x2 = g . x1 & h . x2 = g . x2 ;
F c= 2 -tuples_on the carrier of X & F is one-to-one ;
reconsider w = |. s1 .| as Real_Sequence ;
1 / ( m * m + r ) < p ;
dom f = dom IK1 & dom IK1 = dom IK1 ;
[#] P-17 = [#] ( K ) & [#] P-2 = [#] ( K ) ;
cluster - x -> ExtReal means : Def8 : x <= - y ;
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 W3 implies u in W2
reconsider y = y , z = z as Element of L2 ;
N is full SubRelStr of T |^ the carrier of S ;
sup { x , y } = c "\/" c ;
g . n = n to_power 1 .= n ;
h . J = EqClass ( u , J ) ;
let seq be summable sequence of X , x be Element of X ;
dist ( x `2 , y ) < r / 2 ;
reconsider mm = m , mn = n as Element of NAT ;
x- x0 < r1 - x0 & x - x0 < r2 - x0 ;
reconsider P ` = P ` as strict Subgroup of N ;
set g1 = p * idseq ( q `1 ) ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is lower ;
D2 . ( ID2 . ( k + 1 ) ) in { x } ;
cluster subcondensed for Subset of T ;
let P be compact non empty Subset of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ;
Gik in LSeg ( PI , 1 ) /\ LSeg ( Gik , Gij ) ;
n be Element of NAT , x be Element of X ;
reconsider SS = S , SS = T as Subset of T ;
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 as Element of NAT ;
reconsider d = x `2 as Element of C ( ) ;
let s be 0 -started State of SCMPDS , a be Int-Location ;
let t be 0 -started State of SCMPDS , Q ;
b , b , x , y , z is_collinear ;
assume that i = n \/ { n } and j = k \/ { k } ;
let f be PartFunc of X , Y ;
let N1 >= ( sqrt c ) / sqrt 2 & N2 >= 0 ;
reconsider t7 = T" as TopSpace , T8 = T8 " ;
set q = h * p ^ <* d *> ;
z2 in U . ( y2 ) /\ Q2 & z2 in Q . ( y2 ) /\ Q2 ;
A |^ 0 = { <%> E } & A |^ 1 = A ;
len W2 = len W + 2 & len W2 = len W + 1 ;
len h2 in dom h2 & len h2 in dom h2 ;
i + 1 in Seg ( len s2 + 1 ) ;
z in dom g1 /\ dom f & z in dom g1 /\ dom f ;
assume that p2 = E-max ( K ) and p1 `2 = - 1 ;
len G + 1 <= i1 + 1 ;
f1 (#) f2 is convergent & lim ( f1 (#) f2 ) = x0 ;
cluster s-10 + sT -> summable for Real_Sequence ;
assume j in dom M1 & i + 1 in dom M1 ;
let A , B , C be Subset of X ;
let x , y , z be Point of X , p be Point of p ;
b ^2 - ( 4 * a * c ) >= 0 ;
<* x/y *> ^ <* y *> ^ <* y *> ^ x ` ;
a , b in { a , b } ;
len p2 is Element of NAT & len p1 = len p2 ;
ex x being element st x in dom R & R . x = y ;
len q = len ( K (#) G ) ;
s1 = Initialize Initialized s , P1 = P +* I ;
consider w being Nat such that q = z + w ;
x ` is F of x , L & x ` is Element of L ;
k = 0 & n <> k or k > n ;
then X is discrete for A being Subset of X ;
for x st x in L holds x is FinSequence ;
||. f /. c .|| <= r1 & ||. f /. c .|| <= r2 ;
c in uparrow p & not c in { p } ;
reconsider V = V as Subset of the topology of TOP-REAL n ;
let N , M be being being being being being being being being being being being being being being 0 Element of L ;
then z is_>=_than waybelow x & z is_>=_than compactbelow y ;
M \lbrack f , g .] = f & M [. g , f .] = g ;
( ( >= 1 ) to_power ( 1 + 1 ) ) = TRUE ;
dom g = dom f |^ X & rng g c= dom f ;
mode : il of G is \HM { w.r.t. } ;
[ i , j ] in Indices M & [ i , j ] in Indices M ;
reconsider s = x " , t = y " as Element of H ;
let f be Element of dom Subformulae p & f . f = F ( f ) ;
F1 . ( a1 , - a1 ) = G1 . ( a1 , - a1 ) ;
redefine func E ( a , b , r ) -> compact ;
let a , b , c , d be Real ;
rng s c= dom ( 1 / ( n + 1 ) ) ;
curry ( ( F . -19 ) , k ) is additive additive ;
set k2 = card dom B , k1 = card dom C , k2 = card dom D ;
set G = ( the Sorts of X ) * the Arity of S ;
reconsider a = [ x , s ] as Object of G ;
let a , b be Element of ML , x be Element of ML ;
reconsider s1 = s , s2 = t as Element of S1 ;
rng p c= the carrier of L & rng q c= the carrier of L ;
let d be Subset of the Sorts of A ;
( x .|. x = 0 iff x = 0. W )
I-21 in dom stop I & IY in dom stop I ;
let g be continuous Function of X | B , Y ;
reconsider D = Y , E = X as Subset of TOP-REAL n ;
reconsider i0 = len p1 , i2 = len p2 as Integer ;
dom f = the carrier of S & rng g c= the carrier of T ;
rng h c= union ( the carrier of J ) & rng h c= union ( the carrier of J )
cluster All ( x , H ) -> \cal \widetilde L~ -like ;
d * N1 ^2 > N1 * 1 & d * N2 ^2 > 0 ;
]. a , b .] c= [. a , b .] ;
set g = f " | D1 , h = f " | D2 ;
dom ( p | mm1 ) = mm1 & dom ( p | mm1 ) = mm1 ;
3 + - 2 <= k + - 2 ;
tan is_differentiable_in ( arccot . x ) & tan . x > 0 ;
x in rng ( f /^ n ) /\ rng ( f /^ n ) ;
let f , g be FinSequence of D ;
p ( ) in the carrier of S1 & P [ p ] ;
rng f " = dom f & rng f = rng g ;
( the Target of G ) . e = v & ( the Target of G ) . e = v ;
width G -' 1 < width G -' 1 ;
assume v in rng ( S | E1 ) ;
assume x is root or x is root or x is root ;
assume 0 in rng ( g2 | A ) ;
let q be Point of TOP-REAL 2 , r be Real ;
let p be Point of TOP-REAL 2 , r be Real ;
dist ( O , u ) <= |. p2 .| + 1 ;
assume dist ( x , b ) < dist ( a , b ) ;
<* S7 *> is_the carrier of C-20 & <* C7 *> is non empty ;
i <= len ( G * ( i1 -' 1 , j1 ) ) ;
let p be Point of TOP-REAL 2 , r be Real ;
x1 in the carrier of I[01] & x2 in the carrier of I[01] ;
set p1 = f /. i ;
g in { g2 : r < g2 & g2 < r } ;
Q2 = SdL " Q .= SdL " Q ;
( ( 1 / 2 ) |^ ( n + 1 ) ) is summable ;
- p + I c= - p + A ;
n < LifeSpan ( P1 , s1 ) + 1 ;
CurInstr ( p1 , s1 ) = i .= halt SCM+FSA ;
A /\ Cl { x } \ { x } <> {} ;
rng f c= ]. r - 1 , r + 1 .[ ;
let g be Function of S , V ;
let f be Function of L1 , L2 , g be Function of L2 , L1 ;
reconsider z = z , t = y as Element of CompactSublatt L ;
let f be Function of S , T ;
reconsider g = g as Morphism of c opp , b opp ;
[ s , I ] in [: S , f1 :] ;
len ( the connectives of C ) = 4 & len ( the connectives of C ) = 3 ;
let C1 , C2 be subFunctor of C , D ;
reconsider V1 = V , V2 = V as Subset of X | B ;
attr p is valid means : Def8 : All ( x , p ) is valid ;
assume that X c= dom f and f .: X c= dom g and g . X = g . X ;
H |^ a " is Subgroup of H & H |^ a = H ;
let A1 be p1 such that A1 on E1 , E1 & A2 on E1 ;
p2 , r3 , q2 is_collinear & q2 , q3 , q3 is_collinear ;
consider x being element such that x in v ^ K ;
not x in { 0. TOP-REAL 2 } & not x in { 0. TOP-REAL 2 } ;
p in [#] ( I[01] | B11 ) & p in the carrier of I[01] ;
0 . 0 < M . ( E8 . n ) ;
op ( c , c ) |^ a = c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . s2 ) & a2 in dom ( F . s2 ) ;
cluster -> of *> -* w for F -st L is Let ;
set i1 = the Nat , i2 = the Element of NAT ;
let s be 0 -started State of SCM+FSA , I be Program of SCM+FSA ;
assume y in ( f1 union f2 ) .: A ;
f . ( len f ) = f /. len f .= f /. 1 ;
x , f . x '||' f . x , f . y ;
pred X c= Y means : Def8 : cos | X c= cos | Y ;
let y be upper Subset of Y , x be Element of X ;
cluster -> -> as as NAT real for Element of NAT ;
set S = <* Bags n , ( T /. i ) *> ;
set T = [. 0 , 1 / 2 .] ;
1 in dom mid ( f , 1 , 1 ) ;
( 4 * PI ) / 2 < ( 2 * PI ) / 2 ;
x2 in dom f1 /\ dom f & x2 in dom f1 /\ dom f ;
O c= dom I & { {} } = { {} } ;
( the Target of G ) . x = v & ( the Target 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 be Element of G opp ;
h19 . i = f . ( h . i ) ;
p `1 = p1 `1 & p `2 = p2 `2 or p `1 = p1 `2 ;
i + 1 <= len Cage ( C , n ) ;
len <* P *> @ = len P & len <* P @ *> = 1 ;
set N-26 = the non empty Subset of N , NN2 = the carrier of N ;
len gLet gLet + ( x + 1 ) - 1 <= x ;
a on B & b on B implies not a on B
reconsider r-12 = r * I . v as FinSequence of REAL ;
consider d such that x = d and a [= d ;
given u such that u in W and x = v + u ;
len f /. ( \downharpoonright n , i ) = len ^2 ;
set q2 = N-min L~ Cage ( C , n ) , q2 = W-min L~ Cage ( C , n ) ;
set S =
MaxADSet ( b ) c= MaxADSet ( P /\ Q ) ;
Cl ( G . q1 ) c= F . r2 & Cl ( G . q2 ) c= F . q2 ;
f " D meets h " V & g " D meets 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 ( the carrier of S ) * ;
rng f c= the carrier of S2 & rng g c= the carrier of S2 ;
consider y being Element of X such that x = { y } ;
f1 . ( a1 , b1 ) = b1 & f1 . ( b1 , b2 ) = b2 ;
the carrier' of G `1 = E \/ { E } & the carrier' of G `2 = E ;
reconsider m = len thesis - k as Element of NAT ;
set S1 = LSeg ( n , UMP C ) , S2 = LSeg ( f , n ) ;
[ i , j ] in Indices M1 & [ i , j ] in Indices M1 ;
assume that P c= Seg m and M is \HM { an } and P is { \cdot } ;
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 ;
( p . i ) `1 = ( p . i ) `1 ;
let PA , PA be a_partition of Y , PA be Subset of Y ;
pred 0 < r & r < 1 implies 1 < r & r < 1 ;
rng ( ( \cal E } ) | X ) = [#] X ;
reconsider x = x , y = y , z = z 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 & len ( canFS ( s ) ) = card s ;
reconsider x2 = x1 , y2 = x2 as Element of L2 ;
Q in FinMeetCl ( the topology of X ) & Q c= FinMeetCl the topology of X ;
dom ( fx0 ) c= dom ( ux0 ) /\ dom ( fx0 ) ;
pred n divides m means : Def8 : m divides n & n = m ;
reconsider x = x , y = y , z = z as Point of [: I[01] , I[01] :] ;
a in dom ( the in s of R , T2 ) ;
not y0 in the carrier of f & not y0 in the carrier of g ;
Hom ( ( a [: b , c :] , c ) , ( b * c ) ) <> {} ;
consider k1 such that p " < k1 and k1 < len p ;
consider c , d such that dom f = c \ d ;
[ x , y ] in [: dom g , dom k :] ;
set S1 = a1 +* ( x , y , z ) ;
l1 = m2 & l1 = i2 & l2 = j2 implies l1 = l2
x0 in dom ( u01 /\ A01 ) & x0 in dom ( u01 /\ A01 ) ;
reconsider p = x , q = y , r = z as Point of TOP-REAL 2 ;
I[01] = R^1 | B01 & the carrier of I[01] = the carrier of TOP-REAL 2 ;
f . p4 <= f . p1 & f . p1 <= f . p2 ;
( F `1 ) ^2 / ( F `2 ) ^2 <= ( x `1 ) ^2 / ( F `2 ) ^2 ;
x `2 = ( W7 ) `2 .= ( W8 ) `2 ;
for n being Element of NAT holds P [ n ] ;
let J , K be non empty Subset of I ;
assume 1 <= i & i <= len <* a " *> ;
0 |-> a = <*> the carrier of K & 0 |-> a = <*> the carrier of K ;
X . i in 2 to_power ( A . i \ B . i ) ;
<* 0 *> in dom ( e --> [ 1 , 0 ] ) ;
then P [ a ] & P [ succ a ] implies P [ succ a ] ;
reconsider sl = s/. i , sl = sl /. i as w of D ;
( i - 1 ) <= len thesis & ( i - 1 ) <= len thesis ;
[#] S c= [#] the TopStruct of T & [#] the TopStruct of S = [#] the TopStruct of T ;
for V being strict RealUnitarySpace holds V in
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 n1 , K ;
- a * - b = a * b & - a * b = - a ;
for A being Subset of AS holds A // A & A // C implies A = C
( for o2 being object of o2 st o2 in <^ o2 , o2 ^> holds o2 = o1 ) implies o1 = o2
then ||. x .|| = 0 & x = 0. X ;
let N1 , N2 be strict normal Subgroup of G , a be Element of G ;
j >= len ( upper_volume ( g , D1 ) | indx ( D2 , D1 , j ) ) ;
b = Q . ( len Qb - 1 + 1 ) ;
f2 (#) f1 /* s is divergent_to-infty & f2 (#) f2 /* s is divergent_to-infty ;
reconsider h = f * g as Function of N1 , G ;
assume that a <> 0 and Let a , b , c be Real ;
[ t , t ] in the Relation of A & [ t , t ] in the InternalRel of A ;
( v |-- E ) | n is Element of ( T | n ) ;
{} = the carrier of L1 + L2 & the carrier of L1 + L2 = the carrier of L2 ;
Directed I is_closed_on Initialized s , P & Directed I is_closed_on Initialized s , P ;
Initialized p = Initialize ( p +* q ) .= p +* q +* q ;
reconsider N2 = N1 , N2 = N2 as strict net of R1 , R2 ;
reconsider Y = Y as Element of <* Ids L , \subseteq *> ;
"/\" ( uparrow p \ { p } , L ) <> p ;
consider j being Nat such that i2 = i1 + j and j in dom f ;
not [ s , 0 ] in the carrier of S2 & not [ s , 0 ] in the carrier of S2 ;
mm in ( B '/\' C ) '/\' D \ { {} } ;
n <= len ( P + ) & len ( P + Q ) = len P + len Q ;
x1 `1 = x2 & y1 `2 = y2 & x1 `2 = y2 ;
InputVertices S = { x1 , x2 } & InputVertices S = { x1 , x2 } ;
let x , y be Element of FTT1 ( n ) ;
p = |[ p `1 , p `2 ]| & p `2 = |[ p `2 , p `2 ]| ;
g * 1_ G = h " * g * h ;
let p , q be Element of PFuncs ( V , C ) ;
x0 in dom x1 /\ dom x2 & x1 . x0 = x1 . x0 + x2 . x0 ;
( R qua Function ) " = R " & ( R " ) " = R " ;
n in Seg len ( f /^ ( i -' 1 ) ) ;
for s being Real st s in R holds s <= s2 implies |. s1 .| <= |. s2 .|
rng s c= dom ( f2 * f1 ) /\ dom ( f2 * f1 ) ;
synonym for for for for for for for for for the carrier of X , the carrier of X ;
1_ K * 1_ K = 1_ K & 1_ K * 1_ K = 1_ K ;
set S = Segm ( A , P1 , Q1 ) , P = Segm ( A , P1 , Q1 ) ;
ex w st e = ( w - f ) / 2 & w in F ;
curry ( ( P+* ( i , k ) ) # x ) is convergent ;
cluster open -> open for Subset of [: T , T :] ;
len f1 = 1 .= len f3 + 1 .= len f3 + 1 ;
( i * p ) / p < ( 2 * p ) / p ;
let x , y be Element of OSSub ( U0 ) ;
b1 , c1 // b9 , c9 & b1 , c1 // b9 , c9 ;
consider p be element such that c1 . j = { p } ;
assume that f " { 0 } = {} and f is total ;
assume that IC Comput ( F , s , k ) = n and F . IC Comput ( F , s , k ) = halt SCM+FSA ;
Reloc ( J , card I + 1 ) does not ` not ` ;
Macro ( card I + 1 ) does not ` ;
set m3 = LifeSpan ( p3 , s3 ) , P4 = Comput ( p3 , s3 , 1 ) ;
IC SCMPDS in dom Initialize p & IC Comput ( p , s , k ) in dom I ;
dom t = the carrier of SCM R & dom t = the carrier of SCM R ;
( ( E-max L~ f ) .. f ) .. f = 1 ;
let a , b be Element of thesis of thesis , f , g be Function ;
Cl Int ( union F ) c= Cl Int ( union F ) ;
the carrier of X1 union X2 misses ( ( X1 union X2 ) \/ ( ( X1 union X2 ) \/ ( X2 \/ A ) ) ) ;
assume not LIN a , f . a , g . a ;
consider i being Element of M such that i = d6 and i in D ;
then Y c= { x } or Y = { x } ;
M , v / ( y , x ) / ( y , x ) |= H ;
consider m being element such that m in Intersect ( ( F . 0 ) ) ;
reconsider A1 = support u1 , A2 = support u2 as Subset of X ;
card ( A \/ B ) = k-1 + ( 2 * 1 ) ;
assume that a1 <> a3 and a2 <> a4 and a3 <> a4 and a4 <> a5 ;
cluster s -carrier V -> $ string of S , X , Y be non empty for string of S ;
LL2 /. n2 = LL2 . n2 .= LL2 . n2 .= LL2 /. n2 ;
let P be compact non empty Subset of TOP-REAL 2 , p1 , p2 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 , a be Real ;
assume that [ k , m ] in Indices DD1 and D * ( k , m ) = D * ( k , m ) ;
0 <= ( ( 1 / 2 ) |^ p ) / ( 2 |^ n ) ;
( F . N ) | E8 . x = +infty ;
pred X c= Y & Z c= V implies X \ V c= Y \ Z ;
y `2 * z `2 <> 0. I & y `2 * z `2 = 0. I ;
1 + card X-18 <= card u + card X-18 ;
set g = z :- ( ( L~ z ) .. z ) ;
then k = 1 & p . k = <* x , y *> . k ;
cluster total for Element of C -\mathopen the carrier of X , the carrier of X ;
reconsider B = A , C = B as non empty Subset of TOP-REAL n ;
let a , b , c be Function of Y , BOOLEAN , p be Function of Y , BOOLEAN ;
L1 . i = ( i .--> g ) . i .= g ;
Plane ( x1 , x2 , x3 , x4 ) c= P & Plane ( x1 , x2 , x3 , x4 ) c= P ;
n <= indx ( D2 , D1 , j1 ) + 1 ;
( ( ( g2 ) . O ) `1 ) ^2 = - 1 ;
j + p .. f - len f <= len f - len f ;
set W = W-bound C , E = E-bound C ;
S1 . ( a `2 , e `2 ) = a + e .= a `2 ;
1 in Seg width ( M * ( ColVec2Mx p ) ) ;
dom ( i (#) Im f ) = dom Im ( f ) /\ dom Im ( f ) ;
( ^2 ) . x = W . ( a , *' ( a , p ) ) ;
set Q = ( s Q ) \ { g , h } ;
cluster -> MSsorted for ManySortedSet of U1 , B be MSelement of U2 ;
attr F = { A } means : Def8 : F is discrete ;
reconsider z9 = \hbox { - 1 } as Element of product \overline G ;
rng f c= rng f1 \/ rng f2 & rng f1 c= rng f2 \/ rng g2 ;
consider x such that x in f .: A and x in f .: C ;
f = <*> the carrier of F_Complex & f = <*> the carrier of F_Complex implies f = <*> the carrier of F_Complex
E , j |= All ( x1 , x2 , H ) implies E , j |= H
reconsider n1 = n , n2 = m , n3 = 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 ) /\ B1 ) = 0 & card ( ( x \ B2 ) /\ B2 ) = 1 ;
g + R in { s : g-r < s & s < g + r } ;
set q-17 = ( q , <* s *> ) : q in dom s & s in dom ( p , q ) ;
for x being element st x in X holds x in rng f1 implies x in rng f2
h0 /. ( i + 1 ) = h0 . ( i + 1 ) ;
set mw = max ( B , } , } , mw = max ( B , m ) ;
t in Seg width ( I ^ ( n , n ) ) ;
reconsider X = dom f , C = rng g as Element of Fin NAT ;
IncAddr ( i , k ) = <% x , y %> + k ;
( for q being Point of TOP-REAL 2 st q in L~ f holds q `2 <= q `2 ) implies ( f /. q ) `2 <= N-bound L~ f
attr R is condensed means : |. : R is condensed & Cl R is condensed ;
pred 0 <= a & 1 <= b & b <= 1 implies 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 & len C + - 2 >= 9 ;
x , z , y is_collinear & x , z , x is_collinear implies x = y
a |^ ( n1 + 1 ) = a |^ n1 * a & a |^ ( n1 + 1 ) = a ;
<* \underbrace ( 0 , \dots , 0 *> , x ) *> in Line ( x , a * x ) ;
set yi1 = <* y , c *> ;
FF2 /. 1 in rng Line ( D , 1 ) & FF2 /. len FF2 = D . 1 ;
p . m joins r /. m , r /. ( m + 1 ) ;
p `2 = ( f /. i1 ) `2 & p `2 = ( f /. i1 ) `2 ;
W-bound ( X \/ Y ) = W-bound ( X \/ Y ) & W-bound ( X \/ Y ) = W-bound ( X \/ Y ) ;
0 + ( p `2 ) ^2 <= 2 * r + ( p `2 ) ^2 ;
x in dom g & not x in g " { 0 } implies x in dom ( g " )
f1 /* ( seq ^\ k ) is divergent_to-infty & f2 /* ( seq ^\ k ) is divergent_to-infty ;
reconsider u2 = u , v2 = v as VECTOR of P`1 , V ;
p |-count ( Product Sgm X11 ) = 0 implies p |-count ( p |-count X11 ) = 0
len <* x *> < i + 1 & i + 1 <= len c + 1 ;
assume that I is non empty and { x } /\ { y } = { 0. I } ;
set ii2 = ( card I + 4 ) .--> goto 0 , ii2 = goto 2 ;
x in { x , y } & h . x = {} ( Tx , y ) ;
consider y being Element of F such that y in B and y <= x `2 ;
len S = len ( the charact of A0 ) & len the charact of ( B * the charact of A0 ) = len the charact of B ;
reconsider m = M , i = I , n = N , m = I as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . j ;
set N8 = : : G = : G = ( G . e ) `1 ;
rng F c= the carrier of gr { a } & rng F = the carrier of gr { a } ;
implies for K being \vert , n , r being Nat holds ( for k being Nat holds k <= n implies K . k is a and r = k )
f . k , f . ( mod n ) in rng f ;
h " P /\ [#] T1 = f " P /\ [#] T2 .= f " P /\ [#] T1 ;
g in dom f2 \ f2 " { 0 } & f2 . g in dom f2 \ f2 " { 0 } ;
g+ X /\ dom f1 = g1 " X & g- X = g2 " X ;
consider n being element such that n in NAT and Z = G . n ;
set d1 = \bf dist ( x1 , y1 ) , d2 = dist ( x2 , y2 ) ;
b `2 + 1 / 2 < ( 1 - r ) / 2 + r / 2 ;
reconsider f1 = f , g1 = g as VECTOR of the carrier of X , Y ;
pred i <> 0 means : Def8 : i ^2 mod ( i + 1 ) = 1 ;
j2 in Seg len ( g2 . i2 ) & j2 + 1 in Seg ( len g2 ) ;
dom ( i ) = dom ( ( i - 1 ) * ( a - b ) ) .= dom ( a - b ) ;
cluster sec | ]. PI / 2 , PI / 2 .[ -> one-to-one ;
Ball ( u , e ) = Ball ( f . p , e ) ;
reconsider x1 = x0 , y1 = x1 , y2 = x2 as Function of S , IV ;
reconsider R1 = x , R2 = y , R1 = z as Relation of L ;
consider a , b being Subset of A such that x = [ a , b ] ;
( <* 1 *> ^ p ) ^ <* n *> in Rn ;
S1 +* S2 = S2 +* S1 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2
( ( #Z n ) * ( cos * sin ) ) `| Z = f ;
cluster -> [. 0 , 1 .] -valued for Function of C , REAL ;
set C7 = 1GateCircStr ( <* z , x *> , f3 ) , C8 = 1GateCircStr ( <* z , x *> , f3 ) ;
Ea1 . e2 = E8 . e2 -T & Ea2 . e2 = E8 . e2 -T ;
( ( arctan * ( f1 + f2 ) ) `| Z ) . x = 1 / ( x + a ) ^2 ;
upper_bound A = ( PI * 3 / 2 ) * ( PI / 2 ) & lower_bound A = 0 ;
F . ( dom f , - f ) is_transformable_to F . ( cod f , - f ) ;
reconsider pNAT = q`2 , px1 = q `2 as Point of TOP-REAL 2 ;
g . W in [#] Y0 & [#] Y0 c= [#] Y & g . W in [#] Y0 ;
let C be compact non vertical non horizontal Subset of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) ;
rng s c= dom f /\ ]. x0 - r , x0 .[ & rng s c= dom f /\ ]. x0 , x0 + r .[ ;
assume x in { idseq 2 , Rev ( idseq 2 ) } ;
reconsider n2 = n , m2 = m , m2 = n - 1 as Element of NAT ;
for y being ExtReal st y in rng seq holds g <= y implies g <= y
for k st P [ k ] holds P [ k + 1 ] ;
m = m1 + m2 .= m1 + m2 .= m1 + m2 .= m1 + m2 ;
assume for n holds H1 . n = G . n -H . n ;
set B" = f .: the carrier of X1 , B" = f .: the carrier of X2 ;
ex d being Element of L st d in D & x << d ;
assume that R -Seg ( a ) c= R -Seg ( b ) and R -Seg ( b ) c= R -Seg ( a ) ;
t in ]. r , s .[ or t = r or t = s ;
z + v2 in W & x = u + ( z + v2 ) ;
x2 |-- y2 iff P [ x2 , y2 ] & P [ y2 ] & P [ y2 ] ;
pred x1 <> x2 means : Def8 : |. x1 - x2 .| > 0 & |. x1 - x2 .| > 0 ;
assume that p2 - p1 , p3 - p1 , p2 - p1 is_collinear and p2 - p1 , p3 - p1 is_collinear ;
set q = ( f , f ) ^ <* 'not' A *> ;
let f be PartFunc of REAL-NS 1 , REAL-NS n , g be PartFunc of REAL-NS 1 , REAL-NS n ;
( n mod ( 2 * k ) ) + 1 = n mod k ;
dom ( T * ( succ t ) ) = dom ( succ t ) .= dom ( T * ( succ t ) ) ;
consider x being element such that x in wc iff x in c & x in X ;
assume ( F * G ) . ( v . x3 ) = v . x4 ;
assume that the carrier' of D1 c= the carrier' of D2 and the carrier' of D1 c= the carrier' of D2 ;
reconsider A1 = [. a , b .[ , A2 = [. a , b .] as Subset of R^1 ;
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 = q .. Cage ( C , n ) ;
n1 -' len f + 1 <= len ( - len f + 1 ) - len f + 1 ;
st c= c= { [ u , v , a , b ] , [ u , b , b ] } ;
set C-2 = ( ( Seg k ) .--> 1 ) . ( k + 1 ) ;
Sum ( L * p ) = 0. R * Sum p .= 0. V ;
consider i being element such that i in dom p and t = p . i ;
defpred Q [ Nat ] means 0 = Q ( $1 ) & P [ $1 ] ;
set s3 = Comput ( P1 , s1 , k ) , P3 = P1 +* I ;
let l be variable of k , Al , A-30 be Element of D ;
reconsider U2 = union G-24 , G-24 = union G-24 as Subset-Family of T-24 ;
consider r such that r > 0 and Ball ( p `2 , r ) c= Q ` ;
( h | ( n + 2 ) ) /. ( i + 1 ) = p2 ;
reconsider B = the carrier of X1 , C = the carrier of X2 , D = the carrier of X2 as Subset of X ;
p$ p$ = <* - c , 1 *> ^ ( <* - 1 *> ^ ( <* - 1 *> ^ ( - 1 ) ) ) ;
synonym f is real-valued means : Def6 : rng f c= NAT & f is one-to-one ;
consider b being element such that b in dom F and a = F . b ;
( x < card X0 ) + card Y0 & ( x in card Y0 or x = card X0 ) ;
pred X c= B1 means : Defooo) : X c= succ B1 & X c= B1 ;
then w in Ball ( x , r ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( x-y , 0 , PI ) ;
pred 1 <= len s means : Def21 : for s being Element of NAT holds s . ( 0 + 1 ) = s ;
f-47 c= f . ( k + ( n + 1 ) ) ;
the carrier of { 1_ G } = { 1_ G } & the carrier of { 1_ G } = { 1_ G } ;
pred p '&' q in \cdot ( the carrier of A ) & q '&' p in TAUT ( A ) ;
- ( t `1 ) ^2 < ( t `1 ) ^2 & ( t `2 ) ^2 < ( t `2 ) ^2 ;
UA . 1 = ( 9 /. 1 ) `1 .= ( 9 /. 1 ) `1 .= ( 9 /. 1 ) `1 ;
f .: the carrier of x = the carrier of x & f .: the carrier of x = the carrier of x ;
Indices OO = [: Seg n , Seg n :] & Indices OO = [: Seg n , Seg n :] ;
for n being Element of NAT holds G . n c= G . ( n + 1 ) ;
then V in M @ ex x being Element of M st V = { x } ;
ex f being Element of F-9 st f is H & f is H & f . 1 = F ;
[ h . 0 , h . 3 ] in the InternalRel of G & [ h . 0 , h . 3 ] in the InternalRel of G ;
s +* Initialize ( ( intloc 0 ) .--> 1 ) = s3 +* Initialize ( ( intloc 0 ) .--> 1 ) ;
|[ w1 , v1 ]| - ( - 1 ) * ( w1 + ( - 1 ) * ( w1 + ( - 1 ) * ( w1 + ( - 1 ) * ( w1 + ( - 1 ) * ( w1 + ( - 1 ) * ( w1 + ( - 1 ) * ( w1 + w2 ) )
reconsider t = t , s = s as Element of INT |^ X ;
C \/ P c= [#] ( GX | ( [#] GX \ A ) ) & C /\ A = {} ;
f " V in the topology of ( X ) /\ D & D = the carrier of ( X ) /\ D ;
x in [#] ( the carrier of ( the carrier of A ) ) /\ A & x in [#] ( ( the carrier of A ) ) ;
g . x <= h1 . x & h . x <= h1 . x implies h . x <= h1 . x
InputVertices S = { xy , yz , zx } \/ { xy , yz , zx } \/ { xy , yz } ;
for n being Nat st P [ n ] holds P [ n + 1 ] ;
set R = \cdot ( M , i ) * Line ( M , i ) ;
assume that M1 is being_line and M2 is being_line and M3 is being_line and M3 is being_line and M3 is being_line and M3 is being_line ;
reconsider a = f4 . ( i0 -' 1 ) , b = f4 . ( i0 -' 1 ) as Element of K ;
len B2 = Sum Len ( F1 ^ F2 ) & len B2 = len Len ( F1 ^ F2 ) + len ( F2 ^ F1 ) ;
len ( ( the ` of n ) * ( i , j ) ) = n & len ( ( i , j ) * ( i , j ) ) = n ;
dom max ( - ( f + g ) , f + g ) = dom ( f + g ) ;
( the Sorts of seq ) . n = upper_bound Y1 & ( the Sorts of seq ) . n = upper_bound Y1 ;
dom ( p1 ^ p2 ) = dom f12 & dom ( p1 ^ p2 ) = dom f12 ;
M . [ 1 , y ] = 1 / C * v1 .= y * v1 .= y * v1 ;
assume that W is non trivial and W .vertices() c= the carrier' of G2 and W is non trivial ;
godo /. i1 = G1 * ( i1 , i2 ) & card C = 1 ;
C8 |- 'not' Ex ( x , p ) 'or' p . ( x , y ) ;
for b st b in rng g holds lower_bound rng fnon \lbrace b } <= b
- ( ( q1 `1 / |. q1 .| - cn ) / ( 1 + cn ) ) = 1 ;
( LSeg ( c , m ) \/ NAT ) \/ LSeg ( l , k ) c= R ;
consider p being element such that p in Ball x and p in L~ f and x = f . p ;
Indices ( X @ ) = [: Seg n , Seg 1 :] & Indices ( X @ ) = [: Seg n , Seg 1 :] ;
cluster s => ( q => p ) => ( q => ( s => p ) ) -> valid ;
Im ( ( Partial_Sums F ) . m ) is_measurable_on E & Im ( ( 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 ] ;
p in LSeg ( N-min Z , p2 ) /\ LSeg ( o , p2 ) implies p `1 = p2 `1
set R8 = R / ( 1 - b ) , R8 = R / ( 1 - b ) ;
IncAddr ( I , k ) = SubFrom ( da , db ) .= SubFrom ( da , db ) ;
seq . m <= ( the Sorts of seq ) . k & ( the Sorts of seq ) . m <= ( the Sorts of seq ) . k ;
a + b = ( a ` *' ) ` & ( a + b ) ` = ( a ` *' ) ` ;
id ( X /\ Y ) = id X /\ id Y .= id ( X /\ Y ) ;
for x being element st x in dom h holds h . x = f . x ;
reconsider H = U1 \/ U2 , U2 = U1 \/ U2 , H = U2 \/ U1 as non empty Subset of U0 ;
u in ( ( c /\ ( ( d /\ e ) /\ b ) /\ f ) /\ m ;
consider y being element such that y in Y and P [ y , inf B ] ;
consider A being finite stable set of R such that card A = ( the carrier of R ) \ A ;
p2 in rng ( f |-- p1 ) \ rng <* p1 *> & p2 in rng <* p1 *> ;
len s1 - 1 > 1-1 & len s2 - 1 > 0 implies len s2 - 1 > 0
( N-min ( P ) ) `2 = N-bound ( P ) & ( N-min ( P ) ) `2 = N-bound ( P ) ;
Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) \/ LeftComp Cage ( C , k + 1 ) ;
f . a1 ` = f . a1 ` & f . a2 = f . a2 ` ;
( seq ^\ k ) . n in ]. -infty , x0 + r .[ & ( seq ^\ k ) . n in ]. x0 , x0 + r .[ ;
gg . s0 = g . s0 | G . s0 & gg . s0 = g . s0 ;
the InternalRel of S is symmetric implies the InternalRel of S is transitive & the InternalRel of S is transitive
deffunc F ( Ordinal , Ordinal ) = phi . $1 & phi . $1 = phi . $1 ;
F . s1 . a1 = F . s2 . a1 & F . s2 = F . a1 ;
x `2 = A . o .= Den ( o , A . a ) ;
Cl ( f " P1 ) c= f " ( Cl P1 ) & f " P1 c= f " P2 ;
FinMeetCl ( ( the topology of S ) | D ) c= the topology of T & the topology of ( S | D ) | D = the topology of T ;
synonym o is \bf means : U <> o *' & o <> * & o <> * ;
assume that X = Y |^ + & card X <> card Y and X <> Y and Y <> {} ;
the such that the such of s <= 1 + ( the g1 of s ) * ( the ResultSort of s ) & the {} of s = the carrier of s ;
LIN a , a1 , d or b , c // b1 , c1 & LIN a , c , d ;
e / 2 . 1 = 0 & e / 2 . 2 = 1 & e / 2 . 3 = 0 ;
ES1 in SS1 & not ES1 in { NS1 } & ES2 in SS2 & ES2 in ES2 ;
set J = ( l , u ) If ;
set A1 = .| ( ( a , b , c ) --> ( p , q ) ) ;
set vs = [ <* c , d *> , '&' ] , f3 = [ <* d , c *> , '&' ] , f4 = [ <* c , d *> , '&' ] , I = [ <* d , c *> , '&' ] , J = [ <* c , d *> , '&' ] , I ] ;
x * z `1 * x " in x * ( z * N ) * x " ;
for x being element st x in dom f holds f . x = ( g3 . x ) * ( g2 . x )
Int cell ( f , 1 , G ) c= RightComp f \/ RightComp f \/ RightComp f \/ RightComp f ;
UA is_an_arc_of W-min C , E-max C & W-min C in L~ Cage ( C , n ) implies W-min L~ Cage ( C , n ) in L~ Cage ( C , n )
set f-17 = f @ "/\" g @ ;
attr S1 is convergent means : Def8 : S2 is convergent & lim ( S1 - S2 ) = 0 ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a ;
cluster -> \in for let reflexive transitive RelStr , F be reflexive transitive non empty reflexive RelStr , f be Function ;
consider d being element such that R reduces b , d and R reduces c , d and R reduces d , c ;
not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) & not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) ;
( z + a ) + x = z + ( a + y ) .= z + a + y ;
len ( l (#) ( a |^ 0 ) ) = len l & len ( l (#) ( a |^ 0 ) ) = len l ;
( t of {} , rng t4 ) -valued FinSequence , p be ( {} , rng t4 ) -valued FinSequence ;
t = <* F . t *> ^ ( C . p ^ q ) .= <* F . t *> ^ q ;
set p-2 = W-min L~ Cage ( C , n ) , p`2 = W-min L~ Cage ( C , n ) ;
( k -' 1 ) -' ( i + 1 ) = ( k - 1 ) - ( i + 1 ) ;
consider u being Element of L such that u = u ` "/\" D and u in D ` ;
len ( ( width E ) |-> a ) = width E & width ( ( width E ) |-> a ) = width E ;
( F . x ) . ( ( G * the_arity_of o ) . x ) in dom ( G * the_arity_of o ) ;
set cH2 = the carrier of H2 , cH2 = the carrier of H2 ;
set cH1 = the carrier of H1 , cH2 = the carrier of H2 ;
( Comput ( P , s , 6 ) ) . intpos m = s . intpos m .= s . intpos m ;
IC Comput ( Q2 , t , k ) = ( l + 1 ) + 1 .= ( card I + 1 ) ;
dom ( ( - cos * sin ) `| REAL ) = REAL & dom ( ( - cos * sin ) `| REAL ) = REAL ;
cluster <* l *> ^ phi -> ( 1 + 1 ) -element for string of S ;
set b5 = [ <* that x1 , x2 *> , '&' ] , c5 = [ <* x1 , x2 *> , '&' ] , b5 = [ <* x1 , x2 *> , '&' ] ;
Line ( Segm ( M @ , P , Q ) , x ) = L * Sgm Q .= L ;
n in dom ( ( the Sorts of A ) * the_arity_of o ) & ( the Sorts of A ) . n = ( the Sorts of A ) . n ;
cluster f1 + f2 -> continuous for PartFunc of REAL , the carrier of S ;
consider y be Point of X such that a = y and ||. x-y .|| <= r ;
set x3 = t3 . DataLoc ( ( s . SBP ) , 2 ) , x4 = s . SBP , x4 = s . SBP , x4 = s . SBP , 7 = s . SBP , 8 = s . SBP , 6 = s . SBP , 7 = s . SBP , 8 = s . SBP , 8 = s . SBP
set p-3 = stop I ( ) , ps2 = stop I ( ) ;
consider a being Point of D2 such that a in W1 and b = g . a and a in W2 ;
{ A , B , C , D , E } = { A , B } \/ { C , D , E } ;
let A , B , C , D , E , F , J , M , N , F , J , M , N , N , M , N , F , N , M ;
|. p2 .| ^2 - ( p2 `2 ) ^2 - ( p2 `1 ) ^2 >= 0 & ( p2 `1 ) ^2 - ( p2 `2 ) ^2 >= 0 ;
l -' 1 + 1 = n-1 * ( l + 1 ) + ( 1 + 1 ) ;
x = v + ( a * w1 + b * w2 ) + ( c * w2 ) ;
the TopStruct of L = , the TopStruct of L = , the Scott Scott TopAugmentation of L ;
consider y being element such that y in dom H1 and x = H1 . y and y in H1 . x ;
fH \ { n } = Free All ( v1 , H ) & fH \ { n } = Free H ;
for Y being Subset of X st Y is summable holds Y is summable & Y is \cdot summable implies X \/ Y is summable
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
for s being FinSequence holds len ( the { s } ) = len s & len ( the { s } ) = len s
for x st x in Z holds exp_R * f is_differentiable_in x & ( exp_R * f ) . x > 0 ;
rng ( h2 * f2 ) c= the carrier of R^1 & rng ( h2 * g2 ) c= the carrier of R^1 ;
j + - len f <= len f + ( len - len f ) - len f ;
reconsider R1 = R * I , R2 = ( R * I ) * ( I * J ) as PartFunc of REAL , REAL-NS n ;
C8 . x = s1 . x0 .= C8 . x .= C8 . x .= ( C * ( i , j ) ) . x ;
power F_Complex . ( z , n ) = 1 .= x |^ n .= x |^ n ;
t at ( C , s ) = f . ( the connectives of S ) . t .= s ;
support ( f + g ) c= support f \/ ( support g ) & support ( f + g ) c= support f \/ support g ;
ex N st N = j1 & 2 * Sum ( seq1 | N ) > N & N > 0 ;
for y , p st P [ p ] holds P [ All ( y , p ) ]
{ [ x1 , x2 ] where x1 is Point of X1 , x2 is Point of X2 : x1 in X & x2 in Y } c= X ;
h = ( i = j |-- h , id B . i ) .= H . i ;
ex x1 being Element of G st x1 = x & x1 * N c= A & x1 in A ;
set X = ( ( st ( q , O1 ) `1 , 4 ) ) `1 , Y = ( d , O1 ) `2 ;
b . n in { g1 : x0 < g1 & g1 < a1 . n & g1 < x0 } ;
f /* s1 is convergent & f /. x0 = lim ( f /* s1 ) & f /. x0 = lim ( f /* s1 ) ;
the lattice of the topology of Y = the carrier of the topology of Y & the topology of Y = the topology of X ;
'not' ( a . x ) '&' b . x 'or' a . x '&' 'not' ( b . x ) = FALSE ;
2 = len ( q0 ^ r1 ) + len q1 & len q1 + 1 = len ( q0 ^ r1 ) + len q1 ;
( 1 / a ) (#) ( sec * f1 ) - id Z is_differentiable_on Z & ( ( 1 / a ) (#) ( sec * f1 ) ) `| Z is_differentiable_on Z ;
set K1 = integral ( ( lim H ) || A , ( lim H ) || A ) , K1 = ( ( lim H ) || A ) ;
assume e in { ( w1 - w2 ) `1 : w1 in F & w2 in G } ;
reconsider d7 = dom a `1 , d8 = dom F `1 , d8 = dom G `1 , d8 = dom G `2 as finite set ;
LSeg ( f /^ j , j ) = LSeg ( f , j ) \/ LSeg ( f , j + q .. f ) ;
assume that X in { T . N2 , N2 : h . N2 = N2 } and h . N2 = N2 ;
assume that Hom ( d , c ) <> {} and <: f , g :> * f1 = <: f , g :> * f2 ;
dom S29 = dom S /\ Seg n .= dom L6 .= Seg n /\ Seg n .= dom L6 .= dom L6 ;
x in H |^ a implies ex g st x = g |^ a & g in H & g in H
* ( ( 0 , 1 ) --> ( a , 1 ) ) = a `2 - ( 0 * n ) .= a `2 ;
D2 . j in { r : lower_bound A <= r & r <= D1 . i } ;
ex p being Point of TOP-REAL 2 st p = x & P [ p ] & p `2 <= 0 & p <> 0. TOP-REAL 2 ;
for c holds f . c <= g . c implies f @ = g @ ;
dom ( f1 (#) f2 ) /\ X c= dom ( f1 (#) f2 ) /\ X & dom ( f1 (#) f2 ) /\ X c= dom f1 /\ X ;
1 = ( p * p ) / p .= p * ( p / p ) .= p * 1 ;
len g = len f + len <* x + y *> .= len f + 1 .= len f + 1 ;
dom ( F | [: N1 , S-23 :] ) = [: F | [: N1 , S-23 :] , F | [: N1 , S-23 :] :] ;
dom ( f . t * I . t ) = dom ( f . t * g . t ) ;
assume a in ( "\/" ( ( T |^ the carrier of S ) , F ) ) .: D ;
assume that g is one-to-one and ( the carrier' of S ) /\ rng g c= dom g and g is one-to-one and g is one-to-one ;
( ( x \ y ) \ z ) \ ( ( x \ z ) \ ( y \ z ) ) = 0. X ;
consider f such that f * f `1 = id b and f `2 * f `2 = id a & f `2 = id b ;
( cos | [. 2 * PI * 0 , PI + 2 * PI .] ) is increasing ;
Index ( p , co ) <= len LS - Gij .. LS - 1 & Index ( Gij , LS ) + 1 <= len LS ;
let t1 , t2 , t3 be Element of ( the carrier of S ) * , s be Element of ( the carrier of S ) * ;
"/\" ( ( Frege ( curry H ) ) . h , L ) <= "/\" ( ( Frege G ) . h , L ) ;
then P [ f . i0 ] & F ( f . ( i0 + 1 ) ) < j ;
Q [ ( [ D . x , 1 ] ) `1 , F . [ 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 holds l . i is carrier of G . i
the Sorts of A2 = ( the carrier of S2 ) --> BOOLEAN .= ( the carrier of S1 ) --> BOOLEAN .= the Sorts of A1 +* A2 ;
consider s being Function such that s is one-to-one and dom s = NAT and rng s = F . 0 and rng s = { 1 } ;
dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( a , b2 ) & dist ( a1 , b1 ) < r ;
( Lower_Seq ( C , n ) /. len Lower_Seq ( C , n ) ) `1 = ( W-min L~ Cage ( C , n ) ) `1 ;
q `2 <= ( UMP Upper_Arc L~ Cage ( C , 1 ) ) `2 & ( UMP L~ Cage ( C , 1 ) ) `2 <= ( UMP L~ Cage ( C , 1 ) ) `2 ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} ;
given a being ExtReal such that a <= II and A = ]. a , II .] and a <= II ;
consider a , b being complex number such that z = a & y = b and z + y = a + b ;
set X = { b |^ n where n is Element of NAT : n <= m & b |^ n in n } ;
( ( x * y * z \ x ) \ z ) \ ( x * y \ x ) = 0. X ;
set xy = [ <* xy , y *> , f1 ] , yz = [ <* y , z *> , f2 ] , f4 = [ <* z , x *> , f3 ] , f4 = [ <* z , x *> , f3 ] ;
l /. len l = ( l . len l ) * ( l /. len l ) .= l . len l ;
( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 = 1 ;
( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) ^2 < 1 ;
( ( ( S \/ Y ) `2 ) / 2 ) * ( ( S + Y ) `2 ) = ( ( S + Y ) `2 ) / 2 ;
( seq - seq ) . k = seq . k - seq . k .= seq . k - seq . k .= seq . k - seq . k ;
rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , u0 ) ;
the carrier of X = the carrier of X & the carrier of X0 = the carrier of X & the carrier of X0 = the carrier of X ;
ex p4 st p3 = p4 & |. p4 - |[ a , b ]| .| = r & |. p4 - |[ a , b ]| .| = r ;
set ch = chi ( X , A5 ) , A5 = chi ( X , A5 ) ;
R |^ ( 0 * n ) = I\HM ( X , X ) .= R |^ n |^ 0 .= R |^ 0 ;
( Partial_Sums ( ( curry ( F , n ) ) . 0 ) . n is nonnegative & ( ( curry ( F , n ) ) . 0 ) . n is nonnegative ;
f2 = C7 . ( E8 . ( V , K ) ) .= C8 . ( V , K ) ;
S1 . b = s1 . b .= S2 . b .= S2 . b .= S2 . b .= S2 . b ;
p2 in LSeg ( p2 , p1 ) /\ LSeg ( p2 , p1 ) & p2 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ;
dom ( f . t ) = Seg n & dom ( I . t ) = Seg n & dom ( I . t ) = Seg n ;
assume that o = ( the connectives of S ) . 11 and the carrier' of S = ( the carrier' of S ) . 11 ;
set phi = ( l1 , l2 ) implies phi is ( l1 , l2 ) u u in D ;
synonym p is is is is is is is is / for p , T & p is is is is is is is is is is *' ;
Y1 `2 = - 1 & 0. TOP-REAL 2 <> Y1 & Y1 `2 <> - 1 & Y1 `1 <= 1 implies Y1 `1 = 1 & Y2 `1 = - 1
defpred X [ Nat , set , set ] means P [ $2 , $2 , $2 ] & P [ $2 , $2 , $2 ] ;
consider k being Nat such that for n being Nat st k <= n holds s . n < x0 + g ;
Det ( I |^ ( m -' n ) ) * ( m -' n ) = 1_ K & Det ( I |^ ( m -' n ) ) = 1_ K ;
( - b - sqrt ( b ^2 - 4 * a * c ) ) / ( 2 * a * c ) < 0 ;
Cs . d = C7 . d mod C7 . d & C8 . d = C8 . d mod C8 . d ;
attr X1 is dense means : Defdense : X2 is dense dense & X1 /\ X2 is dense SubSpace of X & X2 /\ X1 is dense SubSpace of X ;
deffunc F ( Element of E , Element of I ) = $1 * $2 & $2 = $1 * $2 & $2 = $1 * $2 ;
t ^ <* n *> in { t ^ <* i *> : Q [ i , T . t ] } ;
( x \ y ) \ x = ( x \ x ) \ y .= y ` \ y .= 0. X ;
for X being non empty set for Y being Subset-Family of X holds the topology of X is Basis of <* X , FinMeetCl Y *>
synonym A , B are_separated means : Def1 : Cl A misses B & A misses Cl B & B misses A ;
len ( M @ ) = len p & width ( M @ ) = width ( M @ ) & width ( M @ ) = width ( M @ ) ;
J = { v where x is Element of K : 0 < v . x & v . x < 0 } ;
( Sgm ( Seg m ) ) . d - ( Sgm ( Seg m ) ) . e <> 0 ;
lower_bound divset ( D2 , k + k2 ) = D2 . ( k + k2 - 1 ) .= D2 . ( k + k2 - 1 ) ;
g . r1 = - 2 * r1 + 1 & dom h = [. 0 , 1 .] & rng h c= [. 0 , 1 .] ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `2 & h . x = ( h . x ) `2 ;
ex w st w in dom B1 & <* 1 *> ^ s = <* 1 *> ^ w & len w = len b1 + 1 ;
[ 1 , {} , <* d1 *> ] in ( { [ 0 , {} , {} ] } \/ S1 ) \/ S2 ;
IC Exec ( i , s1 ) + n = IC Exec ( i , s2 ) .= IC Exec ( i , s2 ) ;
IC Comput ( P , s , 1 ) = IC Comput ( P , s , 9 ) .= 5 + 9 .= 5 ;
( IExec ( W6 , Q , t ) ) . intpos ( e + 1 ) = t . intpos ( e + 1 ) .= t . intpos ( e + 1 ) ;
LSeg ( f /^ q , i ) misses LSeg ( f /^ q , j ) \/ LSeg ( f /^ q , j ) ;
assume for x , y being Element of L st x in C & y in C holds x <= y or y <= x ;
integral ( integral ( f , C ) , x , x0 ) = f . ( upper_bound C ) - f . ( lower_bound C ) ;
for F , G being one-to-one FinSequence st rng F misses rng G holds F ^ G is one-to-one & F ^ G is one-to-one
||. R /. L - R /. h .|| < e1 * ( K + 1 * ||. h .|| ) ;
assume a in { q where q is Element of M : dist ( z , q ) <= r } ;
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 \not c= d ;
for y , x being Element of REAL st y ` in Y & x in X ` holds y <= x ` & y <= x ;
func |. \bullet p .| -> variable of A means : Def10 : for x being element st x in NBI holds it . x = min ( NBI ( p ) , x ) ;
consider t being Element of S such that x `1 , y `2 '||' z `1 , t `2 and x `2 , z `2 '||' y `1 , t `2 ;
dom x1 = Seg len x1 & len x1 = len l1 & len y1 = len l1 & len y1 = len l1 & len y1 = len l1 ;
consider y2 being Real such that x2 = y2 and 0 <= y2 and y2 < 1 / 2 and y2 <= 1 / 2 ;
||. f | X /* s1 .|| = ||. f .|| | X & ||. f .|| | X is convergent & lim ( ||. f .|| | X ) = ||. f .|| /. x0 ;
( the InternalRel of A ) -Seg ( x ` ) /\ Y = {} \/ {} .= {} \/ {} .= {} \/ {} .= {} ;
assume that i in dom p and for j being Nat st j in dom q holds P [ i , j ] and i + 1 in dom p and P [ j , i ] ;
reconsider h = f | X ( ) , g = f | X , h = g | X , f | X ( ) as Function ;
u1 in the carrier of W1 & u2 in the carrier of W2 & u1 in the carrier of W1 implies u1 + u2 in the carrier of W2
defpred P [ Element of L ] means M <= f . $1 & f . $1 <= $1 & f . $1 <= f . $1 ;
^ ( u , a , v ) = s * x + ( - ( s * x ) + y ) .= b ;
- ( x-y ) = - x + - y .= - x + y .= - x + y .= - x + y ;
given a being Point of GX such that for x being Point of GX holds a , x , x , a , x , y C C C ;
fJ = [ [ dom @ f2 , cod f2 ] , h2 ] & fJ = [ cod @ g2 , h2 ] ;
for k , n being Nat st k <> 0 & k < n & n is prime holds k , n are_relative_prime & k , n are_relative_prime
for x being element holds x in A |^ d iff x in ( ( A ` ) |^ d ) ` & x in A `
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 = L9 . ( F . k ) & F . k in dom L9 & F . k = L9 . k ;
set i2 = SubFrom ( a , i , - n ) , i1 = goto - ( n + 1 ) ;
attr B is thesis means : Def8 : for S holds S `1 = B `1 & S `2 = C `2 ;
a9 "/\" D = { a "/\" d where d is Element of N : d in D & d in D } ;
|( exp_R , q9 - q9 )| * |( exp_R , exp_R )| + |( exp_R , exp_R )| * |( exp_R , q2 )| >= |( exp_R , exp_R )| + |( exp_R , q2 )| ;
( - f ) . ( sup A ) = ( ( - f ) | A ) . ( sup A ) .= - ( f | A ) . ( sup A ) ;
GG2 `1 = ( ( G * ( len G , k ) ) `1 ) `1 .= ( ( G * ( 1 , k ) ) `1 ) `1 ;
( Proj ( i , n ) ) . LM = <* ( proj ( i , n ) ) . LM *> .= ( Proj ( i , n ) ) . LM ;
f1 + f2 * reproj ( i , x ) is_differentiable_in ( reproj ( i , x ) . x ) & f2 + ( reproj ( i , x ) . x ) = ( f1 + f2 ) . x ;
pred ( for x st x in Z holds ( tan . x ) ^2 = tan . x ) & for x st x in Z holds ( tan * tan ) . x = tan . x ;
ex t being SortSymbol of S st t = s & h1 . t . x = h2 . t . x & h1 . t = h2 . t ;
defpred C [ Nat ] means ( ( P . $1 ) is D & ( A is D is D empty ) or A is D is D D D D D D D D is non empty ;
consider y being element such that y in dom ( p9 . i ) and ( q9 . i ) . y = ( p9 . y ) . y ;
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 & T . ( id d ) = id d
not ( for p , n holds f . p = ( f | n ) ^ <* p *> ) .= f ^ <* p *> ^ <* 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 `2 - p `2 = ( f | ( n , L ) ) *' - ( f . ( - - p ) ) .= ( f - ( - ( p - g ) ) ) *' ;
consider r be Real such that r in rng ( f | divset ( D , j ) ) and r < m + s ;
f1 . ( |[ r2 `1 , r3 ]| ) in f1 .: W2 & f2 . ( |[ r2 `1 , r3 ]| ) in f1 .: W3 ;
eval ( a | ( n , L ) , x ) = eval ( a | ( n , L ) , x ) .= a ;
z = DigA ( tz , x ) .= DigA ( tz , x ) .= DigA ( tz , x ) .= DigA ( tz , x ) ;
set H = { Intersect S where S is Subset-Family of X : S c= G & S is open } , G = { meet S where S is Subset of X : S is open & S is open } ;
consider S19 being Element of D * , d being Element of D * such that S `1 = S19 ^ <* d *> and S29 = S29 ^ <* d *> ;
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 ) ) ^2 / ( 1 + sn ) ^2 ;
(0). V is Linear_Combination of A & Sum ( L ) = 0. V implies Sum ( L ) = 0. V & Sum ( L ) = 0. V
let k1 , k2 , k2 , k1 , k2 , k2 , k2 , k2 , k2 , k1 , k2 , k2 , k2 , k2 , k2 be Instruction of SCM+FSA , a be Int-Location , k1 be Int-Location ;
consider j being element such that j in dom a and j in g " { k `2 } and x = a . j and a . j = b . j ;
H1 . x1 c= H1 . x2 or H1 . x2 c= H1 . x1 or H1 . x2 c= H1 . x1 & H1 . x2 c= H2 . x2 ;
consider a being Real such that p = -' ( a * p1 + ( a * p2 ) ) and 0 <= a and a <= 1 ;
assume that a <= c & d <= b & [' a , b '] c= dom f and [' a , b '] c= dom g ;
cell ( Gauge ( C , m ) , 1 , width Gauge ( C , m ) -' 1 , 0 ) is non empty ;
Ay in { ( S . i ) `1 where i is Element of NAT : i <= n & not contradiction } ;
( T * b1 ) . y = L * b2 /. y .= ( F /. y ) `1 .= ( F /. y ) `1 .= ( F /. y ) `1 ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x - s . y .| ;
( log ( 2 , k + 1 ) ) ^2 >= ( log ( 2 , k + 1 ) ) ^2 + ( log ( 2 , k + 1 ) ) ^2 ;
then that p => q in S and not x in the carrier of p and p => All ( x , q ) in S ;
dom ( the InitS of r-10 ) misses dom ( the InitS of r-11 ) & dom ( the InitS of r-11 ) misses dom ( the InitS of r-11 ) ;
synonym f is integer means : Def3 : for x being set st x in rng f holds x is Integer ;
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 p3 + len <* x *> .= len p3 + len <* x *> .= len p3 + 1 .= len p1 + 1 ;
l `1 = ( g /. 1 , 3 ) `1 + ( k - 1 ) * ( e /. 1 , 3 ) `1 - ( e /. 1 ) * ( e /. 2 , e /. 3 ) ;
CurInstr ( P2 , Comput ( P2 , s2 , l2 ) ) = halt SCM+FSA .= halt SCM+FSA .= CurInstr ( P2 , Comput ( P2 , s2 , l2 ) ) ;
assume for n be Nat holds ||. seq .|| . n <= Rseq . n & Rseq is summable & Rseq is summable & Rseq is summable ;
sin . ( non empty ) = sin . ( r * cos ( - s ) ) * sin ( s ) .= 0 ;
set q = |[ g1 `1 . t0 , g2 `2 . t0 ]| , g1 `2 = |[ g2 `1 . t0 , g2 `2 . t0 ]| ;
consider G being sequence of S such that for n being Element of NAT holds G . n in implies G in implies for n being Element of NAT holds G . n = F ( n ) ;
consider G such that F = G and ex G1 st G1 in SM & G = width G1 & G1 is finite & G is finite ;
the root of [ x , s ] in ( the Sorts of Free ( C , X ) ) . s & ( the Sorts of C ) . s = ( the Sorts of Free ( C , X ) ) . s ;
Z c= dom ( exp_R * ( f + ( #Z 2 ) * f1 ) ) /\ dom ( ( #Z 2 ) * f1 ) ;
for k be Element of NAT holds seq1 . k = ( ( \HM { Im ( f ) ) . k ) . i & ( Im ( f ) ) . k = 0
assume that - 1 < n ( ) and q `2 > 0 and ( q `1 / |. q .| - cn ) < 0 and ( q `1 / |. q .| - cn ) < 0 ;
assume that f is continuous one-to-one 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 sE = Comput ( P1 , s1 , r ) and r <= q and r <= q ;
LE f /. ( i + 1 ) , f /. j , L~ f , f /. ( len f ) , f /. ( len f ) ;
assume that x in the carrier of K and y in the carrier of K and ex_inf_of { x , y } , L and inf { x , y } in the carrier of L ;
assume that f +* ( i1 , \xi ) in ( proj ( F , i2 ) " ( A . i1 ) ) " ( ( A . i1 ) " ) and f . i1 = ( proj ( F , i2 ) " ) . ( A . i1 ) ;
rng ( ( ( Flow M ) ~ | ( the carrier' of M ) ) c= the carrier' of M & rng ( ( Flow M ) ~ | ( the carrier' of M ) ) c= the carrier' of M ;
assume z in { ( the carrier of G ) \/ { t where t is Element of T : t in A } ;
consider l be Nat such that for m be Nat st l <= m holds ||. s1 . m - x0 .|| < g & g in dom ( f | X ) ;
consider t be VECTOR of product G such that mt = ||. D5 . t .|| & ||. t .|| <= 1 ;
assume that the degree of v = 2 and v ^ <* 0 *> in dom p and v ^ <* 1 *> in dom p and p . 1 = v ;
consider a being Element of the carrier of X39 , A being Element of the lines of X39 such that not a on A and not a on A ;
( ( - x ) |^ ( k + 1 ) ) * ( ( - x ) |^ ( k + 1 ) ) " = 1 ;
for D being set st for i st i in dom p holds p . i in D holds p . i is FinSequence of D & for i st i in dom p holds p . i is FinSequence of D
defpred R [ element ] means ex x , y st [ x , y ] = $1 & P [ x , y ] & P [ x , y ] ;
L~ f2 = union { LSeg ( p0 , p2 ) , LSeg ( p1 , p10 ) } .= { p2 , p1 } \/ { p2 } .= { p2 } \/ { p2 } ;
i -' len h11 + 2 - 1 < i - len h11 + 1 + 1 - 1 + 1 + 1 - 1 + 1 + 1 - 1 ;
for n be Element of NAT st n in dom F holds F . n = |. ( nthesis . ( n -' 1 ) ) .| ;
for r , s1 , s2 holds r in [. s1 , s2 .] iff s1 <= r & r <= s2 & s1 <= s2 & r <= s2
assume v in { G where G is Subset of T2 : G in B2 & G c= z1 & G c= z2 & G c= z1 } ;
let g be .| non-empty element of A |^ X , INT , b be Element of INT |^ X holds ( b |^ X ) |^ b <> 0
min ( g . [ x , y ] , k ) = ( min ( g , k , x , z ) ) . y ;
consider q1 being sequence of CL such that for n holds P [ n , q1 . n ] and P [ q1 ] ;
consider f being Function such that dom f = NAT and for n being Element of NAT holds f . n = F ( n ) and for n being Element of NAT holds P [ n , f . n ] ;
reconsider B-6 = B /\ Y , OI = O , $ $ Z = Z /\ Y , $ Z = { Z } , X = Z /\ X , Y = { Z } ;
consider j being Element of NAT such that x = the b the FinSequence of n and 1 <= j and j <= n and j <= n ;
consider x such that z = x and card ( x . O2 ) in card ( x . O1 ) and x in L1 & x in L2 ;
( C * : T4 . ( k , n2 ) ) . 0 = C . ( ( : 4 . ( k , n2 ) ) . 0 ) ;
dom ( X --> rng f ) = X & dom ( ( X --> f ) . x ) = dom ( X --> f . x ) ;
( ex b being Element of TOP-REAL 2 st b <= ( SpStSeq L~ SpStSeq C ) `2 & ( SpStSeq L~ SpStSeq C ) `2 <= N-bound L~ SpStSeq C ) implies ( SpStSeq L~ SpStSeq C ) `2 <= ( N-bound L~ SpStSeq C ) `2
synonym x , y are_collinear means : Def1 : x = y or ex l being that { x , y } c= l & x in l ;
consider X being element such that X in dom ( f | ( n + 1 ) ) and ( f | ( n + 1 ) ) . X = Y ;
assume that Im k is continuous and for x , y being Element of L for a , b being Element of Im k st a = x & b = y holds x << y iff a << b ;
( 1 / 2 * ( ( - ( x - 0 ) ) * ( AffineMap ( n , 0 ) ) ) ) is_differentiable_on REAL ;
defpred P [ Element of omega ] means ( the partial of A1 ) . $1 = A1 . $1 & ( the partial of A2 ) . $1 = A2 . $1 ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 1 ) .= 6 + 1 .= 6 + 1 .= 6 ;
f . x = f . g1 * f . g2 .= f . g1 * 1_ H .= f . g1 * 1_ H .= f . g1 * ( g . g2 ) .= f . g1 * ( g . g2 ) ;
( M * ( F . n ) ) . n = M . ( ( canFS ( Omega ) ) . n ) .= M . ( { ( canFS ( Omega ) ) . n } ) ;
the carrier of L1 + L2 c= ( the carrier of L1 ) \/ ( the carrier of L2 ) & the carrier of L1 + L2 c= the carrier of L1 & the carrier of L1 + L2 c= the carrier of L2 ;
pred a , b , c , x , y , c , d , x , y , z , y be Element of o , a , b , c , d , x , y ;
( the PartFunc of s , s ) . n <= ( the PartFunc of s , 1 ) . n * s . ( n + 1 ) & ( the 0. of s ) . ( n + 1 ) = ( the 0. of s ) . n ;
pred - 1 <= r & r <= 1 & ( arccot - arccot ) . r = - 1 / ( 1 + r ^2 ) ;
seq in { p ^ <* n *> where n is Nat : p ^ <* n *> in T1 & p ^ <* n *> in T2 } ;
|[ x1 , x2 , x3 , x4 ]| . 2 - |[ y1 , y2 , x4 ]| . 2 = x2 - y2 & |[ x1 , x2 , x3 , x4 ]| . 2 = x2 - y2 ;
attr for m being Nat holds F . m is nonnegative means : Def8 : for n being Nat holds ( Partial_Sums F ) . n is nonnegative & ( Partial_Sums F ) . m <= ( Partial_Sums F ) . n ;
len ( ( G . z ) * ( x , y ) ) = len ( ( ( G . ( x , y ) ) * ( y , z ) ) ) .= len ( ( G . ( y , z ) ) * ( y , z ) ) ;
consider u , v being VECTOR of V such that x = u + v and u in W1 /\ W2 and v in W2 /\ W3 and u in W3 /\ W3 ;
given F being finite Subset of NAT such that F = x and dom F = n & rng F c= { 0 , 1 } & Sum F = k ;
0 = ( - 1 ) * ( - \hbox { - } 1 ) iff 1 = ( ( - 1 ) * ( - \hbox { - } 1 } ) * ( - ( - 1 ) * ( - ( 1 - 0 ) ) ) ;
consider n be Nat such that for m be Nat st n <= m holds |. ( f # x ) . m - lim ( f # x ) .| < e ;
cluster -> being being being being being non empty w w structure means : Def1 : ( the carrier of \mathop { \rm and } _ 2 ( L ) ) is Boolean & ( the carrier of \mathop { \rm and } _ 3 ( L ) ) is Boolean ;
"/\" ( BB , {} ) = Top BB .= the carrier of S .= the carrier of ( S | ( the carrier of B ) ) .= [#] ( ( S | ( the carrier of B ) ) | ( the carrier of B ) ) ;
( r / 2 ) ^2 + ( rbeing Element of REAL ) ^2 + ( rbeing Element of REAL ) ^2 + ( rbeing Element of REAL ) ^2 + ( rbeing Element of REAL ) ^2 ) <= ( r / 2 ) ^2 + ( rbeing Element of REAL ) ^2 ;
for x being element st x in A /\ dom ( f `| X ) holds ( f `| X ) . x >= r2 & ( f `| X ) . x >= r2
2 * r1 - c * |[ a , c ]| - ( 2 * r1 - b ) * |[ b , c ]| = 0. TOP-REAL 2 ;
reconsider p = P * ( exp_R , 1 ) , q = a " * ( ( - a ) |^ n ) , r = a " * ( - a ) * ( - b ) as FinSequence of K ;
consider x1 , x2 being element such that x1 in uparrow s and x2 in uparrow t and x = [ x1 , x2 ] and x = [ x1 , x2 ] ;
for n be Nat st 1 <= n & n <= len q1 holds q1 . n = ( ( upper_volume ( g , M7 ) ) | n ) . ( len ( ( upper_volume ( g , M7 ) ) | n )
consider y , z being element such that y in the carrier of A and z in the carrier of A & i = [ y , z ] and i = [ y , z ] ;
given H1 , H2 being strict Subgroup of G such that x = H1 & y = H2 and H1 is Subgroup of H2 and H2 is Subgroup of H1 and H2 is Subgroup of H2 ;
for S , T being non empty p2 , d being Function of T , S st T is complete holds d is monotone implies d is monotone & d is monotone
[ a + 0. F_Complex , b2 ] in ( the carrier of F_Complex ) /\ ( the carrier of V ) & [ a + 0. F_Complex , b2 ] in [: the carrier of V , the carrier of V :] ;
reconsider mm = max ( len F1 , len ( p . n ) * ( p . n ) ) as Element of NAT ;
I <= width GoB ( ( GoB h ) * ( 1 , width GoB h ) + ( GoB h ) * ( 1 , width GoB h ) ) & I <= width GoB ( ( GoB h ) * ( 1 , width GoB h ) + ( GoB h ) * ( 1 , width GoB h ) ) ;
f2 /* q = ( f2 /* ( f1 /* s ) ) ^\ k .= ( f2 * f1 ) /* s .= ( f2 * f1 ) /* s ;
attr A1 \/ A2 is linearly-independent means : Def8 : A1 : A1 misses A2 & A2 /\ ( A1 \/ A2 ) = (0). ( A1 , A2 ) & ( A1 /\ A2 ) = (0). ( A2 , A1 ) ;
func A -carrier C -> set equals union { A . s where s is Element of R : s in C & s in C } ;
dom ( Line ( v , i + 1 ) (#) ( ( a * ( p , m ) ) * ( \square , 1 ) ) ) = dom ( F ^ <* x *> ) ;
cluster [ x `1 , 4 ] , [ x `1 , 4 ] , [ x `1 , 4 ] , [ x `1 , 4 ] ] , [ x `1 , 4 ] , [ x , 4 ] ] -> reduces x , y ;
E , f |= All ( x1 , x2 ) => ( x2 , x1 ) => ( x1 , x2 ) => ( x1 , x2 ) '&' ( x1 , x2 ) = All ( x1 , x2 ) '&' ( x1 , x2 ) ;
F .: ( id X , g ) . x = F . ( id X , g . x ) .= F . ( x , g . x ) .= F . ( x , g . x ) ;
R . ( h . m ) = F . ( x0 + h . m ) - ( h . m ) + ( h . m ) - ( h . m ) ;
cell ( G , Xs -' 1 , ( Y + 1 ) \ L~ f ) meets ( UBD L~ f ) \/ ( UBD L~ f ) & ( L~ f \/ L~ f ) meets ( UBD L~ f ) \/ ( UBD L~ f ) ;
IC Result ( P2 , s2 ) = IC IExec ( I , P , Initialize s ) .= card I .= card I + card J .= card I + card J + 1 .= card I + 1 .= card I + 1 ;
sqrt ( ( - ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ) ^2 - ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ) > 0 ;
consider x0 being element such that x0 in dom a and x0 in g " { k } and y0 = a . x0 and x0 in g " { k } and a . x0 = a . x0 ;
dom ( r1 (#) chi ( A , C ) ) = dom chi ( A , C ) /\ dom ( chi ( A , C ) ) .= C /\ dom ( chi ( A , C ) ) .= C /\ dom ( chi ( A , C ) ) .= C /\ dom ( ( chi ( A , C ) ) | A ) ;
d-7 . [ y , z ] = ( ( [ y , z ] `2 - ( [ y , z ] `2 ) ) - ( [ y , z ] `2 - ( [ y , z ] `2 - ( y , z ) `2 ) ) ) * ( y , z ) ;
pred for i being Nat holds C . i = A . i /\ B . i means C . i c= A . i /\ B . i ;
assume that x0 in dom f and f is_continuous_in x0 and ||. f .|| is_continuous_in x0 and ||. f .|| is_continuous_in x0 and ||. f .|| is_continuous_in x0 and ||. f .|| is_continuous_in x0 ;
p in Cl A implies for K being Basis of p , Q being Basis of T st Q in K holds A meets Q & A meets Q
for x being Element of REAL n st x in Line ( x1 , x2 ) holds |. y1 - y2 .| <= |. y1 - y2 .| & |. y1 - y2 .| <= |. y1 - y2 .|
func the \times <*> of a -> \rm \rm \hbox { - } number means : Def6 : a in it & for b being id of a st b in b holds it . b c= a ;
[ a1 , a2 , a3 ] in ( [: the carrier of A , the carrier of A :] \/ [: the carrier of A , the carrier of A :] ) & [ a1 , a2 , a3 ] in [: the carrier of A , the carrier of A :] ;
ex a , b being element st a in the carrier of S1 & b in the carrier of S2 & x = [ a , b ] & x = [ a , b ] ;
||. ( vseq . n ) - ( vseq . m ) .|| * ||. x - y .|| < ( e / ( ||. 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 } holds z in x & z in Z ;
sup compactbelow [ s , t ] = [ sup ( { s } ) , sup ( compactbelow [ s , t ] ) ] .= sup ( compactbelow [ s , t ] ) .= sup ( compactbelow [ s , t ] ) ;
consider i , j being Element of NAT such that i < j and [ y , f . j ] in Iz and [ f . i , z ] in Iz and [ y , f . i ] in Iz ;
for D being non empty set , p , q being FinSequence of D st p c= q holds ex p being FinSequence of D st p ^ q = q & p ^ q = p ^ q
consider e19 be Element of the affine of X such that c9 , a9 // b9 , e and a9 <> b9 & b9 <> c9 and c9 <> e and e <> d ;
set U2 = I \! \mathop { \vert S .| } , U1 = U U U U = U | n ;
|. q3 .| ^2 = ( ( q3 `1 ) ^2 + ( q2 `2 ) ^2 ) * ( ( q3 `1 ) ^2 + ( q2 `2 ) ^2 ) .= |. q .| ^2 + ( q `2 ) ^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 implies x = y
dom signature U1 = dom ( the charact of U1 ) & Args ( o , MSAlg U1 ) = dom ( the charact of U1 ) & Args ( o , MSAlg U1 ) = dom ( the charact of U1 ) & dom the charact of U1 = the carrier of U1 ;
dom ( h | X ) = dom h /\ X .= dom ( ||. h .|| ) /\ X .= dom ( ||. h .|| ) /\ X .= dom ( ||. h .|| ) /\ X .= dom ( ||. h .|| ) /\ X .= X ;
for N1 , N1 being Element of ( G . K1 ) holds dom ( h . K1 ) = N & rng ( h . K1 ) = N1 & rng ( h . K1 ) c= N1 & rng ( h . K1 ) c= N1
( mod ( u , m ) + mod ( v , m ) ) . i = ( mod ( u , m ) ) . i + ( mod ( v , m ) ) . i ;
- ( q `1 ) ^2 < - 1 or q `2 >= - ( q `1 ) ^2 & - ( q `1 ) ^2 <= - ( q `1 ) ^2 or - ( q `1 ) ^2 = - ( q `1 ) ^2 ;
pred r1 = fp & r2 = fp & r1 * r2 = fp * ( f1 - f2 ) & r2 * ( f1 - f2 ) = fp * ( f1 - f2 ) ;
vseq . m is bounded Function of X , the carrier of Y & x9 . m = ( for x be Element of X holds vseq . x = ( ( vseq . m ) * ( vseq . m ) ) . x ) & ( vseq . m ) . x = ( ( vseq . m ) * ( vseq . m ) ) . x ;
pred a <> b & b <> c & angle ( a , b , c ) = PI & angle ( b , c , a ) = 0 & angle ( c , a , b ) = 0 ;
consider i , j being Nat , r , s being Real such that p1 = [ i , r ] and p2 = [ j , s ] and i < j and r < s ;
|. p .| ^2 - ( 2 * |( p , q )| ) ^2 + |. q .| ^2 = |. p .| ^2 + |. q .| ^2 - ( 2 * |( p , q )| ) ^2 ;
consider p1 , q1 being Element of X ( ) such that y = p1 ^ q1 and q1 in rng p1 and p1 ^ q1 = p1 ^ q1 and p1 ^ q1 = p2 ^ q1 and p1 ^ q1 = p2 ^ q1 ;
1. ( A , ( r1 , r2 , s1 , s2 ) ) = ( s2 - s1 ) * ( s2 - s1 ) .= ( s2 - s1 ) * ( s2 - s1 ) .= ( s2 - s1 ) * ( s2 - s1 ) ;
( LMP A ) `2 = lower_bound ( proj2 .: ( A /\ /\ /\ /\ /\ /\ E-bound ( w ) ) ) & proj2 .: ( A /\ /\ /\ /\ /\ /\ /\ /\ /\ /\ /\ /\ /\ /\ /\ /\ set ( w , f ) ) is non empty ;
s , k1 |= H1 '&' H2 iff s |= ( H1 , k1 ) implies s |= ( H1 , k2 ) implies s = H1 & s = H2 & s = H2
len ( s + 1 ) = card ( support b1 ) + 1 .= card ( support b2 ) + 1 .= card ( support b1 ) + 1 .= card ( support b2 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) + 1 ;
consider z being Element of L1 such that z >= x and z >= y and for z being Element of L1 st z >= x & z `1 >= y holds z `1 >= y & z `2 >= x ;
LSeg ( UMP D , |[ ( W-bound D + E-bound D ) / 2 , ( N + N-bound D ) / 2 ]| ) /\ D = { UMP D } /\ D .= { UMP D } ;
lim ( ( ( f `| N ) / g ) /* b ) = lim ( ( f `| N ) / ( g `| N ) ) .= lim ( ( f `| N ) / g ) ;
P [ i , pr1 ( f ) . i , pr1 ( f ) . ( i + 1 ) , pr2 ( 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 ) - Rz .|| < r
for X being set , P being a_partition of X , x , a , b being set st x in a & a in P & x in P & b in P & x <> a & x in P holds a = b
Z c= dom ( ( #Z 2 ) * f ) \ ( ( #Z 2 ) * f ) " { 0 } & Z c= dom ( ( #Z 2 ) * f ) \ ( ( #Z 2 ) * f ) " { 0 } ;
ex j being Nat st j in dom ( l ^ <* x *> ) & j < i & y = ( l ^ <* x *> ) . j & i = 1 + len l & z = ( l ^ <* x *> ) . j & i = len l + 1 & j = len l + 1 ;
for u , v being VECTOR of V , r being Real st 0 < r & r < 1 & u in dom _ N holds r * u + ( 1-r * v ) in e
A , Int A , Cl A , Cl ( Int A , Cl ( A , B ) ) , Cl ( Int A , Cl ( A , B ) ) , Cl ( Int A , Cl ( A , B ) ) , Cl ( A , B ) , Cl ( A , B ) ) in F ;
- Sum <* v , u , w *> = - ( v + u + w ) .= - ( v + u ) + w .= - ( v + u ) + w .= - v + u ;
Exec ( ( a := b ) , s ) . IC SCM R = ( Exec ( ( a := b ) , s ) ) . IC SCM R .= succ IC s .= IC s .= IC s .= IC s ;
consider h being Function such that f . a = h and dom h = I and for x being element st x in I holds h . x in ( the carrier of J ) . x and h . x = ( the carrier of J ) . x ;
for S1 , S2 , D being non empty reflexive RelStr , D being non empty directed Subset of S1 , x being Element of S1 , y being Element of S2 holds x is directed & x <= y implies cos . ( x , y ) is directed
card X = 2 implies ex x , y st x in X & y in X & x <> y & for z st z in X holds z = x or z = y or z = y or z = x or z = y
E-max L~ Cage ( C , n ) in rng ( Cage ( C , n ) :- W-min L~ Cage ( C , n ) ) & E-max L~ Cage ( C , n ) in rng Cage ( C , n ) ;
for T , T being DecoratedTree , p , q being Element of dom T st p ^ q = q holds ( T , p ) . q = T . q & ( T , p ) . q = T . q
[ i2 + 1 , j2 ] in Indices G & [ i2 , j2 ] in Indices G & f /. k = G * ( i2 + 1 , j2 ) & f /. k = G * ( i2 + 1 , j2 ) ;
cluster -> commutative means : such : k divides it & n divides it & n divides it & for m being Nat st k divides m & m divides it holds it divides m & it divides m ;
dom F " = the carrier of X2 & rng F " = the carrier of X1 & F " is one-to-one & F " is one-to-one & dom F = the carrier of X2 & rng F = the carrier of X2 & F " is one-to-one & rng F = the carrier of X2 & F " is one-to-one ;
consider C being finite Subset of V such that C c= A and card C = \cdot and the carrier of V = Lin ( BM \/ C ) and C = Lin ( BM \/ C ) and C = Lin ( BM \/ C ) ;
V is prime implies for X , Y being Element of [: the topology of T , the topology of T :] st X /\ Y c= V holds X c= V or Y c= V
set X = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } , Y = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } ;
angle ( p1 , p3 , p4 ) = 0 .= angle ( p2 , p3 , p2 ) .= angle ( p2 , p3 , p2 ) .= angle ( p2 , p3 , p2 ) .= angle ( p2 , p3 , p2 ) ;
- sqrt ( ( - ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ) = - sqrt ( ( - ( q `1 / |. 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 . 0 = p1 & f . 1 = p2 & f . 0 = p2 & f . 1 = p3 & f . 1 = p4 ;
attr f is partial differentiable of REAL , u0 means : Def8 : SVF1 ( 2 , pdiff1 ( f , 1 ) , u0 ) is_continuous_in ( 2 , pdiff1 ( f , 3 ) , u0 ) is continuous & SVF1 ( 2 , pdiff1 ( f , 3 ) , u0 ) is_continuous_in ;
ex r , s st x = |[ r , s ]| & G * ( len G , 1 ) `1 < r & r < G * ( 1 , 1 ) `2 & s < G * ( 1 , 1 ) `2 ;
assume that f is_sequence_on G and 1 <= t & t <= width G and G * ( t , width G ) `2 >= N-bound L~ f and f /. len f = ( f /. len f ) `2 and f /. len f = ( f /. len f ) `2 ;
pred i in dom G means : Def8 : r * ( f * reproj ( i , x ) ) = r * f * reproj ( i , x ) ;
consider c1 , c2 being bag of o1 + o2 such that ( decomp c ) /. k = <* c1 , c2 *> and c = c1 + c2 and c1 = c2 + c2 and c2 = c2 + c1 and c2 = c2 + c2 ;
u0 in { |[ r1 , s1 ]| : r1 < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < s1 & s1 < G * ( 1 , 1 ) `2 } ;
Cl ( X ^ Y ) . k = the carrier of X . k2 .= C4 . k .= C4 . k .= C4 . k .= C4 . k .= C4 . k .= C4 . k ;
pred M1 = len M2 means : Def8 : len M1 = width M2 & M1 = M2 - M1 & M1 = M2 - M2 & M2 = - M2 + M1 ;
consider g2 be Real such that 0 < g2 and { y where y is Point of S : ||. y - x0 .|| < g2 & y in dom f & f /. y = f /. ( y - x0 ) + g2 & y in dom f } c= N2 ;
assume x < ( - b + sqrt ( x0 - a , b ) / ( 2 * a ) ) or x > ( - b - sqrt ( x0 - a , b ) / ( 2 * a ) ) / ( 2 * a ) ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ G1 ) . i & ( H1 '&' H2 ) . i = ( <* 3 *> ^ G1 ) . i & ( H1 '&' H2 ) . i = ( <* 3 *> ^ G1 ) . i ;
for i , j st [ i , j ] in Indices ( M3 + M1 ) holds ( M3 + M1 ) * ( i , j ) < M2 * ( i , j ) & ( M3 + M1 ) * ( i , j ) < M2 * ( i , j )
for f being FinSequence of NAT , i being Element of NAT st for j being Element of NAT st j in dom f holds i divides f /. j holds i divides Sum f & for j being Element of NAT st j in dom f holds f . j = Sum f
assume F = { [ a , b ] where a , b is Subset of X : for c st c in B39 & a c= c & b c= c } & F c= c & c c= d & a c= b & b c= c ;
b2 * q2 + ( b3 * q3 ) + - ( ( a1 * q2 ) + ( - ( a2 * q3 ) ) ) = 0. TOP-REAL n + ( ( a1 * q2 ) + ( - ( a2 * q3 ) ) ) .= 0. TOP-REAL n + ( ( a1 * q2 ) + ( - ( a2 * q3 ) ) ) .= 0. TOP-REAL n ;
Cl Cl F = { D where D is Subset of T : ex B being Subset of T st D = Cl B & B in F & B in F } & F is open & A c= Cl F & B c= Cl F ;
attr seq is summable means : Def8 : seq is summable & seq is summable & Partial_Sums ( seq ) . n = Partial_Sums ( seq ) . n & Partial_Sums ( seq ) . n = Partial_Sums ( seq ) . n + Partial_Sums ( seq ) . n ;
dom ( ( cn " ) | 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 , Z ]| is full full SubRelStr of ( Omega Z ) |^ the carrier of X & |[ X , Y ] is full full SubRelStr of ( Omega Z ) |^ the carrier of X & [: X , Y :] is full SubRelStr of ( Omega Z ) |^ the carrier of X ;
G * ( 1 , j ) `2 = G * ( i , j ) `2 & G * ( 1 , j ) `2 <= G * ( 1 , j ) `2 & G * ( 1 , j ) `2 <= G * ( 1 , j + 1 ) `2 ;
synonym m1 c= m2 means : Def3 : for p being set st p in P holds the non empty set of m1 <= p & the carrier of m1 <= the carrier of m2 & p is non empty ;
consider a being Element of B ( ) such that x = F ( a ) and a in { G ( b ) where b is Element of A ( ) : P [ b ] } and P [ a ] ;
synonym IT is multiplicative loop structure means : Defst : : : for a being Element of L1 , b being Element of the carrier of L1 holds ( the multF of L1 ) . ( a , b ) = [ a , b ] ) & the multF of L1 = the multF of L2 & the multF of L1 = the multF of L2 ;
sequence ( a , b ) + 1 + sequence ( c , d ) = b + sequence ( c , d ) .= b + d + 1 .= b + d + 1 .= sequence ( a + c , b + d ) ;
cluster + ( i1 , i2 ) -> NAT for Element of INT means : Def1 : for i , j being Element of NAT holds it . ( i1 , i2 ) = + ( i1 , i2 ) & it . ( i1 , i2 ) = + ( i2 , i1 ) ;
( - s2 ) * p1 + ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - s2 * p2 ) ) ) ) * p2 ) = ( - r2 ) * p1 + ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - ( s2 * p2 - s2 * p2 ) ) ) * p2 ) ;
eval ( ( a | ( n , L ) ) *' , x ) = eval ( ( a | n ) *' , x ) * 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 directed Subset of Omega S , V being open Subset of Omega S st sup D in V & V is open holds V is open and for V being open Subset of Omega S st V in V & V is open holds V meets V ;
assume that 1 <= k & k <= len w + 1 and T-7 . ( ( q11 , w ) -a9 , w ) = ( T11 . k , w ) -a9 and T11 . k = ( T11 . k , w ) -a9 ;
2 * ( a |^ ( n + 1 ) + ( 2 * b |^ ( n + 1 ) ) ) >= a |^ ( n + 1 ) + ( ( a |^ n ) * b + ( b |^ n ) * a ) + ( ( b |^ n ) * b ) ;
M , v2 |= All ( x. 3 , Ex ( x. 0 , All ( x. 4 , H ) ) ) implies M , v / ( x. 0 , All ( x. 4 , H ) ) / ( x. 4 , All ( x. 0 , H ) ) / ( x. 0 , H ) ) |= H
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f /. x0 or for x0 st x0 in l holds f /. x0 - f /. x0 < 0 & f /. x0 < 0 ;
for G1 being _Graph , W being Walk of G1 , e being set , G2 being Walk of G1 , W being Walk of G2 st not e in W & W is Walk of G1 & e in W holds W is Walk of G2
not c9 is empty iff ( for y1 , y2 st y1 is not empty & y2 is not empty & y1 is not empty & y2 is not empty & y2 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 q1 is not empty & q2 is not empty & not q2 is not empty ;
Indices GoB f = [: dom GoB f , Seg width GoB f :] & i1 + 1 in Seg ( len GoB f ) & i2 + 1 in Seg ( len GoB f ) & 1 in Seg ( len GoB f ) & 1 in Seg ( len GoB f ) & 1 in Seg ( len GoB f ) ;
for G1 , G2 , G3 , G3 being strict Subgroup of O , O being stable Subgroup of G1 st G1 is stable & G2 is stable & G1 is stable & G2 is stable & G2 is stable holds G1 * G2 is stable Subgroup of G1 * the carrier of G2 * the carrier of G1 = ( G1 * the Subgroup of G2 * the carrier of G1 ) * the stable Subgroup of G1
UsedIntLoc ( inint f ) = { intloc 0 , intloc 1 , intloc 2 , intloc 3 , intloc 4 , intloc 5 , intloc 6 , 6 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 } ;
for f1 , f2 be FinSequence of F st f1 ^ f2 is p -element & Q [ f1 ^ f2 ] & Q [ f2 ^ f1 ] & Q [ f1 ^ f2 ] holds Q [ f1 ^ f2 ]
( p `1 ) ^2 / ( sqrt ( 1 + ( p `1 / p `2 ) ^2 ) ) = ( q `1 ) ^2 / ( sqrt ( 1 + ( q `1 / q `2 ) ^2 ) ) .= ( q `1 ) ^2 / ( sqrt ( 1 + ( q `1 / q `2 ) ^2 ) ) ;
for x1 , x2 , x3 , x4 being Element of REAL n holds |( x1 - x2 , x3 )| = |( x1 , x3 )| + |( x2 , x3 )| + |( x3 , x4 )| + |( x2 , x3 )| + |( x3 , x4 )| + |( x2 , x3 )| + |( x3 , x4 )|
for x st x in dom ( ( F | A ) | A ) holds ( ( F | A ) | A ) . ( - x ) = - ( ( F | A ) . ( - x ) )
for T being non empty TopSpace , P being Subset-Family of T , x being Point of T st P c= the topology of T & for B being Basis of x ex P being Basis of T st B c= P & P is Basis of x & P is Basis of x
( a 'or' b 'imp' c ) . x = 'not' ( ( a 'or' b ) . x ) 'or' c . x .= 'not' ( ( a . x ) 'or' b . x ) 'or' c . x .= TRUE 'or' TRUE .= TRUE ;
for e being set st e in [: A , Y1 :] ex X1 being Subset of Y , Y1 being Subset of X st e = [: X1 , Y1 :] & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y2 is open & Y1 is open & Y1 is open & Y2 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y2 is open & Y1 is open & Y1 is open
for i be set st i in the carrier of S for f be Function of [: S , S1 :] , S1 . i st f = H . i & F . i = f | ( F . i ) holds F . i = f | ( F . i )
for v , w st for y st x <> y holds w . y = v . y holds Valid ( VERUM ( Al ( ) , J ) , v ) . w = Valid ( VERUM ( Al ( ) , J ) , v ) . w
card D = card D1 + card D1 - card { i , j } .= ( c1 + 1 ) + ( 1 - 1 ) - 1 .= c1 + ( 1 - 1 ) + ( 1 - 1 ) .= 2 * c1 + 1 - 1 .= c1 + ( 1 - 1 ) - 1 .= c1 + ( 1 - 1 ) ;
IC Exec ( i , s ) = ( s +* ( 0 .--> succ ( s . 0 ) ) ) . 0 .= ( 0 .--> succ ( s . 0 ) ) . 0 .= ( 0 .--> succ ( s . 0 ) ) . 0 .= succ ( IC s ) .= IC s ;
len f /. ( \downharpoonright i1 -' 1 ) -' 1 + 1 = len f -' 1 + 1 - 1 .= len f -' 1 + 1 - 1 .= len f -' 1 + 1 - 1 .= len f -' 1 + 1 - 1 .= len f -' 1 + 1 ;
for a , b , c being Element of NAT st 1 <= a & 2 <= b & k <= a holds k < ( a + b-2 ) or k = a + b-2 or k = a + b-2 or k = a + b-2 or k = a + b-2 or k = a + b-2 or k = a + b-2 or k = a + b-2
for f being FinSequence of TOP-REAL 2 , p being Point of TOP-REAL 2 , i being Element of NAT st p in LSeg ( f , i ) & p in LSeg ( f , i ) holds Index ( p , f ) <= i
lim ( ( curry ( P+* ( i , k + 1 ) ) # x ) = lim ( ( curry ( P+* ( i , k ) ) # x ) + lim ( ( curry ( F+* ( i , k ) ) # x ) ) .= lim ( ( curry ( F+* ( i , k + 1 ) ) # x ) ) ;
z2 = g /. ( \downharpoonright n1 -' n2 + 1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) .= g . ( i -' n2 + 1 ) ;
[ f . 0 , f . 3 ] in id ( the carrier of G ) \/ ( the InternalRel of G ) or [ f . 0 , f . 3 ] in the InternalRel of C & [ f . 0 , f . 2 ] in the InternalRel of G & [ f . 0 , f . 3 ] in the InternalRel of G ;
for G being Subset-Family of B st G = { R [ X ] where R is Subset of [: A , B :] , R : R in ( F . X ) & R in ( F . X ) & R [ Y ] } holds ( Intersect ( F ) ) . X = Intersect ( G )
CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) = CurInstr ( P1 , Comput ( P1 , s1 , m2 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m2 ) ) .= halt SCMPDS .= halt SCMPDS .= halt SCMPDS .= ( CurInstr ( P1 , Comput ( P1 , s1 , m2 ) ) ) ;
assume that a on M and b on M and c on N and d on N and p on N and a on P and c on P and d on P and a on P and b on P and c on P and d on P and a on P and b on P and c on Q and d on Q and d on Q and a <> b and a <> b and b <> c and a <> c and b <> c ;
assume that T is \hbox of 4 , T2 and F is closed and ex F being Subset-Family of T st F is closed , countable & ind F <= 0 & ind F <= 0 & ind F <= 0 & ind F <= 0 ;
for g1 , g2 st g1 in ]. r^ - 1 , r .[ & g2 in ]. r1 , r2 .[ holds |. f . g1 - g . g2 .| <= ( g1 - g2 ) / ( |. g1 .| - 1 ) * ( |. g2 .| - 1 ) * ( |. g2 .| - 1 ) * ( |. g2 .| - 1 ) * ( |. g2 .| - 1 ) * ( |. g2 .| - 1 ) * ( |. g2 .| - 1 ) * ( |. g2 .| - 1 ) * ( |. g2 .| - |. g2 .| - ( |. g2 .| - ( |. g2 .| - ( |. g2 .| - ( |. g2 .| - ( |. g2 .| - ( |. g2 .| - ( |. g2 .| - ( |. g2 .| - ( |. g2 .| - ( |. g2 .| - ( |. g2 .| - ( |. g2 .| - ( |. g2 .| - ( |. g2 .| - ( |. g2 .| - (
cosh /. ( z1 + z2 ) = ( cosh /. z1 ) * ( cosh /. z2 ) + ( ( cosh /. z1 ) * ( cosh /. z2 ) + ( ( cosh /. z2 ) * ( cosh /. z2 ) ) * ( cosh /. z2 ) .= ( ( cosh /. z1 ) * ( cosh /. z2 ) + ( cosh /. z2 ) * ( cosh /. z2 ) ) ;
F . i = F /. i .= 0. R + r2 .= b |^ ( n + 1 ) .= <* b |^ ( n + 1 ) , \dots , b |^ ( n + 1 ) *> .= <* ( n + 1 ) |^ ( n + 1 ) , \dots , b |^ ( n + 1 ) *> .= <* ( n + 1 ) |^ ( n + 1 ) , \dots , b *> ;
ex y being set , f being Function st y = f . n & dom f = NAT & f . 0 = A ( ) & for n holds f . ( n + 1 ) = RB ( ) . ( n , f . n ) & for n holds f . ( n + 1 ) = R ( n , f . n ) ;
func f (#) F -> FinSequence of V means : Def6 : len it = len F & for i be Nat st i in dom it holds it . i = F /. i * f /. i & it . i = F /. i * F /. i ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , 7 , 8 , 8 , 7 , 8 } = { x1 , x2 } \/ { x3 , x4 , x5 , 8 } \/ { x4 , x5 , 7 , 8 } .= { x1 , x2 } \/ { x4 , x5 , 8 } \/ { x5 , 8 , 7 , 8 } ;
for n being Nat , x being set st x = h . n holds h . ( n + 1 ) = o . ( x , n ) & x in InputVertices S ( x , n ) & o . ( x , n ) in InnerVertices S ( x , n ) & o . ( x , n ) in InnerVertices S ( x , n ) ;
ex S1 being Element of CQC-WFF ( Al ) st SubP ( P , l , e ) = S1 & ( S1 is Element of CQC-WFF ( Al ) & ( S1 is Element of CQC-WFF ( Al ) implies S1 is Element of CQC-WFF ( Al ) ) & ( S1 is Element of CQC-WFF ( Al ) implies S1 is Element of CQC-WFF ( Al ) ) & ( not S1 is Element of CQC-WFF ( Al ) implies S1 is Element of CQC-WFF ( Al ) )
consider P being FinSequence of Gs2 such that p7 = Product P and for i st i in dom P ex t7 being Element of the carrier of G st P . i = t7 & t7 = t & t7 = t & t7 = t & t7 = t ;
for T1 , T2 being non empty TopSpace , P being Basis of T1 , P being Basis of T2 st the carrier of T1 = the carrier of T2 & P is Basis of T1 & P is Basis of T2 & P = the topology of T1 & P = the topology of T2 & P = the topology of T1 & P = the topology of T2 holds P = the topology of T1
assume that f is_\mathbin { \lbrack u0 , u0 .] and r (#) pdiff1 ( f , 3 ) is_partial_differentiable_in u0 , 2 and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 ) = r (#) pdiff1 ( f , 3 ) and partdiff ( f , 3 ) = r (#) pdiff1 ( f , 3 ) , u0 ) ;
defpred P [ Nat ] means for F , G being FinSequence of ExtREAL for s being Permutation of ExtREAL , G st len F = $1 & G = F * s & not G = F * s & not F = G * s holds Sum F = Sum G & Sum G = Sum F ;
ex j st 1 <= j & j < width GoB f & ( GoB f ) * ( 1 , j ) `2 <= s & s <= ( GoB f ) * ( 1 , j + 1 ) `2 & ( GoB f ) * ( 1 , j + 1 ) `2 <= s & s <= ( GoB f ) * ( 1 , j + 1 ) `2 ;
defpred U [ set , set ] means ex F-23 be Subset-Family of T st $2 = F-23 & union ( F-23 is open & union ( F-23 ) = $1 & ( for x being Element of T st x in Fholds Fst . x = x ) & ( x in F . x ) & ( x in F . x ) & ( x in F . x ) & ( x in F . x ) & ( x in F . x ) ;
for p4 being Point of TOP-REAL 2 st LE p4 , p1 , P , p1 , p2 & LE p2 , p1 , P , p1 , p2 & LE p4 , p1 , P , p1 , p2 & LE p4 , p1 , P , p1 , p2 & LE p4 , p1 , P , p1 , p2 & LE p2 , p1 , P , p1 , p2 & LE p4 , p1 , P , p2 & LE p2 , p1 , P , p1 , p2 & LE p2 , p1 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 & LE p1 , p2 , P , p2 & LE p1 , p2 , P , p2 & LE p1 , p2 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 & LE p1 , p2 , P , P , p1 , p2 & LE
f in D ( H ) & for g st g . y <> f . y holds x = y & g in D ( ) implies f in D ( ) & g in E ( ) & f . x = f . ( All ( x , H ) ) implies f in D ( )
ex 8 being Point of TOP-REAL 2 st x = 8 & ( ( ( 8 - 1 ) / ( 2 * ( 1 + ( p `2 / p `1 ) ) ) ^2 + ( ( 8 - 1 ) / ( 2 * ( 1 + ( p `2 / p `1 ) ) ^2 ) ) & ( 8 - 1 ) * ( 1 + ( p `2 / p `1 ) ^2 ) <= 0 ) ;
assume for d7 being Element of NAT st d7 <= max ( d7 , ( n + 1 ) -NAT ) holds s1 . ( ( d + 1 ) -t7 ) = s2 . ( ( d + 1 ) -t7 ) & s2 . ( ( d + 1 ) -t7 ) = s2 . ( ( d + 1 ) -t7 ) ;
assume that s <> t and s is Point of Sphere ( x , r ) and s is Point of Sphere ( x , r ) and ex e being Point of E st { e } = Sphere ( s , t ) /\ Sphere ( x , r ) & e = Sphere ( s , t ) /\ Sphere ( x , r ) ;
given r such that 0 < r and for s holds 0 < s and for x1 , x2 be Point of CNS st x1 in dom f & x2 in dom f & ||. x1 - x2 .|| < s holds |. f /. x1 - f /. x2 .| < r ;
( p | x ) | ( p | ( ( x | x ) | ( x | x ) ) ) = ( ( ( x | x ) | ( x | x ) ) | p ) | ( ( x | x ) | p ) ;
assume that x , x + h in dom sec and ( for x st x in dom sec holds ( ( sec * sec ) `| Z ) . x = ( 4 * sin . ( x + h ) * sin . ( x + h ) ) / ( cos . ( x + h ) * sin . ( x + h ) ) ^2 ) and ( sec * sec ) `| Z ) . x = ( 4 * sin . ( x + h ) ) / ( cos . ( x + h ) ;
assume that i in dom A and len A > 1 and for B st B > 1 & B c= dom A & A * ( i , j ) = ( A * ( i , j ) ) * ( A * ( i , j ) ) and len A = len ( A * ( i , j ) ) and width A = width ( A * ( i , j ) ) ;
for i be non zero Element of NAT st i in Seg n holds ( i divides n or i = n or i = <* 1_ F_Complex *> ) & ( i <> n implies h . i = <* 1_ F_Complex *> ) & ( i <> n implies h . i = 1. F_Complex ) & ( i <> n implies h . i = 1. F_Complex )
( ( ( b1 'imp' b2 ) '&' ( c1 'imp' c2 ) ) '&' ( ( a1 'or' b1 ) '&' ( a2 'or' c1 ) ) '&' 'not' ( a2 '&' c1 ) ) '&' 'not' ( ( a1 '&' b1 ) '&' ( a2 '&' c1 ) ) '&' 'not' ( ( a1 '&' b1 ) '&' 'not' ( a2 '&' c1 ) ) '&' 'not' ( a2 '&' c1 ) '&' 'not' ( a1 '&' b1 ) '&' 'not' ( a2 '&' c1 ) '&' 'not' ( a2 '&' c1 ) ;
assume that for x holds f . x = ( ( cot * ( sin . x ) ) `| Z ) . x and x - h / ( sin . ( sin . x ) ) ^2 and for x st x in Z holds ( ( ( cot * ( cot * ( sin . x ) ) ) `| Z ) . x = cos . ( x- . x ) / ( sin . ( - h / ( sin . x ) ) ^2 ) ) ;
consider R8 , I-8 be Real such that R8 = Integral ( M , Re ( F . n ) ) and Integral ( M , Im ( F . n ) ) = Integral ( M , Im ( F . n ) ) and Integral ( M , Im ( F . n ) ) = Integral ( M , Im ( F . n ) ) + Integral ( M , Im ( F . n ) ) ;
ex k being Element of NAT st k0 = k & 0 < d & for q being Element of product G st q in X & ||. q- x .|| < d holds ||. partdiff ( f , q , k ) - partdiff ( f , x , k ) .|| . q = r * partdiff ( f , x , k ) . q
x in { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , 7 , 8 , 8 , 7 , 8 } iff x in { x1 , x2 , x3 , x4 , 8 } \/ { x5 , 7 , 8 } & x in { x1 , x2 , x3 , x4 , 8 } \/ { x5 , 8 , 7 , 8 } \/ { x5 , 8 } ;
G * ( j , i ) `2 = G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , j ) `2 .= G * ( 1 , j + 1 ) `2 .= G * ( 1 , j + 1 ) `2 .= G * ( 1 , j + 1 ) `2 .= G * ( 1 , j + 1 ) `2 .= G * ( 1 , j + 1 ) `2 ;
f1 * p = p .= ( ( the Arity of S1 ) +* ( the Arity of S2 ) ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o ;
func tree ( T , P , T1 ) -> DecoratedTree means : Def1 : q in it iff q in T & for p , r st p in P & r in T holds p ^ r in T1 & p ^ r in T1 & q ^ r in T1 & p ^ r in T1 & p ^ r in T1 ;
F /. ( k + 1 ) = F . ( k + 1 ) .= Fholds F . ( p . ( k + 1 -' 1 ) , k + 1 -' 1 ) = FJ . ( p . ( k + 1 -' 1 ) , k -' 1 ) .= FJ . ( p . ( k + 1 -' 1 ) , k -' 1 ) ;
for A , B , C being Matrix of K st len B = len C & len B = width C & len B = width C & len A > 0 & len B > 0 & len A > 0 & len C > 0 & len A > 0 & len B > 0 & len A > 0 & len C > 0 & len A > 0 & len B > 0 holds A * ( B * A ) = A * BC
seq . ( k + 1 ) = 0. F_Complex + seq . ( k + 1 ) .= ( Partial_Sums seq ) . ( k + 1 ) + seq . ( k + 1 ) .= ( Partial_Sums seq ) . ( k + 1 ) + seq . ( k + 1 ) .= ( Partial_Sums seq ) . ( k + 1 ) + seq . ( k + 1 ) ;
assume that x in ( the carrier of CQ ) /\ ( the carrier of CQ ) and y in ( the carrier of CQ ) /\ ( the carrier of CQ ) and [ x , y ] in the carrier of CQ and [ x , y ] in the InternalRel of CQ and [ y , x ] in the InternalRel of CQ ;
defpred P [ Element of NAT ] means for f st len f = $1 holds ( VAL g ) . ( k + 1 ) = ( VAL g ) . ( ( VAL f ) . ( k + 1 ) ) '&' ( VAL g ) . ( ( VAL f ) . ( k + 1 ) ) ;
assume that 1 <= k and k + 1 <= len f and f is_sequence_on G and [ i , j ] in Indices G and f /. k = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) ;
assume that sn < 1 and q `1 > 0 and q `2 / |. q .| >= sn and p `1 / |. q .| - cn and p `2 / |. q .| - cn and p `1 / |. q .| - cn and p `1 / |. q .| - cn and p `1 / |. q .| - cn and p `1 / |. q .| - cn and p `2 / |. q .| - cn and p `1 / |. q .| - cn and p `1 / |. q .| = 0 ;
for M being non empty TopSpace , x being Point of M , x being Point of M st x = x `1 holds ex f being sequence of B\HM ( M ) st for n being Element of NAT holds f . n = Ball ( x `1 , ( 1 / ( n + 1 ) ) * ( f . n ) )
defpred P [ Element of omega ] means f1 is differentiable Z implies ( f1 + f2 ) (#) ( f1 - f2 ) is_differentiable_on Z & for x st x in Z holds ( f1 - f2 ) `| Z . ( $1 + 1 ) = f1 . ( x - a ) + f2 . ( x - a ) / ( ( x - a ) * ( x - a ) ) ;
defpred P1 [ Nat , Point of CNS ] means $2 in Y & ||. s1 . $1 - ( f /. $1 ) .|| < r & ||. ( f /. $1 - f /. ( $1 + 1 ) ) .|| < r & ||. ( f /. $1 - f /. ( $1 + 1 ) ) .|| < r ;
( f ^ mid ( g , 2 , len g ) ) . i = ( mid ( g , 2 , len g ) ) . ( i - len f + 1 ) .= g . ( i - len f + 1 ) .= g . ( i - len f + 1 ) .= g . ( i - len f + 1 ) .= g . ( i - len f + 1 ) ;
( 1 / 2 * n0 + 2 * n0 ) * ( 2 * n0 + 2 * n0 ) = ( ( 1 / 2 * n0 + 2 * n0 ) * ( 2 * n0 + 1 ) ) * ( 2 * n0 + 1 ) * ( 2 * n0 + 1 ) * ( 2 * n0 + 1 ) * ( 2 * n0 + 1 ) .= ( 1 / 2 * n0 + 1 ) * ( 2 * n0 + 1 ) * ( 2 * n0 + 1 ) ;
defpred P [ Nat ] means for G being non empty finite strict symmetric RelStr , H being symmetric RelStr st G is \rm free & card the carrier of G = $1 & the carrier of H = the carrier of G & the carrier of H = the carrier of H & the carrier of H = the carrier of H & the carrier of H = the carrier of H & the carrier of H = the carrier of H ;
assume that not f /. 1 in Ball ( u , r ) and 1 <= m and m <= len - ( f /. i ) and for i st 1 <= i & i <= len f & LSeg ( f , i ) /\ Ball ( u , r ) <> {} holds not f /. i in Ball ( u , r ) & not f /. i in Ball ( u , r ) ;
defpred P [ Element of NAT ] means ( Partial_Sums ( ( cos * ( ]. - r , r .[ ) ) ) ) . $1 = ( Partial_Sums ( ( cos * ( ]. - r , r .[ ) ) ) ) . ( 2 * $1 ) & ( Partial_Sums ( ( cos * ( ]. - r , r .[ ) ) ) ) . ( 2 * $1 ) = ( Partial_Sums ( ( cos * ( ( - r , r ) ) ) ) . ( 2 * $1 ) ;
for x being Element of product F holds x is FinSequence of G & dom x = I & x = ( the carrier of F ) . i & for i being set st i in dom ( the support of F ) holds x . i = ( the Sorts of F ) . i & x . i = ( the Sorts of F ) . i
( x " ) |^ ( n + 1 ) = ( ( x " ) |^ n ) * x " .= ( x |^ ( n + 1 ) ) * x " .= ( x |^ ( n + 1 ) ) * x " .= ( x |^ ( n + 1 ) ) * x " .= ( x |^ ( n + 1 ) ) * x .= ( x |^ ( n + 1 ) ) * x ;
DataPart Comput ( P +* ( a , I ) , Initialized s ) = DataPart Comput ( P +* I , Initialize s , LifeSpan ( P +* I , Initialize s ) + 3 ) .= DataPart Comput ( P +* I , Initialize s , LifeSpan ( P +* I , Initialize s ) + 3 ) .= DataPart Comput ( P +* I , Initialize s , LifeSpan ( P +* I , Initialize s ) + 1 ) ;
given r such that 0 < r and ]. x0 - r , x0 .[ c= ( dom f1 /\ dom f2 ) and for g st g in ]. x0 - r , x0 .[ /\ dom f2 holds f1 . g <= f2 . g & f2 . g <= f2 . g & f2 . g <= f2 . g ;
assume that X c= dom f1 /\ dom f2 and f1 | X is continuous and f2 | X is continuous and ( f1 - f2 ) | X is continuous and ( f1 - f2 ) | X is continuous and ( f1 - f2 ) | X is continuous and ( f1 - f2 ) | X is continuous and f2 | X is continuous and ( f1 - f2 ) | X is continuous ;
for L being continuous complete LATTICE for l being Element of L st for X being Subset of L holds l = sup X & for x being Element of L st x in X holds x is compact holds for x being Element of L st x in X holds x is compact & x is compact & x is compact holds l is compact
Support ( e ) in { Support ( m *' p ) where m is Polynomial of n , L : i = Support ( m *' p ) & ex p being Polynomial of n , L st p in Support ( m *' p ) & p in Support ( m *' q ) & p in Support ( m *' q ) } ;
( f1 - f2 ) /. ( lim s1 ) = lim ( f1 /* s1 ) - lim ( f2 /* s1 ) .= lim ( f1 /* s1 ) - lim ( f2 /* s1 ) .= lim ( f1 /* s1 ) - lim ( f2 /* s1 ) .= lim ( f1 /* s1 ) - lim ( f2 /* s1 ) .= lim ( f2 /* s1 ) - lim ( f2 /* s1 ) ;
ex p1 being Element of CQC-WFF ( Al ) st F . p1 = g `2 & for g being Function of [: \hbox { 0 } , D :] , D st P [ g , p1 , ( len p1 ) qua Nat ] & P [ g , p1 , ( len p1 ) qua Nat ] holds P [ p1 , p2 , g , p1 ] ;
( mid ( f , i , len f -' 1 ) ^ <* f /. j *> ) /. j = ( mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) ) . j .= ( mid ( f , j , len f -' 1 ) ) /. j .= f /. j ;
( ( p ^ q ) . ( len p + k ) ) = ( ( 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 ) . ( len p + k ) ;
len mid ( upper_volume ( D2 , D1 ) , 1 ) + indx ( D2 , D1 , j1 ) = indx ( D2 , D1 , j ) + indx ( D2 , D1 , j ) .= indx ( D2 , D1 , j ) + 1 .= indx ( D2 , D1 , j ) + 1 ;
x * y * z = Mz . ( ( y * z ) * z9 ) .= ( x * ( y * z ) ) * ( y * z ) .= ( x * ( y * z ) ) * ( x * z ) .= ( x * ( y * z ) ) * ( x * z ) .= ( x * ( y * z ) ) * ( x * z ) ;
v . <* x , y *> + ( <* x0 , y0 *> ) * i = partdiff ( v , ( x - x0 ) * ( x - x0 ) + ( ( proj ( 1 , 1 ) * ( x - x0 ) ) * ( x - x0 ) ) + ( ( proj ( 1 , 1 ) * ( x - x0 ) + ( proj ( 1 , 1 ) * ( x - x0 ) ) * ( x - x0 ) ) ;
i * i = <* 0 * ( - 1 ) - ( 0 * 0 ) , 0 * 0 + ( 0 * 0 ) , 0 * 0 + ( 0 * 0 ) + ( 0 * 0 ) + ( 0 * 0 ) , 0 * 0 + ( 0 * 0 ) + ( 0 * 0 ) + ( 0 * 0 ) * 0 + ( 0 * 0 ) * 0 .= ( - 1 ) * 0 + 0 * 0 ;
Sum ( L (#) F ) = Sum ( L (#) ( F1 ^ F2 ) ) .= Sum ( ( L (#) F1 ) ^ ( L (#) F2 ) ) .= Sum ( ( L (#) F1 ) ^ ( L (#) F2 ) ) .= Sum ( ( L (#) F1 ) ^ ( L (#) F2 ) ) .= Sum ( ( L (#) F1 ) ^ ( L (#) F2 ) ) .= Sum ( L (#) F1 ) + Sum ( L (#) F2 ) .= Sum ( L (#) F1 ) ;
ex r be Real st for e be Real st 0 < e ex Y0 be finite Subset of X st Y0 is non empty & Y0 c= Y & for Y1 be finite Subset of X st Y1 c= Y & Y1 c= Y holds |. ( union Y1 ) - ( union Y2 ) .| < r
( GoB f ) * ( i , j ) = f /. ( k + 2 ) & ( GoB f ) * ( i , j + 1 ) = f /. ( k + 1 ) or ( GoB f ) * ( i , j + 1 ) = f /. ( k + 1 ) & ( GoB f ) * ( i , j + 1 ) = f /. ( k + 1 ) ;
( ( r / 2 ) * ( sin . x ) ^2 ) = ( ( r / 2 ) * ( sin . x ) ) ^2 .= ( ( r / 2 ) * ( sin . x ) ) ^2 .= ( ( r / 2 ) * ( sin . x ) ) ^2 .= ( ( r / 2 ) * ( sin . x ) ) ^2 .= ( ( r / 2 ) * ( sin . x ) ) ^2 .= ( r / 2 ) ^2 ;
x- ( ( - b ) + sqrt ( x0 - a ) ) / ( 2 * a ) < 0 & x- ( - b ) - sqrt ( x0 - a ) < 0 or - ( ( - b ) + sqrt ( x0 - a ) ) / ( 2 * a ) > 0 & x- ( - b ) + sqrt ( x0 - a ) > 0 ;
assume that ex_inf_of uparrow "\/" ( X , L ) , L and ex_sup_of X , L and "\/" ( X , L ) = "/\" ( ( uparrow "\/" ( X , L ) ) , L ) and "\/" ( X , L ) = "/\" ( ( uparrow "\/" ( X , L ) ) , L ) and not "\/" ( X , L ) in C ;
( ( for j holds B . ( j , i ) = ( j = i = j = j ) |-- id the Sorts of ( B . i , j ) ) , ( j = i implies ( B . i = j = i ) implies ( B . i = j ) ) & ( B . i = j implies B . i = j )