thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; contradiction ; contradiction ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; contradiction ; contradiction ; contradiction ; contradiction ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; contradiction ; contradiction ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; contradiction ; contradiction ; thesis ; contradiction ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; contradiction ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; contradiction ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; contradiction ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; contradiction ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; contradiction ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; 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 = 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 rng ; b in A ; d divides a ; i < n ; s <= b ; b in B ; let r ; B is one-to-one ; R is total ; x = 2 ; d in D ; let c ; let c ; b = Y ; 0 < k ; let b ; let n ; r <= b ; x in X ; i >= 8 ; let n ; let n ; y in f ; let n ; 1 < j ; a in L ; C is boundary ; a in A ; 1 < x ; S is finite ; u in I ; z << z ; x in V ; r < t ; let t ; x c= y ; a <= b ; m in NAT ; assume f is prime ; not x in Y ; z = +infty ; k be Nat ; K is being_line ; assume n >= N ; assume n >= N ; assume X is .= Y ; assume x in I ; q is as Nat ; assume c in x ; 1-p > 0 ; assume x in Z ; assume x in Z ; 1 <= k2a ; assume m <= i ; assume G is rng ; assume a divides b ; assume P is closed ; b-a > 0 ; assume q in A ; W is non bounded ; f is Assume f is that X 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 `1 <= c `1 ; A meets W ; i `1 <= j `1 ; assume H is universal ; assume x in X ; let X be set ; let T be Tree ; let d be element ; let t be element ; let x be element ; let x be element ; let s be element ; k <= 5 - 1 ; let X be set ; let X be set ; let y be element ; let x be element ; P [ 0 ] let E be set , A be Subset of E ; let C be category ; let x be element ; 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 , & n2 is , ; Q halts_on s ; x in \in \in that x in \in of S ; M < m + 1 ; T2 is open ; z in b id 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 TOP-REAL 2 ; 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 , x ; let E be Ordinal ; o : o1 , o2 o2 ; O <> O2 ; let r be Real ; let f be FinSeq-Location ; let i be Nat ; let n be Nat ; Cl A = A ; L c= Cl L ; A /\ M = B ; let V be RealUnitarySpace , v be VECTOR of V ; not s in Y to_power 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 , M be Subset of V ; P [ 1 ] ; P [ {} ] ; C1 is component ; H = G . i ; 1 <= i `1 + 1 ; F . m in A ; f . o = o ; P [ 0 ] ; abeing <= being Real ; R [ 0 ] ; b in f .: X ; assume q = q2 ; x in [#] V ; f . u = 0 ; assume e1 > 0 ; let V be RealUnitarySpace , M be Subset of V ; s is trivial & s is non empty ; dom c = Q P [ 0 ] ; f . n in T ; N . j in S ; let T be complete LATTICE , x be Element of T ; the ObjectMap of F is one-to-one sgn x = 1 ; k in support a ; 1 in Seg 1 ; rng f = X ; len T in X ; vbeing < n ; ST is bounded ; assume p = p2 ; len f = n ; assume x in P1 ; i in dom q ; let U2 , U1 , U2 ; pp `1 = c ; j in dom h ; let k ; f | Z is continuous ; k in dom G ; UBD C = B ; 1 <= len M ; p in Ball ( x , r ) ; 1 <= jj & jj <= len f ; set A = z1 ; card a [= c ; e in rng f ; cluster B \oplus A -> empty ; H is has no or H is non empty ; assume n0 <= m ; T is increasing ; e2 <> e1 & e2 <> e1 ; 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 ; X0 be set ; c = sup N ; R is_connected implies union M is connected 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 <> {} ; Element of Y ; let f be ) ; 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 + dom \vert v .| ; - y in I ; let A be non empty set , B be non empty Subset of A ; P0 = 1 ; assume r in F . k ; assume f is simple ; let A be as as as as countable set ; rng f c= NAT \/ { x } assume P [ k ] ; f6 <> {} ; o be Ordinal ; assume x is sum of p ; assume not v in { 1 } ; let IX , Y ; 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 <> {} & mm <> {} ; M + N c= M + M ; assume M is connected hhh) ; assume f is G -brr\rm is closed ; 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 & k1 <= len f ; f | A is non empty continuous ; f . x - a <= b ; assume y in dom h ; x * y in B1 ; set X = Seg n ; 1 <= i2 + 1 ; k + 0 <= k + 1 ; p ^ q = p ; j |^ y divides m ; set m = max A ; [ x , x ] in R ; assume x in succ 0 ; a in sup phi ; CH in X ; q2 c= C1 & q2 c= C2 ; a2 < c2 & a2 < b2 ; s2 is 0 -started & s2 is 0 -started ; IC s = 0 & IC s = 0 ; s4 = s4 & s4 = s4 ; let V ; let x , y be element ; let x be Element of T ; assume a in rng F ; x in dom T `1 ; let S be as <> of L ; y " <> 0 ; y " <> 0 ; 0. V = uw ; y2 , y , w be Element of V ; R8 ; let a , b be Real , x be Point of TOP-REAL 2 ; let a be Object of C ; let x be Vertex of G ; let o be Object of C , a be Object of C ; r '&' q = P \lbrack l , P .] ; let i , j be Nat ; let s be State of A , x be Element of A ; s4 . n = N ; set y = ( x `1 ) / 2 ; mi in dom g & mi in dom g ; l . 2 = y1 ; |. g . y .| <= r ; f . x in CX0 ; V1 is non empty & V2 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 N-22 ; reconsider i = i as Ordinal ; r * v = 0. X ; rng f c= INT & rng f c= INT ; G = 0 .--> goto 0 ; let A be Subset of X ; assume that A is dense and A is open ; |. f . x .| <= r ; 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 , Y = Vars ; let o be OperSymbol of S ; let R be connected non empty Poset ; n + 1 = succ n ; x9 c= Z1 & x9 c= Z1 ; dom f = C1 & dom g = C2 ; assume [ a , y ] in X ; Re ( seq . n ) is convergent ; assume a1 = b1 & a2 = b2 ; A = sInt A & B = sInt B ; a <= b or b <= a ; n + 1 in dom f ; let F be Instruction of S , i be Element of NAT ; assume that r2 > x0 and x0 < r2 ; let Y be non empty set , f be Function of Y , BOOLEAN ; 2 * x in dom W ; m in dom ( g2 | X ) ; n in dom ( g1 | X ) ; k + 1 in dom f ; the still of S in { s } ; assume that x1 <> x2 and x2 <> x3 ; v1 in [: V1 , V2 :] ; not [ b `1 , b `2 ] in T ; ( i + 1 ) + 1 = i ; T c= T & T c= T ; ( l `1 ) ^2 = 0 ; n be Nat ; ( t `2 ) ^2 = r ; A\in is_integrable_on M & A\in is_integrable_on M ; set t = Top t ; let A , B be real-membered set ; k <= len G + 1 ; cC misses cC ; Product seq 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 \/ G Int ( G1 \/ G2 ) <> {} ; ( z `2 ) ^2 = 0 ; p01 <> p1 & p01 <> 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 transitive 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 non empty full full ; A \/ { a } c= B ; 0. V = 0. Y & 0. V = 0. Y ; let I be be be be halting Instruction of S , s be Element of S ; f-24 . x = 1 ; assume z \ x = 0. X ; C4 = 2 to_power n ; let B be SetSequence of Sigma ; assume X1 = p .: D ; n + l2 in NAT & n + l2 in NAT ; f " P is compact & f " P is compact ; assume x1 in [: REAL , REAL :] ; p1 = K1 & p2 = K1 or p2 = K1 ; M . k = <*> REAL ; phi . 0 in rng phi ; OSMComput A is closed ; assume z0 <> 0. L & 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 ; [: R , S :] is stable ; set cR = Vertices R , cR = Vertices R ; p0 c= P3 & p1 c= P3 ; x in [. 0 , 1 .[ ; f . y in dom F ; let T be Scott Scott Scott Scott Scott of S ; ex_inf_of the carrier of S , S ; \vert \vert a \vert = downarrow b & a <= 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 ~ ] , b ~ are_equipotent ; assume a in A ( ) ; k in dom ( q | k ) ; p is non empty \HM { x } ; i -' 1 = i-1 - 1 ; f | A is one-to-one ; assume x in f .: X ( ) ; i2 - i1 = 0 & i2 - i1 = 0 ; j2 + 1 <= i2 & j2 + 1 <= width G ; g " * a in N ; K <> { [ {} , {} ] } ; cluster strict for } : cluster strict for \rm \mathbb Z ; |. q .| ^2 > 0 ; |. p4 .| = |. p .| ; s2 - s1 > 0 & 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 I = Seg n ; assume that k in dom C and k <> i ; 1 + 1-1 <= i + - 1 ; dom S = dom F & rng S c= dom G ; let s be Element of NAT , n 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 non empty non void holds S is non empty ; 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= [: { x } , V :] ; assume I is_closed_on s , P & I is_halting_on s , P ; U2 = U2 & U2 = U2 implies U2 is strict & U2 is strict M /. 1 = z /. 1 ; x9 = x9 & y9 = y9 or x9 = y9 ; i + 1 < n + 1 + 1 ; x in { {} , <* 0 *> } ; f9 <= f9 & f9 <= g9 implies f9 <= g9 let l be Element of L ; x in dom ( F . n ) ; let i be Element of NAT ; seq1 is COMPLEX & seq2 is COMPLEX implies seq1 - seq2 is bounded 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 non empty Subset-Family of Omega ; set r = { \cdot ( k + 1 ) } ; y = W . ( 2 * PI ) ; assume dom g = cod f & cod g = cod f ; let X , Y be non empty TopSpace , f be Function of X , Y ; x \oplus A is interval ; |. <*> A .| . a = 0 ; cluster strict for subLattice of L ; a1 in B . s1 & a2 in B . s2 ; let V be finite finite strict z over F , v be Element of V ; A * B on B & A on B ; f-3 = NAT --> 0 .= fg ; let A , B be Subset of V ; z1 = P1 . j & z2 = P2 . j ; assume f " P is closed & 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 , 1 ) ; z in dom ( A --> y ) ; P [ y , h . y ] ; { x0 } c= dom f & { x0 } c= dom g ; let B be non-empty ManySortedSet of I , A be non empty set ; ( PI / 2 ) * PI < Arg z ; reconsider z9 = 0 , z9 = 1 as Nat ; LIN a , d , c ; [ y , x ] in IF ; Q * ( 1 , 3 ) `1 = 0 ; set j = x0 div m , i = m mod n ; assume a in { x , y , c } ; j2 - jj > 0 & j2 - jj > 0 ; I the Element of I the Element of phi = 1 ; [ y , d ] in FF ; let f be Function of X , Y ; set A2 = ( B - C ) / ( B - C ) ; s1 , s2 be Element of L & s1 <= s2 ; j1 -' 1 = 0 & j2 -' 1 = 0 ; set m2 = 2 * n + j ; reconsider t = t as bag of n ; I2 . j = m . j ; i |^ s , n are_relative_prime & i |^ s , n are_relative_prime ; set g = f | D-21 ; assume that X is lower and 0 <= r ; p1 `1 = 1 & p2 `1 = 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 <= len f ; 1 <= i1 -' 1 & i1 <= len f ; i + i2 <= len h & i + 1 <= len h ; x = W-min ( P ) & y = W-min ( P ) ; [ x , z ] in [: X , Z :] ; assume y in [. x0 , x .] ; assume p = <* 1 , 2 , 3 *> ; len <* A1 *> = 1 & width <* A1 *> = 1 ; set H = h . g9 , I = h . ( g . O ) ; card b * a = |. a .| ; Shift ( w , 0 ) |= v ; set h = h2 (*) h1 , h1 = h2 (*) h2 ; assume x in X3 /\ ( X1 union X2 ) ; ||. h .|| < d1 & ||. h .|| < g ; not x in the carrier of f & x in the carrier of g ; f . y = F ( y ) ; for n holds X [ n ] ; k - l = k\overline ( A ) ; <* p , q *> /. 2 = q ; let S be Subset of the carrier of Y ; let P , Q be \langle 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 reflexive RelStr , x be Element of L ; S-20 is x -f1 -basis i ; let r be non positive Real ; M , v / ( x , y ) |= H ; v + w = 0. ( Z1 , p ) ; P [ len F ( ) ] & P [ len F ( ) ] ; assume InsCode ( i , 5 ) = 8 & InsCode ( i , 6 ) = 7 ; the zero of M = 0 & the zero of M = 0 ; cluster z * seq -> summable for Real_Sequence ; let O be Subset of the carrier of C ; ||. f .|| | X is continuous ; x2 = g . ( j + 1 ) ; cluster -> [#] for Element of S ; reconsider l1 = l-1 as Nat ; v4 is Vertex of r2 & v4 is Vertex of r2 ; T2 is SubSpace of T2 & T1 is SubSpace of T2 ; Q1 /\ Q19 <> {} & Q1 /\ Q29 <> {} ; k be Nat ; q " is Element of X & q " is Element of X ; F . t is set of \leq M ; assume that n <> 0 and n <> 1 ; set en = EmptyBag n , en = EmptyBag n ; let b be Element of Bags n , x be Element of Bags n ; assume for i holds b . i is commutative ; x is root & p `2 is root implies x is root not r in ]. p , q .[ ; let R be FinSequence of REAL , x be Element of REAL ; SS does not destroy b1 & not I does not destroy b1 ; IC SCM R <> a & IC SCM R <> a ; |. - ( |[ x , y ]| ) .| >= r ; 1 * seq = seq & 1 * seq = seq ; let x be FinSequence of NAT , n be Nat ; let f be Function of C , D , g be Function of C , D ; for a holds 0. L + a = a IC s = s . NAT & IC s = s . NAT ; H + G = F- ( GG ) ; CS1 . x = x2 & CS2 . x = y2 ; f1 = f .= f2 .= ( f | X ) . x ; Sum <* p . 0 *> = p . 0 ; assume v + W = v + u + W ; { a1 } = { a2 } & { a2 } = { a2 } ; a1 , b1 _|_ b , a & b1 , b2 _|_ a , a ; d3 , o _|_ o , a3 & d1 , o _|_ a , b ; IO is reflexive & IO is transitive ; IO is antisymmetric implies [: O , O :] is antisymmetric sup rng H1 = e & sup rng H1 = e ; x = ( a * ( - 1 ) ) * ( a * ( - 1 ) ) ; |. p1 .| ^2 >= 1 ^2 ; assume that j2 -' 1 < 1 and j2 + 1 < width G ; rng s c= dom f1 /\ dom f2 & rng s c= dom f1 ; assume that support a misses support b and support b c= support b ; let L be associative non empty doubleLoopStr , x be Element of L ; s " + 0 < n + 1 ; p . c = ( f " ) . 1 ; R . n <= R . ( n + 1 ) ; Directed ( I1 , card I + 1 ) = I1 ; 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 NAT -defined Function ; let X be non empty directed Subset of S ; let S be non empty full SubRelStr of L ; cluster <* [ ] , 0 ] *> -> complete non trivial ; ( 1 - a ) " = a " ; ( q . {} ) `1 = o ; ( - i ) - 1 > 0 ; assume ( 1 - 2 ) * ( 1 - 2 ) <= 1 ; card B = k + - 1 ; x in union rng ( f | X ) ; assume x in the carrier of R & y in the carrier of R ; d in dom f ; f . 1 = L . ( F . 1 ) ; the vertices of G = { v } & not v in V ; let G be let G be let finite wfinite _Graph ; e , v6 be set , v be set ; c . ( i - 1 ) in rng c ; f2 /* q is divergent_to+infty & f2 /* q is divergent_to+infty ; set z1 = - z2 , z2 = - z1 , z2 = - z2 , z1 = - z2 , z2 = - z1 , z2 = - z2 ; assume w is_llst S , G ; set f = p |-count ( t - p ) , g = p |-count ( t - p ) ; let c be Object of C ; assume ex a st P [ a ] ; let x be Element of REAL m , y be Element of REAL m ; let IF be Subset-Family of X , B be Subset of Y ; reconsider p = p as Element of NAT ; let v , w be Point of X ; let s be State of SCM+FSA , P be Subset of SCM+FSA ; p is FinSequence of ( the carrier of SCM+FSA ) \/ { IC SCM+FSA } ; stop I ( ) c= P-12 & stop I ( ) c= P-12 set ci = f^ /. i , cj = fj ; w ^ t ^ t ^ s ^ t ^ s ^ t ^ t ^ s ^ t ^ t ^ s ^ t ^ t ^ t ^ t ^ s ^ t ^ t ^ t ^ t ^ W1 /\ W = W1 /\ W ` & W1 /\ W2 = W2 /\ W ` ; f . j is Element of J . j ; let x , y be \mathopen of T2 , a be Element of T2 ; ex d st a , b // b , d ; a <> 0 & b <> 0 & c <> 0 ord x = 1 & x is positive implies x is positive set g2 = lim ( seq ^\ k ) , g1 = 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 & L1 . F-21 = 0 ; thesis ; ( sin . x ) ^2 <> 0 & ( sin . x ) ^2 <> 0 ; ( #Z 2 ) . x > 0 & ( #Z 2 ) . x > 0 ; o1 in ( X /\ O2 ) /\ ( X /\ O2 ) ; e , v6 be set , v be set ; r3 > ( 1 - 2 ) * 0 ; x in P .: ( F -ideal of L ) ; let J be closed non empty Subset of R ; h . p1 = f2 . O & h . p2 = g2 . O ; Index ( p , f ) + 1 <= j ; len ( q | i ) = width M & width ( q | i ) = width M ; the carrier of CK c= A & the carrier of CK c= A ; dom f c= union rng ( F | X ) ; k + 1 in support ( thesis ( n ) --> x ) ; let X be ManySortedSet of the carrier of S ; [ x `1 , y `2 ] in ( an \/ 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 . x1 = g . x2 ; F c= 2 -tuples_on the carrier of X & F c= 2 -tuples_on X reconsider w = |. s1 .| as Real_Sequence ; ( 1 - m ) * m + r < p ; dom f = dom ( I . i ) & dom ( I . i ) = I . i ; [#] P-17 = [#] ( ( TOP-REAL 2 ) | K1 ) ; cluster - x -> ExtReal -> ExtReal for ExtReal ; then { d } c= A & A is closed ; cluster ( TOP-REAL n ) | A -> finite-ind for non empty Subset of TOP-REAL n ; let w1 be Element of M , w2 be Element of M ; let x be Element of dyadic ( n ) ; u in W1 & v in W3 implies u + v in W2 reconsider y = y as Element of L2 ; N is full SubRelStr of T |^ ( the carrier of S ) ; sup { x , y } = c "\/" c ; g . n = n to_power 1 .= n ; h . J = EqClass ( u , J ) ; let seq be summable summable sequence of X , x be Point of X ; dist ( x `1 , 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 . x ) in { x } ; cluster subcondensed -> subcondensed for Subset of T ; let P be compact non empty Subset of TOP-REAL 2 , p1 , p2 , q1 , q2 be Point of TOP-REAL 2 ; Gij in LSeg ( PI , 1 ) /\ LSeg ( Gik , Gij ) ; n be Element of NAT , x be Element of NAT ; reconsider SS = S , SS = T as Subset of T ; dom ( i .--> X `1 ) = { 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 as Element of C ( ) ; let s be 0 -started State of SCMPDS , a be Element of SCMPDS ; let t be 0 -started State of SCMPDS , Q ; b , b , x , y is_collinear ; assume that i = n \/ { n } and j = k \/ { k } ; let f be PartFunc of X , Y ; N1 >= ( sqrt ( c ^2 + sqrt ( d ^2 ) ) / ( 2 * d ) ) ; reconsider t9 = T" as TopSpace , TT = T as TopSpace ; set q = h * p ^ <* d *> ; z2 in U . ( y2 ) /\ Q2 & z2 in Q . ( y2 ) /\ Q2 ; A |^ 0 = { <%> E } & A |^ 0 = { <%> E } ; len W2 = len W + 2 & len W = len W + 2 ; len h2 in dom h2 & len h2 in dom h2 ; i + 1 in Seg len s2 & i + 1 in Seg len s2 ; z in dom g1 /\ dom f & z in dom f1 /\ dom f2 ; assume that p2 = E-max ( K ) and p1 `2 = p2 `2 ; len G + 1 <= i1 + 1 ; f1 (#) f2 is convergent & lim ( f1 (#) f2 ) = x0 ; cluster seq + s-10 -> summable for Real_Sequence ; assume that j in dom M1 and i <= j ; let A , B , C be Subset of X ; x , y , z be Point of X , p be Point of X ; b ^2 - ( 4 * a * c ) >= 0 ; <* xC /y *> ^ <* y *> << x ; a , b in { a , b } ; len p2 is Element of NAT & len p2 = len p1 ; ex x being element st x in dom R & y = R . x ; len q = len ( K (#) G ) .= len G ; s1 = Initialize ( Initialized s ) , P1 = P +* I ; consider w being Nat such that q = z + w ; x ` ` is Element of x & y ` is Element of L ; k = 0 & n <> k or k > n ; then X is discrete for X is closed ; for x st x in L holds x is FinSequence ; ||. f /. c .|| <= r1 & ||. f /. c .|| <= 1 ; c in uparrow p & not c in { p } ; reconsider V = V as Subset of the topology of X ; N , M be being being being being being being being being being being being being being being being being being being being being being being being being being being Element of L holds N , M |= N then z is_>=_than waybelow x & z is_>=_than compactbelow y ; M \lbrack f , g .] = f & M \lbrack g , f .] = g ; ( ( L to_power 1 ) ) /. 1 = TRUE ; dom g = dom f /\ X & dom g = X ; mode : of G is ^ of W , G ; [ 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 in dom Subformulae p ; F1 . ( a1 , - 1 ) = G1 & F1 . ( - 1 ) = G2 ; redefine func being set equals Ball ( a , b , r ) ; let a , b , c , d be Real ; rng s c= dom ( 1 - r ) & rng s c= dom ( 1 - r ) ; curry ( F-19 , k ) is additive & curry ( F-19 , k ) is additive ; set k2 = card dom B , k1 = card dom C , k2 = card dom D ; set G = DTConMSA ( X ) ; reconsider a = [ x , s ] as the \mathopen of G ; let a , b be Element of ML , x be Element of ML ; reconsider s1 = s , s2 = t as Element of ( the carrier of S ) ; rng p c= the carrier of L & p . ( len p ) = x ; let d be Subset of the bound of A ; ( x .|. x ) = 0 iff x = 0. W I-21 in dom stop I & Ik in dom stop I ; let g be continuous Function of X | B , Y ; reconsider D = Y as Subset of ( TOP-REAL n ) | D ; reconsider i0 = len p1 , i2 = len p2 as Integer ; dom f = the carrier of S & f . x = f . x ; rng h c= union ( ( Carrier J ) . i ) ; cluster All ( x , H ) -> carrier of and All ( x , H ) -> carrier ; d * N1 ^2 > N1 * 1 / ( d * 1 ) ; ]. a , b .[ c= [. a , b .] ; set g = f " D1 , h = f " D2 , f = g " D2 ; dom ( p | ( mm1 ) ) = mm1 & dom ( p | ( mm1 ) ) = dom p ; 3 + - 2 <= k + - 2 & k + - 2 <= k + - 2 ; tan is_differentiable_in ( arccot - arccot ) . x & tan . x > 0 ; x in rng ( f /^ ( p -' 1 ) ) ; let f , g be FinSequence of D ; cp in the carrier of S1 & cp in the carrier of S2 ; rng f " { x } = dom f & rng f = dom g ; ( the Source of G ) . e = v & ( the Source of G ) . e = v ; width G - 1 < width G - 1 ; assume that v in rng ( S | E1 ) and u in rng ( S | E1 ) ; assume x is root or x is root or x is root ; assume that 0 in rng ( g2 | A ) and 0 < r ; let q be Point of TOP-REAL 2 , r be Real ; let p be Point of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ; dist ( O , u ) <= |. p2 .| + 1 ; assume dist ( x , b ) < dist ( a , b ) ; <* SS *> is_the carrier of C-20 & <* D *> is_\! ] ; i <= len ( G * ( i1 -' 1 , k ) ) ; let p be Point of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ; x1 in the carrier of [: I[01] , I[01] :] & x2 in the carrier of I[01] ; set p1 = f /. i , p2 = f /. ( i + 1 ) ; g in { g2 : r < g2 & g2 < x0 + r } ; Q2 = Sthesis " ( Q /\ R ) & Q2 = Sy ; ( ( 1 / 2 ) (#) ( 1 / 2 ) ) is summable ; - p + I c= - p + A & - p + I c= A ; n < LifeSpan ( P1 , s1 ) + 1 & I . ( n + 1 ) = P1 . ( n + 1 ) ; CurInstr ( p1 , s1 ) = i & CurInstr ( p1 , s1 ) = i ; A /\ ( Cl { x } ) <> {} ; rng f c= ]. r , r + 1 .[ ; let g be Function of S , V ; let f be Function of L1 , L2 , x be Element of L1 ; reconsider z = z as Element of CompactSublatt L , x be Element of L ; let f be Function of S , T ; reconsider g = g as Morphism of c opp , b opp ; [ s , I ] in [: S , T :] ; len ( the connectives of C ) = 4 & len ( the connectives of C ) = 4 ; let C1 , C2 be subfunctor of C , a be Element of C1 ; reconsider V1 = V , V2 = V as Subset of X | B ; attr p is valid means : Def1 : All ( x , p ) is valid ; assume that X c= dom f and f .: X c= dom g and g .: X c= dom f ; H |^ a " |^ a is Subgroup of H & H |^ a = H |^ a ; let A1 be ( a , b ) on E1 & A1 c= A2 ; p2 , r3 , q2 is_collinear & p1 , p2 , q1 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 ) | B11 ) ; 0 . 0 < M . E8 & M . E8 < M . E8 ; ^ ( c , c ) @ = c ; consider c being element such that [ a , c ] in G ; a1 in dom ( F . s2 ) & a2 in dom ( F . s2 ) ; cluster -> 0. for > the 0. the lattice of L is ) ; set i1 = the Nat , i2 = the Element of NAT , i1 = the Element of NAT , i2 = the Element of NAT ; let s be 0 -started State of SCM+FSA , P be Subset of SCM+FSA ; assume y in ( f1 \/ f2 ) .: A & x in f1 .: A ; f . ( len f ) = f /. ( len f ) ; x , f . x '||' f . x , f . y ; pred X c= Y means : Def1 : cos ( X ) c= cos ( Y ) ; let y be upper Subset of Y , x be Element of X ; cluster -> -> -> -> -> -> -> \mathbb that for Element of A ; set S = <* Bags n , i *> , T = <* i *> , S = <* i *> , T = <* i *> , S = <* i *> , T = <* i *> , T = <* i *> , S = <* i set T = [. 0 , 1 / 2 .] , S = [. 0 , 1 .] ; 1 in dom mid ( f , 1 , 1 ) ; ( 4 * PI ) / 2 < ( 2 * PI ) / 2 ; x2 in dom f1 /\ dom f & x1 in dom f1 /\ dom f2 ; O c= dom I & { {} } = { {} } ; ( the Source of G ) . x = v & ( the Source of G ) . x = v ; { HT ( f , T ) } c= Support f ; reconsider h = R . k as Polynomial of n , L ; ex b being Element of G st y = b * H ; let x , y , z be Element of G opp ; h19 . i = f . ( h . i ) ; p `1 = p1 `1 & p `2 = p2 `2 or p `1 = p2 `2 & p `2 = p2 `2 ; i + 1 <= len Cage ( C , n ) ; len <* P *> @ = len P & width <* P *> = 1 ; set N-26 = the \subseteq of N , Nw = the Element of N ; len gLet + ( x + 1 ) - 1 <= x ; a on B & b on B implies 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 ) = len ex n st f /. n = p /. ( n + 1 ) set q2 = N-min L~ Cage ( C , n ) , q1 = W-min L~ Cage ( C , n ) , q2 = W-min L~ Cage ( C , n ) ; set S = \leq ( S1 , S2 ) , T = ( S2 , S1 ) , E = ( S1 , S2 ) ; MaxADSet ( b ) c= MaxADSet ( P /\ Q ) ; Cl ( G . q1 ) c= F . r2 & Cl ( G . q2 ) c= G . q2 ; f " D meets h " V & f " D meets h " V ; reconsider D = E as non empty directed Subset of L1 , L be Subset of L2 ; H = ( the_left_argument_of H ) '&' ( the_right_argument_of H ) & H = ( the_right_argument_of H ) '&' ( the_right_argument_of H ) ; assume that t is Element of ( F . s ) . ( X . s ) ; rng f c= the carrier of S2 & f . x = f . x ; 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 } .= { E } ; reconsider m = len ( thesis - k ) as Element of NAT ; set S1 = LSeg ( n , UMP C ) , S2 = LSeg ( n , UMP C ) ; [ i , j ] in Indices M1 & [ i , j ] in Indices M1 ; assume that P c= Seg m and M is \HM { i } ; 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 .= L . p ; p-7 . i = pp1 . i & pp2 . i = pp2 . i ; let PA , PA , G be a_partition of Y , a be Element of Y ; pred 0 < r & r < 1 implies 1 / r < 1 / r ; rng ( - ( a , X ) ) = [#] X ; reconsider x = x , y = y as Element of K ; consider k such that z = f . k and n <= k ; consider x being element such that x in X \ { p } ; len ( canFS ( s ) ) = card ( s ) .= card ( s ) ; reconsider x2 = x1 , y2 = x2 as Element of L2 ; Q in FinMeetCl ( ( the topology of X ) \/ { x } ) ; dom ( f . 0 ) c= dom ( u | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( pred n divides m & m divides n implies n = m ; reconsider x = x as Point of [: I[01] , I[01] :] ; a in ' not y0 in the still of f & not y in the carrier of f ; Hom ( ( a ~ ) , c ) <> {} ; consider k1 such that p " < k1 and k1 < p " ; consider c , d such that dom f = c \ d and f . c = d ; [ x , y ] in [: dom g , dom k :] ; set S1 = Let ( x , y , z ) ; l2 = m2 & l1 = i2 & l2 = j2 implies l1 = i2 x0 in dom ( u01 /\ A9 ) & x0 in dom ( u | A9 ) ; reconsider p = x , q = y as Point of TOP-REAL 2 ; I[01] = ( R^1 ) | B01 & ( TOP-REAL 2 ) | B01 = ( TOP-REAL 2 ) | B01 ; f . p4 <= f . ( f . p1 ) & f . p2 <= f . p1 ; ( Fx ) `1 <= ( x `1 ) / ( 1 + ( x `2 / ( 1 + x `1 ) ) ) `1 ; x `2 = ( W7 ) `2 & ( W8 ) `2 = ( W8 ) `2 ; for n being Element of NAT holds P [ n ] implies P [ n + 1 ] let J , K be non empty Subset-Family of I ; assume 1 <= i & i <= len <* a " *> ; 0 |-> a = <*> ( the carrier of K ) & a = <*> ( the carrier of K ) ; X . i in 2 -tuples_on A . i \ B . i ; <* 0 *> in dom ( e --> [ 1 , 0 ] ) ; then P [ a ] & P [ succ a ] & P [ succ a ] ; reconsider sbeing being N , sst t = s\subseteq t , sA = t as ' of D ; ( - i - 1 ) <= len ( - j ) ; [#] S c= [#] T & [#] T c= [#] T implies [#] T c= [#] T for V being strict RealUnitarySpace holds V in and V c= the carrier of V assume k in dom mid ( f , i , j ) ; let P be non empty Subset of TOP-REAL 2 , p1 , p2 , q1 , q2 be Point of TOP-REAL 2 ; let A , B be square Matrix of n1 , K , n , m be Nat ; - a * - b = a * b & - a * b = b ; for A being Subset of A9 holds A // A & A // A ( for o2 being Element of o2 holds o2 in <^ o2 , o2 ^> iff o2 in <^ o2 , o2 ^> ) then ||. x .|| = 0 & x = 0. X ; let N1 , N2 be strict normal Subgroup of G , x be Element of G ; j >= len upper_volume ( g , D1 ) & j <= len upper_volume ( g , D2 ) ; b = Q . ( len Q - 1 + 1 ) ; f2 * f1 /* s is divergent_to+infty & f2 * f1 /* s is divergent_to+infty ; reconsider h = f * g as Function of [: N1 , N2 :] , G ; assume that a <> 0 and Let a , b , c ; [ t , t ] in the InternalRel of A & [ t , t ] in the InternalRel of A ; ( v |-- E ) | n is Element of T7 & ( v |-- E ) | n = v ; {} = the carrier of L1 + L2 & the carrier of L1 + L2 = the carrier of L1 + L2 ; Directed I is_closed_on Initialized s , P & Directed I is_halting_on Initialized s , P ; Initialized p = Initialize ( p +* q ) , p = p +* q , q = p +* q ; reconsider N2 = N1 , N2 = N2 as strict net of R2 ; reconsider Y = Y as Element of [: Ids L , Ids L :] ; "/\" ( 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 + Q ) & n <= len ( P ^ Q ) ; x1 `1 = x2 & y1 `1 = x2 & x1 `2 = y2 ; InputVertices S = { x1 , x2 } & InputVertices S = { x1 , x2 } ; let x , y be Element of FT1 ( n ) ; p = |[ p `1 , p `2 ]| & p = |[ p `1 , p `2 ]| ; g * 1_ G = h " * g * h .= h " * g ; let p , q be Element of being Element of being is Element of being Element of being set ; 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 s <= 1 rng s c= dom ( f2 * f1 ) /\ dom ( f1 * f2 ) ; synonym \mathop { \rm } X for X \/ { x } ; 1. K * 1. K = 1_ K & 1. K * 1. K = 1_ K ; set S = Segm ( A , P1 , Q1 ) , T = Segm ( A , P1 , Q1 ) ; ex w st e = ( w - f ) & w in F ; curry ( P+* ( x , k ) ) # x is convergent ; cluster open -> open for Subset of T\sigma ( T ) ; len f1 = 1 .= len f3 .= len f3 + 1 .= len f3 + 1 ; ( i * p ) / p < ( 2 * p ) / p ; let x , y be Element of OSSub U0 , a be Element of S ; b1 , c1 // b9 , c9 & b1 , c1 // b9 , c ; consider p being element such that c1 . j = { p } ; assume that f " { 0 } = {} and f is total ; assume that IC Comput ( F , s , k ) = n and IC Comput ( F , s , k ) = k ; Reloc ( J , card I + 3 ) does not destroy a ; ( goto ( card I + 1 ) ) not a in dom ( a .--> ( card I + 1 ) ) ; set m3 = LifeSpan ( p3 , s3 ) , P4 = LifeSpan ( p3 , s3 ) ; IC SCMPDS in dom ( Initialize p ) & IC SCMPDS in dom p ; dom t = the carrier of SCM & dom t = the carrier of SCM & t . a = s . a ; ( ( N-min L~ f ) .. f ) .. f = 1 & ( ( E-max L~ f ) .. f ) .. f = ( ( E-max L~ f ) .. f ) .. f ; let a , b be Element of being Element of being Element of being Element of being Element of being set ; Cl ( union ( Int F ) ) c= Cl ( Int ( union F ) ) ; the carrier of X1 union X2 misses ( ( X1 union X2 ) \/ ( X2 union X1 ) ) ; assume not LIN a , f . a , g . a , f . a ; consider i being Element of M such that i = d6 and i in X ; then Y c= { x } or Y = { x } ; M , v / ( y , x ) / ( y , x ) / ( y , x ) |= H ; consider m being element such that m in Intersect ( FF . m ) and x = [ m , m ] ; reconsider A1 = support u1 , A2 = support ( v1 - v2 ) 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 . ( s , X ) -> ( S , X ) -valued for string of S ; LG2 /. n2 = LG2 . n2 & LG2 /. n1 = LG2 . n1 ; let P be compact non empty Subset of TOP-REAL 2 , p1 , p2 , q1 , q2 be Point of TOP-REAL 2 ; assume that r-7 in LSeg ( p1 , p2 ) and rg2 in LSeg ( p1 , p2 ) ; let A be non empty compact Subset of TOP-REAL n , x be Point of TOP-REAL n ; assume that [ k , m ] in Indices DD1 and [ k , m ] in Indices DD1 ; 0 <= ( ( 1 / 2 ) |^ p ) / ( 2 |^ p ) ; ( F . N ) | E8 . x = +infty ; pred X c= Y & Z c= V implies X \ V c= Y \ Z ; ( y `2 ) * ( z `2 ) * ( - 1 ) * ( 1 + ( y `2 / z `1 ) ) ) <> 0. I ; 1 + card X-18 <= card u & card Xk <= card Xk ; set g = z :- ( E-max L~ z ) , M = z .. z , N = .. z , S = L~ z , N = .. z , S = L~ z , N = .. z , S = L~ z , N = L~ z , S = L~ z then k = 1 & p . k = <* x , y *> . k ; cluster total for Element of C -\mathopen the carrier of X , the carrier of Y ; reconsider B = A as non empty Subset of TOP-REAL n , C = { p } ; let a , b , c be Function of Y , BOOLEAN , x be Element of Y ; L1 . i = ( i .--> g ) . i .= g . i ; Plane ( x1 , x2 , x3 , x4 ) c= P & Plane ( x1 , x2 , x3 , x4 ) c= P ; n <= indx ( D2 , D1 , j1 ) & n <= len D2 ; ( ( ( g2 ) . O ) `1 ) `1 = - 1 & ( ( g2 ) . O ) `2 = 1 ; j + p .. f -' len f <= len f - 1 + p .. f ; set W = W-bound C , S = S-bound C , E = E-bound C , N = E-bound C , N = E-bound C , S = S-bound C , N = E-bound C , S = E-bound C , N = E-bound C , N = E-bound C , S = E-bound S1 . ( a `1 , e ) = a + e .= a `1 ; 1 in Seg width ( M * ColVec2Mx ( p ) ) ; dom ( i (#) Im ( f ) ) = dom Im ( f ) ; ^2 . ( x `1 ) = W . ( a , *' ( a , p ) ) ; set Q = ( \rm \rm \rm \rm \rm <* ( g , f , h ) , g , h ) ; cluster MSbeing ManySortedSet of U1 -> MS[ for ManySortedSet of U1 , ( the Sorts of U1 ) . s ; attr F = { A } means : Def1 : F is discrete ; reconsider z9 = \hbox { z } , z9 = z as Element of product G ; rng f c= rng f1 \/ rng f2 & f . 1 = f1 . ( len f1 + 1 ) ; 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 ) ; E , j |= All ( x1 , x2 , H ) implies E , j |= H reconsider n1 = n , n2 = m , n1 = n as Morphism of o1 , o2 ; assume that P is idempotent and R is idempotent and P ** R = R ** P ; card ( B2 \/ { x } ) = k-1 + 1 ; card ( ( x \ B1 ) /\ B1 ) = 0 implies card ( x \ B1 ) = 0 ; g + R in { s : g-r < s & s < g + r } ; set qrng = ( q , <* s *> ) -\subseteq ( q , <* s *> ) -\subseteq ( q , <* s *> ) -\subseteq ( q , <* s *> ) -\subseteq ( q , <* s *> ) -\subseteq ( q , <* s for x being element st x in X holds x in rng f1 implies x in X h0 /. ( i + 1 ) = h0 . ( i + 1 ) ; set mw = max ( B , dom ( R | NAT ) ) ; t in Seg width ( ( I ^ ( n , n ) ) @ ) ; reconsider X = dom f /\ C as Element of Fin ( NAT , the carrier of NAT ) ; IncAddr ( i , k ) = <% ( a , b ) . ( k + 1 ) %> ; S-bound L~ f <= q `2 & q `2 <= q `2 & q `2 <= q `2 ; attr R is condensed means : Def1 : Int R is condensed & Cl R is condensed ; pred 0 <= a & a <= 1 & b <= 1 implies a * b <= 1 ; u in ( ( c /\ ( ( d /\ b ) /\ e ) ) /\ f ) /\ j ; u in ( ( c /\ ( ( d /\ e ) /\ b ) ) /\ f ) /\ j ; len C + - 2 >= 9 + - 3 & len C + - 2 >= 0 ; x , z , y is_collinear & x , z , y is_collinear ; a |^ ( n1 + 1 ) = a |^ n1 * a |^ n1 ; <* \underbrace ( 0 , \dots , 0 ) *> in Line ( x , a * x ) ; set ya1 = <* y , c *> , ya1 = <* c , x *> ; FF2 /. 1 in rng Line ( D , 1 ) & FF2 /. 1 in rng Line ( 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 } ; f1 /* ( seq ^\ k ) is divergent_to+infty & f1 /* ( seq ^\ k ) is divergent_to+infty ; reconsider u2 = u , v2 = v as VECTOR of P`1 , u1 = v as VECTOR of X ; p |-count ( Product Sgm ( X11 ) ) = 0 & p |-count ( Product Sgm ( 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 0 , ii2 = goto 0 ; x in { x , y } & h . x = {} ( Tx , y ) ; consider y being Element of F such that y in B and y <= x `1 ; len S = len ( the charact of A2 ) & len ( the charact of A1 ) = len A1 ; reconsider m = M , i = I , n = N as Element of X ; A . ( j + 1 ) = B . ( j + 1 ) \/ A . j ; set N8 = : ( G-15 ) . e = ( G-15 ) . e ; rng F c= the carrier of gr { a } & F . a = { a } ; P is , and P . ( K , n , r ) is a ^2 ; f . k , f . ( mod n ) in rng f & ( f . k ) mod n in rng f ; h " P /\ [#] T1 = f " P /\ [#] T1 .= f " P /\ [#] T1 ; g in dom f2 \ f2 " { 0 } & ( f1 - f2 ) . g = 0 ; gX /\ dom f1 = g1 " { x } & X /\ dom f2 = { x } ; consider n being element such that n in NAT and Z = G . n ; set d1 = \bf dist ( x1 , y1 ) , d2 = dist ( x2 , y2 ) , d1 = dist ( x2 , y2 ) ; b `1 + ( 1 / 2 ) < ( 1 / 2 ) + ( 1 / 2 ) ; reconsider f1 = f , g1 = g as VECTOR of the carrier of X ; attr i <> 0 means : Def1 : i ^2 mod ( i + 1 ) = 1 ; j2 in Seg len ( g2 . i2 ) & 1 <= j2 & j2 <= len g2 ; dom ( i * ( - 1 ) ) = dom ( i * ( - 1 ) ) .= dom ( - i ) ; cluster sec | ]. PI / 2 , PI / 2 .[ -> one-to-one ; Ball ( u , e ) = Ball ( f . p , e ) ; reconsider x1 = x0 , y1 = x1 as Function of S , IU ; 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 RL ; S1 +* S2 = S2 +* S1 & S2 +* S2 = S1 +* S2 +* S2 & S1 +* S2 = S2 +* S2 +* S2 +* S2 +* S2 +* S2 ; ( ( #Z 2 ) * ( cos - sin ) ) is_differentiable_on Z & ( ( #Z 2 ) * ( cos - sin ) ) `| Z = f ; cluster [. 0 , 1 .] -> [. 0 , 1 .] -valued ; set C7 = 1GateCircStr ( <* z , x *> , f3 ) , C8 = 1GateCircStr ( <* x , y *> , f3 ) ; Ea1 . e2 = ( ( e . e2 ) -T ) . ( e2 . e1 ) ; ( ( arctan * ( f1 + f2 ) ) `| Z ) = ( ( arctan * ( f1 + f2 ) ) `| Z ) ; upper_bound A = ( PI * 3 / 2 ) * 2 & lower_bound A = 0 ; F . ( dom f , - F . ( cod f ) ) = F . ( cod f , - F . ( cod f ) ) ; reconsider pbeing = qbeing Point of TOP-REAL 2 , p8 = ( - 1 ) * ( ( - 1 ) * ( 1 - r ) ) as Point of TOP-REAL 2 ; g . W in [#] Y0 & [#] Y0 c= [#] Y & g . W in [#] Y ; 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 /\ ]. -infty , x0 + r .[ & rng s c= dom f /\ ]. x0 , x0 + r .[ ; assume x in { idseq ( 2 ) , Rev ( idseq 2 ) } ; reconsider n2 = n , m2 = m , n1 = n as Element of NAT ; for y being ExtReal st y in rng seq holds g <= y ; 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 ( a ) c= R -Seg ( b ) ; t in ]. r , s .[ or t = s or t = s ; z + v2 in W & x = u + ( z + v2 ) ; x2 |-- y2 iff P [ x2 , y2 ] & P [ x2 , y2 ] ; pred x1 <> x2 means : Def1 : |. x1 - x2 .| > 0 & |. x1 - x2 .| > 0 ; assume that p2 - p1 , p3 - p1 , p2 - p1 is_collinear and p2 - p1 , p3 - p1 , p3 - p1 is_collinear ; set q = non \cap <* 'not' 'not' A *> , r = 'not' 'not' 'not' 'not' 'not' A ; let f be PartFunc of REAL-NS 1 , REAL-NS n , x be Point of REAL-NS 1 , r be Real ; ( n mod ( 2 * k ) ) + 1 = n mod k ; dom ( T * ( succ t ) ) = dom ( \mathop { 0 } , 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 . x3 & ( F * G ) . x3 = v . x3 ; assume that the carrier of D1 c= the carrier of D2 and for x being Element of D1 holds x in the carrier of D1 ; 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 `1 , s = q `1 , w = q `1 , e = q `1 , s = q `1 , w = q `2 , s = q `2 , w = q `2 , e = q `1 , w n1 -' len f + 1 <= len ( - 1 ) + 1 - 1 + 1 ; EqClass ( q , O1 ) = [ u , v , a `1 , b `2 ] ; set C-2 = ( ( n , n ) .--> ( G . ( k + 1 ) ) ) ; Sum ( L (#) p ) = 0. R * Sum p .= 0. V * p ; consider i being element such that i in dom p and t = p . i ; defpred Q [ Nat ] means 0 = Q ( $1 ) & ( $1 <= m implies $2 = 1 ) ; set s3 = Comput ( P1 , s1 , k ) , P3 = P1 , s4 = P1 , P4 = P1 , P4 = P1 , P4 = P1 , P4 = P2 , P4 = P2 , P4 = P2 , P4 = Comput ( P1 , s1 , k ) , P4 = P1 , P4 = P1 , let l be -> -> -> and of k , Al , P be Subset of l ; reconsider U2 = union ( G-24 ) , U2 = union ( rng ( G | k ) ) as Subset-Family of TL ; consider r such that r > 0 and Ball ( p `1 , r ) c= Q ` ; ( h | ( n + 2 ) ) /. ( i + 1 ) = p29 ; reconsider B = the carrier of X1 , C = the carrier of X2 as Subset of X ; p\in = <* - vs , 1 , 1 *> & p9 = <* - vs , 1 *> ; synonym f is real-valued means : Def1 : rng f c= NAT & for x st x in NAT holds f . x = x ; consider b being element such that b in dom F and a = F . b ; x9 < card X0 + card ( Y0 \/ ( Y0 \/ ( Y0 \/ ( Y1 \/ Y2 ) ) ) ) & x9 < card ( X0 \/ ( Y1 \/ Y2 \/ ( Y1 \/ Y2 ) ) ) ; attr X c= B1 means : Def1 ooo) : X c= succ B1 & X c= B ; then w in Ball ( x , r ) & dist ( x , w ) <= r ; angle ( x , y , z ) = angle ( x-y , 0 , z ) ; pred 1 <= len s means : Def1 : for x being Element of NAT holds ( the mapping of s ) . x = s . x ; fJ c= f . ( k + ( n + 1 ) ) ; the carrier of { 1_ G } = { 1_ G } & the carrier of G = { 1_ G } ; pred p '&' q in \cdot ( carrier \ { p } ) & q '&' p in TAUT ( A ) ; - ( t `1 ) / ( 1 + ( t `2 / t `1 ) ^2 ) < ( t `1 ) / ( 1 + ( t `2 / t `1 ) ^2 ) ; U2 . 1 = U2 /. 1 .= ( U2 /. 1 ) * ( U2 /. 1 ) .= ( U2 /. 1 ) * ( U2 /. 1 ) ; f .: ( the carrier of x ) = the carrier of x & f .: ( the carrier of x ) = { x } ; Indices ( O @ ) = [: Seg n , Seg n :] & dom ( ( n , n ) --> ( n , 1 ) ) = Seg n ; for n being Element of NAT holds G . n c= G . ( n + 1 ) ; then V in M @ \square ex x being Element of M st V = { x } ; ex f being Element of F-9 st f is \cup ( A \/ B ) & f is is \setminus & f is is is is finite ; [ 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 - |[ w , v1 ]| `1 <> 0. TOP-REAL 2 & |[ w1 , v1 ]| `1 - |[ w , v1 ]| `2 = v1 ; reconsider t = t as Element of ( INT * ) * ; C \/ P c= [#] ( GX | ( [#] GX \ A ) ) ; f " V in ( the topology of X ) /\ D & f " V in D /\ ( the topology of X ) ; x in [#] ( ( the carrier of A ) /\ A ) & x in [#] ( ( the carrier of A ) /\ A ) ; g . x <= h1 . x & h . x <= h1 . x ; InputVertices S = { xy , y , z } & InputVertices S = { xy , y , z } ; for n being Nat st P [ n ] holds P [ n + 1 ] ; set R = ( Line ( M , i ) , a * Line ( M , i ) ) ; assume that M1 is being_line and M2 is being_line and 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 ) & width B2 = len ( F1 ^ F2 ) + len ( F2 ^ F2 ) ; len ( ( the H of n ) * ( i , j ) ) = n & len ( ( - 1 ) * ( i , j ) ) = n ; dom max ( - ( f + g ) , f + g ) = dom ( f + g ) ; ( the Sorts of seq ) . n = upper_bound Y1 & ( for n holds seq . n = ( seq . n ) + ( seq . n ) ) ; dom ( p1 ^ p2 ) = dom ( f12 ^ <* x *> ) .= dom ( f12 ^ <* x *> ) ; M . [ 1 / y , y ] = 1 / ( 1 / y ) * v1 .= y ; assume that W is non trivial and W .vertices() c= the carrier' of G2 and not e in the carrier' of G2 ; C6 * ( i1 , i2 ) `1 = G1 * ( i1 , i2 ) `1 .= G1 * ( i1 , i2 ) `1 ; C8 |- 'not' Ex ( x , p ) 'or' p . ( x , y ) ; for b st b in rng g holds lower_bound rng fbeing Element of REAL st b <= b & b <= a - ( ( q1 `1 / |. q1 .| - cn ) / ( 1 + cn ) ) = 1 ; ( LSeg ( c , m ) \/ [: l , k :] ) \/ [: { l } , k :] c= R ; consider p being element such that p in Ball ( x , r ) and p in L~ f ; Indices ( X @ ) = [: Seg n , Seg 1 :] & dom ( X @ ) = Seg 1 ; cluster s => ( q => p ) => ( q => ( s => p ) ) -> valid ; Im ( ( Partial_Sums F ) . m ) is Element of S & Im ( ( Partial_Sums F ) . m ) is Element of S ; 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 , \hbox { u : not contradiction } & p in LSeg ( NW-corner Z , p ) } ; set R8 = R / ( 1 - b ) , R8 = ( 1 - b ) / ( 1 - b ) ; IncAddr ( I , k ) = AddTo ( da , db ) .= goto ( card I + k ) ; seq . m <= ( the Element of ( seq ^\ k ) ) . ( n + k ) ; a + b = ( a ` ) *' ( b ` ) ` .= ( a ` ) *' ( b ` ) ; id ( X /\ Y ) = id ( X /\ Y ) .= id ( X /\ Y ) ; for x being element st x in dom h holds h . x = f . x ; reconsider H = U2 \/ U2 , U2 = U2 \/ U1 , U1 = 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 such that card A = len R and card A = card A ; p2 in rng ( f |-- p1 ) \ rng <* p1 *> & p2 in rng ( f |-- p1 ) ; len s1 - 1 > 0 & len s2 - 1 > 0 implies len s1 - 1 > 0 ( N-min P ) `2 = N-bound ( P ) & ( E-max P ) `2 = N-bound ( P ) ; Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) & Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) ; f . a1 ` ` = f . a1 ` & f . a2 = ( f . a1 ) ` ; ( seq ^\ k ) . n in ]. -infty , x0 + r .[ & ( seq ^\ k ) . n in dom ( seq ^\ k ) ; gg . s0 = g . s0 | G . s0 .= g . s0 | G . s0 ; the InternalRel of S is non empty implies the InternalRel of S is non empty & the InternalRel of S is non empty deffunc F ( Ordinal , Ordinal ) = phi . ( $1 , $2 ) & phi . ( $2 , $2 ) = phi . ( $2 , $2 ) ; F . s1 . a1 = F . s2 . a1 & F . s2 . a1 = F . s2 . a1 ; x `1 = A . a .= Den ( o , A . a ) .= Den ( o , A . a ) ; Cl ( f " P1 ) c= f " ( Cl P1 ) & Cl ( f " P1 ) c= f " ( Cl P1 ) ; FinMeetCl ( ( the topology of S ) \/ { x } ) c= the topology of T & FinMeetCl ( ( the topology of S ) \/ { x } ) c= the topology of T ; synonym o is \bf means : Def1 : o <> *' & o <> {} & o <> {} ; assume that X = Y + = Y + 1 and card X <> card Y and card Y <> 0 ; the finite implies the \hbox { \boldmath $ s $ } c= 1 + ( the \hbox { \boldmath $ s $ } ) & ( the carrier of S ) \/ { s } c= the carrier of S LIN a , a1 , d or b , c // b1 , c1 & a , c // a1 , b1 ; e / 2 . 1 = 0 & e / 2 . 2 = 1 & e / 2 . 3 = 0 ; ES1 in SS1 & ES2 in { NS1 } implies ES1 in SS2 set J = ( l , u ) If , K = I " ; set A1 = } , A2 = ( ( a , b , c , d ) --> ( A1 , A2 , c ) ) ; set vs = [ <* c , d *> , '&' ] , xy = [ <* d , c *> , '&' ] , $ = [ <* c , d *> , '&' ] , } = [ <* c , d *> , '&' ] , } , { c , d } = [ <* d , c *> , '&' ] , c = [ <* d , d *> x * z `1 * x " in x * ( z * N ) * x " ; for x being element st x in dom f holds f . x = g3 . x & f . x = g1 . x ; Int cell ( f , 1 , G ) c= RightComp f \/ L~ f \/ L~ f \/ L~ f \/ L~ f ; U2 is_an_arc_of W-min C , E-max C & P = W-min C or P = W-min C & P = E-max C & P = E-max C ; set f-17 = f @ "/\" ( g @ ) ; attr S1 is convergent means : Def1 : S2 is convergent & ( for n holds S1 . n = S2 . n ) implies S1 is convergent & lim ( S2 ) = lim ( S2 ) ; f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a + ( 0 qua Ordinal ) .= a + ( 0 qua Ordinal ) ; cluster -> \in -> \in \in \in \in \in cluster -> reflexive transitive transitive for non empty reflexive transitive RelStr -symmetric , 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 \lbrack ( a - b ) / ( 0 , 1 ) ) = len l ; t4 } is ( {} \/ rng t4 ) -valued ( rng t4 \/ { {} } ) -valued ; t = <* F . t *> ^ ( C . p ^ q ) .= ( C . p ^ q ) ^ ( C . q ^ q ) ; set p-2 = W-min L~ Cage ( C , n ) , pw2 = W-min L~ Cage ( C , n ) , pw2 = W-min L~ Cage ( C , n ) , pwhich = W-min L~ Cage ( C , n ) , pw2 = W-min L~ Cage ( C , n ) , pwhich = W-min L~ Cage ( C , n ) ( k -' ( i + 1 ) ) - ( i + 1 ) = ( k - ( i + 1 ) ) - ( i + 1 ) ; consider u being Element of L such that u = u ` and u in D ` and u in D ; len ( ( width aG ) |-> a ) = width ( ( len aG ) |-> a ) .= len ( a * ( width G ) ) ; FM . x in dom ( ( G * the_arity_of o ) . x ) & ( ( G * the_arity_of o ) . x ) . x = x ; set cH2 = the carrier of H2 , cH1 = the carrier of H1 , H2 = the carrier of H2 ; set cH1 = the carrier of H1 , cH2 = the carrier of H2 ; ( Comput ( P , s , 6 ) ) . intpos ( m + 6 ) = s . intpos ( m + 6 ) ; IC Comput ( Q2 , t , k ) = ( l + 1 ) .= ( card I + 1 ) ; dom ( ( cos * sin ) `| REAL ) = REAL & dom ( ( cos * sin ) `| REAL ) = dom ( ( cos * sin ) `| REAL ) ; cluster <* l *> ^ phi -> ( 1 + 1 ) -element for string of S ; set b9 = [ <* A1 , cin *> , '&' ] , c = [ <* cin , cin *> , '&' ] , d = [ <* A1 , cin *> , '&' ] ; Line ( Segm ( M @ , P , Q ) , x ) = L * Sgm Q ; n in dom ( ( the Sorts of A ) * ( the_arity_of o ) ) & ( ( the Sorts of A ) * ( the_arity_of o ) ) . n = ( the Sorts of A ) . ( ( the_arity_of o ) /. 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 = t2 . DataLoc ( s2 . SBP , 2 ) , x4 = s2 . DataLoc ( s2 . SBP , 2 ) , P4 = s2 . DataLoc ( s2 . SBP , 3 ) ; set p-3 = stop I ( ) , p-3 = stop I ( ) , p-3 = stop I ( ) , p-3 = stop I ( ) , p-3 = stop I ( ) , p-3 = stop I ( ) , p-3 = stop I ( ) , p\! = stop I ( ) , p\! = stop consider a being Point of D2 such that a in W1 and b = g . a and a in W1 ; { A , B , C , D , E } = { A , B , C , D } let A , B , C , D , E , F , J , M , N , M , N , N , M , N , N , M , N , N , M , N , N , N , M , N , N , N , M , N , N , M , N , N , M , N , |. p2 .| ^2 - ( p2 `2 ) ^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 topology of L :] & the TopStruct of L = [: the topology of L , the topology of L :] ; consider y being element such that y in dom H1 and x = H1 . y and y in H ; f9 \ { n } = \mathop { \rm Free ( All ( v1 , H ) , E ) } & f . ( f . ( f . ( f . ( f . ( f . ( f . ( f . ( f . ( f . ( f . ( f . ( f . ( f . ( f . ( f . ( f . ( f . ( f . for Y being Subset of X st Y is summable holds Y is not summable implies Y is not summable 2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ; for s being FinSequence holds len ( the { w } ) = len s & for i being Nat st i in dom s holds s . i = ( the { w } ^ <* i *> ) . ( s . i ) 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 ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( K ) ) ) ) ) ) ) ) ; j + ( len f ) - len f <= len f + ( len f ) - len f ; reconsider R1 = R * I , R2 = R * I as PartFunc of REAL n , REAL-NS n ; C8 . x = s1 . ( a - 1 ) .= C8 . x - 1 .= C8 . x ; power F_Complex . ( z , n ) = 1 .= ( x |^ n ) * ( x |^ n ) .= ( x |^ n ) |^ n ; t at ( C , s ) = f . ( ( the connectives of S ) . t ) .= f . ( ( the connectives of S ) . t ) ; 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 & for n st n in N holds |. seq . n - seq . n .| < r ; for y , p st P [ p ] holds P [ All ( y , p ) ] ; { [ x1 , x2 ] where x1 , x2 is Point of [: X1 , X2 :] : x1 in X1 & x2 in X2 } c= X1 h = ( i , j |-- h ) . ( i , id B ) .= H . i ; ex x1 being Element of G st x1 = x & x1 * N c= A & N c= A ; set X = ( ( |^ ( q , O1 ) ) ) `1 , Y = ( ( a , b ) |^ 4 ) , Z = { ( a , b ) |^ 4 } ; b . n in { g1 : x0 < g1 & g1 < a1 . n & a1 . n < x0 } ; f /* s1 is convergent & f /. x0 = lim ( f /* s1 ) & lim ( f /* s1 ) = lim ( f /* s1 ) ; the lattice of Y = the lattice of ( the topology of Y ) & the carrier of ( X | Y ) = the carrier of ( X | Y ) ; 'not' ( a . x ) '&' b . x 'or' a . x '&' 'not' ( b . x ) '&' b . x = FALSE ; 2 = len ( q0 ^ r1 ) + len ( q1 ^ q2 ) & len ( q2 ^ q1 ) = len ( q2 ^ q1 ) + len ( q2 ^ q2 ) ; ( 1 / a ) (#) ( sec * f1 ) - ( id Z ) (#) ( ( 1 / a ) (#) ( sec * f1 ) ) is_differentiable_on Z ; set K1 = integral ( ( lim H ) || A , ( lim H ) || A ) , D2 = integral ( ( lim H ) || A , ( lim H ) || A ) ; assume e in { ( w1 - w2 ) / 2 : w1 in F & w2 in G } ; reconsider d7 = dom a `1 , d6 = dom F `1 , d7 = dom F `1 , d8 = dom F `1 , d8 = dom F `1 , d7 = dom F `1 , d8 = dom F `1 , d8 = dom F `1 , d8 = dom F `1 , d7 = dom F `1 , d8 = dom F `1 LSeg ( f /^ q , j ) = LSeg ( f , j + q .. f ) .= LSeg ( f , j + q .. f ) ; assume X in { T . ( N2 , K1 ) : h . ( N2 , K1 ) = N2 } ; assume that Hom ( d , c ) <> {} and <* f , g *> * f1 = <* f , g *> * f2 ; dom S29 = dom S /\ Seg n .= dom L6 .= dom L6 /\ Seg n .= dom L6 /\ Seg n .= dom ( L6 | n ) ; x in H |^ a implies ex g st x = g |^ a & g in H & a in H * ( 0. ( Z , n ) ) = a `1 - ( 0 * n ) .= a `1 - ( 0 * n ) ; 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 ; for c holds f . c <= g . c implies f @ = g @ @ c dom ( f1 (#) f2 ) /\ X c= dom ( f1 (#) f2 ) /\ X & dom ( f1 (#) f2 ) = dom f1 /\ dom f2 ; 1 = ( p * p ) / p .= p * ( p / p ) .= p * 1 / p ; len g = len f + len <* x + y *> .= len f + 1 .= len f + 1 .= len f + 1 ; dom F-11 = dom ( F | [: N1 , { x } :] ) & dom ( F | [: N1 , { x } :] ) = [: N1 , { x } :] ; dom ( f . t ) * I . t = dom ( f . t ) * g . t ; assume a in ( "\/" ( ( ( T |^ the carrier of S ) ) .: D , the carrier of S ) ) ; assume that g is one-to-one and ( the carrier' of S ) /\ rng g c= dom g and for x being Element of S holds g . x = f . x ; ( ( x \ y ) \ z ) \ ( ( x \ z ) \ ( y \ z ) ) = 0. X ; consider f such that f * f `1 = id b and f * f `2 = id a and f * f `2 = id b ; ( cos | [. 2 * PI * 0 , PI + ( 2 * PI * 0 ) ) | [. 0 , PI + ( 2 * PI * 0 ) .] is increasing ; Index ( p , co ) <= len LS - Gij .. LS & Index ( Gij , LS ) + 1 <= len LS ; t1 , t2 , t2 be Element of ( T . s ) , s be Element of S . s ; "/\" ( ( Frege ( curry H ) ) . h , L ) <= "/\" ( ( Frege ( ( Frege H ) . h ) , L ) , L ) ; then P [ f . i0 ] & F ( f . ( i0 + 1 ) ) < j & j + 1 <= len f ; Q [ [ [ D . x , 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 ; the Sorts of A2 = ( the carrier of S2 ) --> ( the carrier of S2 ) .= ( the carrier of S1 ) --> ( the carrier of S2 ) ; consider s being Function such that s is one-to-one and dom s = NAT and rng s = F and for n being Nat st n in NAT holds P [ n , s . n ] ; dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( a , b2 ) & dist ( a , b ) <= dist ( a , b ) ; ( Lower_Seq ( C , n ) /. len Lower_Seq ( C , n ) ) `1 = W.. Cage ( C , n ) ; q `2 <= ( UMP Upper_Arc C ) `2 & ( UMP C ) `2 <= ( UMP C ) `2 ; LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} & LSeg ( f | i2 , j ) /\ LSeg ( f , i ) = {} ; given a being ExtReal such that a <= Ia and A = ]. a , Ia .[ and a <= b ; 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 <= k & b |^ n in B } ; ( ( x * y * z ) \ x ) \ ( x * y \ x ) = 0. X ; set xy = [ <* xy , y *> , f1 ] , yz = [ <* y , z *> , f2 ] , xy = [ <* z , x *> , f3 ] , yz = [ <* z , x *> , f3 ] , yz = [ <* x , y *> , f3 ] , zx = [ <* z , x *> , f3 ] , zx = [ <* z , x *> Uk /. len lk = ( l . ( len lk ) ) * ( ( l . ( len lk ) ) ) ; ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 = 1 ; ( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) * ( 1 + sn ) < 1 ; ( ( ( S \/ Y ) `2 ) / 2 ) * ( ( S \/ Y ) `2 ) = ( ( S \/ Y ) `2 ) / 2 ; ( seq - seq ) . k = seq . k - seq . k & ( seq - seq ) . k = seq . k - seq . k ; rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , u0 ) /\ dom SVF1 ( 1 , f , u0 ) ; the carrier of X is the carrier of X & the carrier of X = the carrier of X & the carrier of X = the carrier of Y ; ex p4 st p3 = p4 & |. p4 - |[ a , b ]| .| = r & |. p4 - |[ a , b ]| .| = 1 ; set ch = chi ( X , A ) , Ah = chi ( X , A ) ; R / ( 0 * n ) = I\HM ( X , X ) .= R / ( n * 0 ) ; ( Partial_Sums ( ( curry ( F-19 , n ) ) . n ) ) . n is nonnegative ; f2 = CK . ( EK , len H ) .= CK . ( len H + 1 ) .= H . ( len H + 1 ) ; S1 . b = s1 . b .= S2 . b .= S2 . b .= ( S1 * S2 ) . b ; p2 in LSeg ( p2 , p1 ) /\ LSeg ( p1 , p2 ) & p2 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ; dom ( f . t ) = Seg n & dom ( I . t ) = Seg n & rng ( I . t ) = Seg n ; assume that o = ( the connectives of S ) . 11 and ( the connectives of S ) . 11 = ( the connectives of S ) . 11 ; set phi = ( l1 , l2 ) len phi , phi = ( l1 , l2 ) . ( len phi ) , C = ( l1 , l2 ) . ( len phi ) , D = ( l1 , l2 ) . ( len phi ) , E = ( l1 , l2 ) . ( len phi ) , F = ( l1 , l2 ) . ( len phi ) , D = ( l1 , l2 ) . ( len phi synonym p is is invertible means : Def1 : p is invertible & p is invertible & p is is invertible ; ( Y1 `2 = - 1 ) & 0. TOP-REAL 2 <> 0. TOP-REAL 2 & ( for q st q in Y1 holds q `1 >= 0 ) implies ( not q `1 = 0 & q <> 0. TOP-REAL 2 or q `1 = 0 & q <> 0. TOP-REAL 2 ) defpred X [ Nat , set , set ] means P [ $1 , $2 , , ] & $2 = [ $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 ) ) = 1. ( K , n -' n ) & Det ( I @ ( m -' n ) ) = 1. K ; ( - b - sqrt ( b ^2 - 4 * a * c ) ) / ( 2 * a * c ) < 0 ; Cd . d = Cd . d mod Cd . d & Cd . e = Cd . d mod Cd . e ; attr X1 is dense means : Def1 : X2 is dense & X1 is dense & X2 is dense & X1 is dense implies X2 is dense SubSpace of X ; deffunc FF ( Element of E , Element of I ) = $1 * ( $2 * ( $1 , $2 ) ) ; t ^ <* n *> in { t ^ <* i *> : Q [ i , T . t ] } ; ( x \ y ) \ x = ( x \ x ) \ y .= y ` \ 0. X .= 0. X ; for X being non empty set for F being Subset-Family of X holds for Y being Subset-Family of X holds union Y c= union { X , UniCl Y } synonym A , B are_separated means : Def1 : ( Cl A misses B & A misses B & B misses A & B misses A & B misses A ; len ( M @ ) = len p & width ( M @ ) = width ( M @ ) & width ( M @ ) = width ( M @ ) ; J . v = { x where x is Element of K : 0 < v . x & x < 1 } ; ( Sgm ( \bf m ) ) . d - ( Sgm ( support 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= dom ( g * f ) ; |. a .| * ||. f .|| = 0 * ||. f .|| .= ||. a * f .|| .= ||. a * ( f /. a ) .|| ; 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 & for i st i in dom B1 holds P [ i , w . i ] ; [ 1 , {} , <* d1 *> ] in ( { [ 0 , {} , {} ] } \/ S1 \/ S2 ) \/ S2 ; IC Exec ( i , s1 ) + n = IC Exec ( i , s2 ) .= IC Exec ( i , s2 ) .= IC Exec ( i , s2 ) ; IC Comput ( P , s , 1 ) = succ IC s .= ( 5 + 9 ) .= ( 5 + 9 ) ; ( IExec ( W6 , Q , t ) ) . intpos ( ( len Q ) + 1 ) = t . intpos ( ( len Q ) + 1 ) .= t . intpos ( ( len Q ) + 1 ) ; LSeg ( f /^ q , i ) misses LSeg ( f /^ q , j ) & LSeg ( f /^ q , i ) /\ LSeg ( f /^ q , j ) = { f /. ( i + 1 ) } ; assume for x , y being Element of L st x in C & y in C holds x <= y or y <= x ; integral ( f , C ) . x = 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 . 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 c= d ; for y , x being Element of REAL st y in Y ` & x in X holds y `1 <= x `1 & x `2 <= x `2 func |. p .| -> -> -> variable of A means : Def1 : for x being element holds it . x = min ( NBI . p , x ) ; consider t being Element of S such that x `1 , y `1 '||' z `1 , t `2 and x `1 , z `2 '||' y `1 , t `2 ; dom x1 = Seg len x1 & len x1 = len l1 & for k st k in dom x1 holds x1 . k = ( x1 . k ) * ( x2 /. k ) ; 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 /. s1 .|| = ||. f /. s1 .|| & ||. f /. s1 .|| = ||. f /. s1 .|| ; ( the InternalRel of A ) -Seg ( x ` ) /\ Y = {} \/ ( {} \/ Y ) .= {} \/ ( {} \/ 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 j + 1 in dom p ; reconsider h = f | X ( ) as Function of X ( ) , rng f , rng f :] , rng f ; u1 in the carrier of W1 & u2 in the carrier of W2 & u1 in the carrier of W1 & u2 in the carrier of W2 implies u1 + u2 in the carrier of W2 defpred P [ Element of L ] means M <= f . $1 & f . $1 <= g . $1 & f . $1 <= g . $1 ; \mathbin ( 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 is_collinear ; fSet = [ [ dom ( @ f2 ) , cod ( @ f2 ) ] , [ cod ( @ f2 ) , cod ( @ f2 ) ] ] ; 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 ` ) |^ f ) ` & x in ( ( A ` ) |^ f ) ` 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 = Ln . ( F . k ) & F . k in dom Ln & F . ( k + 1 ) = Ln . ( F . k ) ; set i2 = AddTo ( a , i , - n ) , i1 = - ( n + 1 ) , i2 = - ( n + 1 ) ; attr B is thesis means : Def1 : for S being Subuniversal set holds S is ( B , S ) `1 = ( B , S ) `1 ; a9 "/\" D = { a "/\" d where d is Element of N : d in D & a "/\" d in D } ; |( \square , q29 )| * |( \square , q29 )| >= |( \square , exp_R )| * |( \square , A29 )| ; ( - f ) . ( upper_bound A ) = ( ( - f ) | A ) . ( upper_bound A ) .= ( ( - f ) | A ) . ( upper_bound A ) ; GG2 * ( 1 , k ) `1 = G * ( len G , k ) `1 .= G * ( 1 , k ) `1 .= G * ( 1 , k ) `1 ; ( Proj ( i , n ) * ( L . x ) ) = <* ( proj ( i , n ) * ( L . x ) ) . ( x - x0 ) *> ; f1 + f2 * reproj ( i , x ) is_differentiable_in ( ( the reproj of i , x ) + f2 * reproj ( i , x ) ) . x ; pred ( cos . x ) <> 0 & ( tan . x ) <> 0 & ( tan . x ) <> 0 & ( tan . x ) <> 0 ) ; ex t being SortSymbol of S st t = s & h1 . t . x = h2 . t . x & ( for x being Element of S holds x in rng h1 ) ; defpred C [ Nat ] means P8 . $1 is non empty & A is as D -the carrier of X & A is D [ $1 ] ; consider y being element such that y in dom ( p | i ) and ( q | i ) . y = ( p | i ) . y ; reconsider L = product ( { x1 } +* ( index B , l ) ) as Subset of ( the carrier of SCMPDS ) ; for c being Element of C ex d being Element of D st T . ( id c ) = id d & for d being Element of D st d in C holds d <= c ( f , n ) = ( f | n ) ^ <* p *> .= f ^ <* p *> ^ <* p *> .= f ^ <* p *> ; ( f (#) g ) . x = f . ( g . x ) & ( f (#) h ) . x = f . ( h . x ) ; p in { 1 / 2 * ( G * ( i + 1 , j ) + G * ( i + 1 , j + 1 ) ) } ; f `1 - cp = ( f | ( n , L ) ) *' - ( f - ( - ( g - p ) ) ) .= ( f - ( - ( g - p ) ) ) ) *' ; consider r being Real such that r in rng ( f | divset ( D , j ) ) and r < m + s ; f1 . ( |[ ( r2 - 1 ) / ( 2 |^ ( n + 1 ) ) , ( r2 - 1 ) / ( 2 |^ ( n + 1 ) ) ]| ) in f1 .: ( W1 ) ; eval ( a | ( n , L ) , x ) = ( a | ( n , L ) ) . x .= a . x ; z = DigA ( tz , x9 ) .= DigA ( tz , x9 ) .= DigA ( tz , x ) .= DigA ( tz , x ) ; set H = { Intersect S where S is Subset-Family of X : S c= G & S c= G } , F = { Intersect S where S is Subset-Family of X : S c= G } ; consider S19 being Element of D * , d being Element of D * such that S `1 = S19 ^ <* d *> and S `2 = 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 <= ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 ; 0. ( V ) is Linear_Combination of A & Sum ( ( - L ) (#) ( L (#) ( - L ) ) ) = 0. V implies Sum ( L (#) ( - L (#) ( - L ) ) ) = 0. V let k1 , k2 , k2 , k1 , k2 , k2 , k1 , k2 , k2 be Element of NAT , a be Int-Location , k1 , k2 , k2 be Element of NAT ; consider j being element such that j in dom a and j in g " { k } and x = a . j ; H1 . x1 c= H1 . x2 or H1 . x2 c= H1 . x1 or H1 . x2 c= H1 . x2 or H1 . x2 c= H1 . x1 & H1 . x2 c= H1 . x2 ; consider a being Real such that p = ( - 1 ) * p1 + ( a * p2 ) and 0 <= a and a <= 1 ; assume that a <= c & c <= d & [' 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 ; A9 in { ( S . i ) `1 where i is Element of NAT : not contradiction } & A9 c= { ( S . i ) `1 where i is Element of NAT : not contradiction } ; ( T * b1 ) . y = L * ( b2 /. y ) .= ( F `1 * b1 ) . y .= ( F `1 * b1 ) . y ; g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x - s . y .| ; ( log ( 2 , k + 1 ) ) ^2 >= ( log ( 2 , k + 1 ) ) ^2 ; then that p => q in S and not x in the still of p and not 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 ) = dom ( the InitS of r-11 ) ; synonym f is integer means : Def1 : for x being set st x in rng f holds x is ExtReal ; assume for a being Element of D holds f . { a } = a & for X being Subset-Family of D holds f . ( f .: X ) = f . union X ; i = len p1 .= len p3 + len <* x *> .= len p3 + len <* x *> .= len p3 + 1 .= len p1 + 1 ; l ( ) `1 = ( g . ( k + 3 ) ) `1 + ( k + 1 ) - ( e . ( k + 3 ) ) `1 .= ( g . ( k + 3 ) ) `1 + ( e . ( k + 3 ) ) `1 ; CurInstr ( P2 , Comput ( P2 , s2 , l2 ) ) = halt SCM+FSA .= CurInstr ( P2 , Comput ( P2 , s2 , l2 ) ) ; assume for n be Nat holds ||. seq .|| . n <= ( ||. seq .|| ) . n & ( ||. seq .|| ) . n <= ( ||. seq .|| ) . n ; sin ( 0. ) = sin ( r ) * cos ( ( - cos ( r ) ) * sin ( s ) ) .= 0 ; set q = |[ g1 . t0 , g2 . t0 ]| , r = |[ g2 . t0 , g2 . t0 ]| , s = |[ r , g2 ]| , t = |[ r , s ]| , s = |[ r , t ]| , t = |[ r , s ]| , s = |[ r , t ]| , t = |[ r , s ]| , s = |[ r , t ]| , t = |[ r , s ]| , t = |[ r , s ]| ; consider G being sequence of S such that for n being Element of NAT holds G . n in implies G in implies for x being Element of NAT holds x in G ; consider G such that F = G and ex G1 st G1 in SM & G = ( X \/ G1 ) & G is one-to-one ; the root of [ x , s ] in ( the Sorts of Free ( C , X ) ) . s & ( the Sorts of C ) . s c= ( the Sorts of Free ( C , X ) ) . s ; Z c= dom ( exp_R * ( f + ( #Z 3 ) * ( f1 + f2 ) ) ) & Z c= dom ( exp_R * ( f1 + f2 ) ) ; for k be Element of NAT holds seq1 . k = ( sum ( Im ( f , S ) ) ) . k & ( Im ( f , S ) ) . k = ( Im ( f , S ) ) . k assume that - 1 < n and ( q `2 ) ^2 > 0 and ( q `1 ) ^2 / ( |. q .| ) ^2 < 1 ; assume that f is continuous and a < b and a < d and f . a = c and f . b = d and f . a = d ; consider r being Element of NAT such that s-> Element of NAT such that s-> Element of NAT and r <= q and r <= q ; LE f /. ( i + 1 ) , f /. j , L~ f & f /. 1 = 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 x <> y and y <> x ; assume that f +* ( i1 , \xi ) in ( proj ( F , i2 ) ) " ( A . ( i + 1 ) ) and f . ( i + 1 ) = ( proj ( F , i2 ) ) . ( i + 1 ) ; 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 & t <> {} } ; consider l be Nat such that for m be Nat st l <= m holds ||. ( s1 . m ) - ( lim s1 ) .|| < g ; consider t be VECTOR of product G such that mt = ||. D5 . t .|| and ||. t .|| <= 1 ; assume that the carrier of v = 2 and v ^ <* 0 *> in dom p and v ^ <* 1 *> in dom p and p . 1 = 1 ; consider a being Element of the lines of Ximplies a on A & not a on A & not a on B ; ( - x ) |^ ( k + 1 ) * ( ( - x ) |^ ( k + 1 ) ) " = 1 ; for D being set for i st i in dom p holds p . i in D & p . i in D defpred R [ element ] means ex x , y st [ x , y ] = $1 & P [ x , y ] & P [ y , x ] ; L~ f2 = union { LSeg ( p0 , p10 ) , LSeg ( p1 , p10 ) } .= { LSeg ( p1 , p10 ) , LSeg ( p1 , p10 ) } ; i -' len h11 + 2 - 1 < i -' len h11 + 2 - 1 + 1 - 1 + 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 ; for n being Element of NAT st n in dom F holds F . n = |. ( F . n ) . ( n - 1 ) .| ; for r , s1 , s2 , s3 holds r in [. s1 , s2 .] iff s1 <= s2 & s2 <= s3 & s1 <= s2 & s2 <= s3 assume v in { G where G is Subset of T2 : G in B2 & G c= z1 & G c= z2 & G c= z1 & G c= z1 & G c= z1 & G c= z2 } ; let g be \cap element of A , Z , ( ( 0 , 1 ) --> ( b , 0 ) ) = 0 & ( ( 0 , 1 ) --> ( b , 1 ) ) = 0 ; min ( g . [ x , y ] , k ) . [ y , z ] = ( min ( g , k , x ) ) . y ; consider q1 being sequence of CL such that for n holds P [ n , q1 . n ] and P [ n + 1 ] ; consider f being Function such that dom f = NAT & for n being Element of NAT holds f . n = F ( n ) ; reconsider B-6 = B /\ B , OO = O , Z = { Z where Z is Subset of B : Z in F & Z c= F } as Subset of B ; consider j being Element of NAT such that x = the ` of n and 1 <= j and j <= n and f . j = i ; consider x such that z = x and card ( x . O2 ) in card ( x . O2 ) and x in L1 . ( x . O2 ) ; ( C * _ T4 . ( k , n2 ) ) . 0 = C . ( ( dom ( T4 . k , n2 ) ) . 0 ) .= C . ( ( dom ( T4 . k , n2 ) ) . 0 ) ; dom ( X --> rng f ) = X & dom ( ( X --> f ) . x ) = dom ( X --> f . x ) ; S-bound L~ SpStSeq C <= ( b - ( SpStSeq C ) ) / 2 & ( b - ( S-bound C ) ) / 2 <= ( ( b - ( S-bound C ) ) / 2 ) / 2 ; synonym x , y are_collinear means : Def1 : x = y or ex l being Subset of S st { x , y } c= 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 , a , b being Element of Im k st a = x & b = y & x << b holds a << b ; ( 1 / 2 * ( ( ( #Z 2 ) * ( ( #Z 2 ) * ( 1 / 2 ) ) ) ) ) `| REAL ) = ( ( 1 / 2 ) * ( ( #Z 2 ) * ( 1 / 2 ) ) ) ; defpred P [ Element of omega ] means ( for n holds ( for x st x in A1 holds x in A ) implies x in A1 . n ) & ( for x st x in A1 . n holds x in A2 . x ) ; 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 ) * ( f . g2 ) ; ( M * F-4 ) . n = M . ( F-4 . 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 , z be set means : Def1 : a , b , c , x is_collinear & a , b , c , x is_collinear & b , c , y is_collinear ; ( the partial of s ) . n <= ( the partial of s ) . n * ( ( n + 1 ) * s . n ) & ( ( n + 1 ) * s ) . n >= 0 ; attr - 1 <= r & r <= 1 implies ( arccot * ( 1 , 1 ) `1 ) `1 = - 1 & ( arccot * ( 1 , 1 ) `1 ) `2 = 1 ; seq in { p ^ <* n *> where n is Nat : p ^ <* n *> in T1 & p ^ <* n *> in T1 } implies for n being Nat holds p ^ <* n *> in T1 |[ x1 , x2 , x3 , x4 ]| . 2 - |[ y1 , y2 , x4 ]| . 2 = x2 - y2 & |[ x1 , x2 , x3 , x4 ]| . 2 = x2 - x3 ; attr m is nonnegative means : Def1 : F . m is nonnegative & ( for n be Nat holds F . n = ( Partial_Sums ( F ) ) . n ) ; len ( ( G . z ) * ( x , y ) ) = len ( ( ( G . x9 ) * ( y , z ) ) ) .= len ( ( G . x9 ) * ( 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 W1 /\ W3 and v in W2 /\ W3 ; given F being FinSequence of NAT such that F = x and dom F = n and rng F c= { 0 , 1 } and Sum F = k ; 0 = \vert \vert ] * ( 1- \hbox { - 1 } ) iff 1 = ( ( - 1 ) * ( - ( 1 - ( 0 - 1 ) ) ) / ( ( 0 - ( 0 - 1 ) ) ) ) ; consider n be Nat such that for m be Nat st n <= m holds |. ( f # x ) . m - lim ( f # x ) .| < e ; cluster -> as being being being non empty iff iff ( for } , \hbox { $ ( let L ) , x being Element of L holds ( ( the carrier of L ) --> x ) is Boolean ) & ( ( the carrier of L ) --> x ) is Boolean "/\" ( BB , L ) = Top BB .= the carrier of S .= [#] ( S | ( [#] T ) ) .= "/\" ( I , L ) .= "/\" ( I , L ) ; ( r / 2 ) ^2 + ( r / 2 ) ^2 <= ( r / 2 ) ^2 + ( r / 2 ) ^2 ; for x being element st x in A /\ dom ( f `| X ) holds ( f `| X ) . x >= r2 & ( f `| X ) . x >= r2 2 * r1 - ( 2 * r1 - ( 2 * r1 - 1 ) ) * ( b - ( 2 * r1 - 1 ) ) = 0. TOP-REAL 2 ; reconsider p = P * ( \square , 1 ) " , q = a " * ( ( - ( - 1 ) ) * ( ( - 1 ) * ( 1 , 1 ) ) ) as FinSequence of K ; consider x1 , x2 being element such that x1 in uparrow s and x2 in uparrow t and x = [ x1 , x2 ] and y = [ x1 , x2 ] ; for n being Nat st 1 <= n & n <= len q1 holds q1 . n = ( upper_volume ( g , M7 ) ) . ( n + 1 ) & ( for k be Nat st k in dom ( ( g | M7 ) | M7 ) holds ( ( g | M7 ) | M7 ) . k = ( ( g | M7 ) | M7 ) . k consider y , z being element such that y in the carrier of A and z in the carrier of A and i = [ y , z ] ; given H1 , H2 being strict Subgroup of G such that x = H1 & y = H2 and H1 , H2 |^ ( n + 1 ) are_relative_prime and H2 , H1 |^ ( n + 1 ) K K ; for S , T being non empty < d , T being Function of T , S st T is complete holds d is monotone & d is monotone & d is monotone [ a + 0. F_Complex , b2 ] in ( the carrier of F_Complex ) /\ ( the carrier of F_Complex ) & [ a + 0. F_Complex , b2 ] in [: the carrier of F_Complex , the carrier of F_Complex :] ; reconsider mm = max ( len F1 , len ( p . n ) * ( x |^ n ) ) , mm = max ( len F1 , len F2 ) * ( x |^ n ) as Element of NAT ; I <= width GoB ( ( GoB h ) * ( 1 , 1 ) `1 , ( GoB h ) * ( 1 , 1 ) `2 ) & I <= width GoB ( ( GoB h ) * ( 1 , 1 ) `2 ) `2 ; f2 /* q = ( f2 /* ( f1 /* s ) ) ^\ k .= ( ( f2 * f1 ) /* s ) ^\ k .= ( ( f2 * f1 ) /* s ) ^\ k .= ( ( f2 * f1 ) /* s ) ^\ k ; attr A1 \/ A2 is linearly-independent means : Def1 : A1 misses A2 & ( Lin A1 ) /\ Lin A2 = { 0. V } & ( Lin A1 ) /\ Lin ( A2 ) = { 0. V } ; 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 ) (#) ( ( Line ( p , m ) ) * ( \square , 1 ) ) ) = dom ( F ^ <* 1 *> ) .= dom ( F ^ <* 1 *> ) ; cluster [ x `1 , 4 ] , x `2 , x `2 , 4 ] -> non empty & [ x `1 , 4 ] , x `2 , x `2 ] `1 = x & [ x `1 , 4 ] , x `2 ] `1 = x ; E , All ( x1 , All ( x2 , x2 ) ) |= All ( x2 , All ( x1 , x2 ) '&' ( x1 '&' ( x1 '&' x2 ) ) ) & E , u |= All ( x2 , x1 ) '&' ( x1 '&' ( 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 - h . m ) ; cell ( G , Xs -' 1 , ( Y + 1 ) \ ( t + 1 ) ) meets ( UBD L~ f ) & ( ( L~ f ) \ ( t + 1 ) ) meets ( ( L~ f ) \ ( t + 1 ) ) ; IC Result ( P2 , s2 ) = IC IExec ( I , P , Initialize s ) .= card I .= card I .= card I + card J .= card I + card J + 3 .= card I + 1 ; sqrt ( ( - ( ( - ( 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 a . x0 = b . x0 and a . x0 = c . x0 ; dom ( r1 (#) chi ( A , C ) ) = dom chi ( A , C ) .= dom ( chi ( A , C ) ) .= dom ( ( r1 (#) chi ( A , C ) ) | A ) .= dom ( ( r1 (#) chi ( A , C ) ) | A ) .= dom ( ( r1 (#) chi ( A , C ) ) ) ) ; d-7 . [ y , z ] = ( ( y , z ) `2 - ( z , y ) `2 ) * ( ( y , z ) `2 - ( z , y ) `2 ) * ( ( y , z ) `2 - ( z , y ) `2 ) ) ; attr i being Nat means C . i = A . i /\ B . i & C . i c= C . ( i + 1 ) ; assume that x0 in dom f and f is_continuous_in x0 and ||. f /. x0 .|| <= ( ||. f .|| ) * ( ||. x0 .|| ) and ||. f /. x0 .|| <= ( ||. f .|| ) * ( ||. 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 for x being Element of REAL n st x in Line ( x1 , x2 ) holds |. y1 - y2 .| <= |. y1 - y2 .| & |. y1 - y2 .| <= |. y1 - y2 .| func the _ -> strict \cup { <*> } -> Ordinal means : Def1 : a in it & for b being Ordinal st a in b holds it . b c= b ; [ 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 ] & P [ a , b ] ; ||. ( vseq . n ) - ( vseq . m ) .|| * ||. x - x0 .|| < ( e * ||. x - x0 .|| ) * ||. x - x0 .|| ; then for Z being set st Z in { Y where Y is Element of I7 : F c= Y & z in Z } holds z in x & z in Z ; sup compactbelow [ s , t ] = [ sup ( ( compactbelow s ) . [ s , t ] ) , sup ( ( compactbelow s ) . [ s , t ] ) ] ; consider i , j being Element of NAT such that i < j and [ y , f . j ] in IF and [ f . i , z ] in IF and [ f . i , f . j ] in IF ; for D being non empty set , p , q being FinSequence of D st p c= q ex p being FinSequence of D st p ^ q = q & p ^ q = p ^ q consider e19 being Element of the affine of X such that c9 , a9 // a9 , e and a9 <> c9 & b9 <> c9 & c9 <> a9 & c9 <> a9 & c9 , e // c9 , e ; set U2 = I \! \mathop { {} } , U2 = I \! \mathop { {} } , SS = I \! \mathop { {} } , SS = I \! \mathop { {} } , SS = { {} } , SS = { {} } , SS = { {} } , SS = { {} } , SS = { {} } , SS = { {} } , SS = { {} } , SS = { {} } , SS = { {} } , SS = { {} } , SS = { {} } , SS = { {} } , SS = { {} } , SS = { {} } , SS = { {} } , SS = { {} } |. q3 .| ^2 = ( q3 `1 ) ^2 + ( q2 `2 ) ^2 .= ( q `1 ) ^2 + ( q2 `2 ) ^2 .= ( q `1 ) ^2 + ( q2 `2 ) ^2 .= ( q `1 ) ^2 ; for T being non empty TopSpace , x , y being Element of [: the topology of T , the topology of T :] holds x "\/" y = x /\ y & x "/\" y = x /\ y dom 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 ) = dom ( the charact of U1 ) ; dom ( h | X ) = dom h /\ X .= dom ( ( ||. h .|| | X ) | X ) .= dom ( ( ||. h .|| | X ) | X ) .= dom ( ( ||. h .|| | X ) | X ) .= dom ( ( ||. h .|| | X ) | X ) ; for N1 , N1 being Element of GX holds dom ( h . K1 ) = N & rng ( h . K1 ) c= 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 ) ) . i ; - ( q `1 ) ^2 / ( |. q .| ) ^2 < - ( q `1 ) ^2 / ( |. q .| ) ^2 & ( q `1 ) ^2 / ( |. q .| ) ^2 <= - ( q `1 ) ^2 / ( |. q .| ) ^2 ; attr r1 = f9 & r2 = f9 & r1 * ( ( - 1 ) * ( f - g ) ) = f9 * ( ( - 1 ) * ( f - g ) ) ; vseq . m is bounded Function of X , the carrier of Y & x9 . m = ( seq_id ( vseq . m , X , Y ) ) . x & x9 . m = ( vseq . m ) . x ; attr a <> b & b <> c & angle ( a , b , c ) = PI implies angle ( b , c , a ) = 0 & angle ( c , a , b ) = 0 ; consider i , j being Nat , r 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 ^ p1 = p1 ^ q1 and p1 ^ q1 = p2 ^ q1 and p1 ^ q1 = p2 ^ q1 and p1 ^ q1 = p2 ^ q2 ; ( ( for r1 , r2 , s1 , s2 being Element of A holds ( for r1 , r2 st r1 in A & r2 in B & r1 < r2 holds r1 < r2 ) ) implies A = B ( LMP A ) `2 = lower_bound ( proj2 .: ( A /\ Vertical_Line w ) ) & ( proj2 .: ( A /\ Vertical_Line w ) ) `2 = upper_bound ( proj2 .: ( A /\ Vertical_Line w ) ) & ( proj2 .: ( A /\ Vertical_Line w ) ) `2 = upper_bound ( proj2 .: ( A /\ Vertical_Line w ) ) ; s |= ( H1 , k1 ) \bf ( H , k2 ) iff s |= ( H1 , k1 ) . ( len ( H , k2 ) ) & ( s |= ( H1 , k2 ) ) . ( len ( H , k1 ) ) ; len s5 + 1 = card ( support b1 ) + 1 .= card ( support b2 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) + 1 .= card ( support b1 ) ; consider z being Element of L1 such that z >= x and z >= y and for z being Element of L1 st z >= x & z >= y holds z `1 >= y `1 ; LSeg ( UMP D , |[ ( W-bound D + E-bound D ) / 2 , ( E-bound D + E-bound D ) / 2 ]| ) /\ D = { UMP D } ; lim ( ( ( f `| N ) / ( g `| N ) ) /* b ) = lim ( ( f `| N ) / ( g `| N ) ) .= lim ( ( f `| N ) / ( g `| N ) ) ; P [ i , pr1 ( f ) . i , pr1 ( f ) . ( i + 1 ) , pr1 ( f ) . ( i + 1 ) ] ; for r be Real st 0 < r ex m be Nat st for k be Nat st m <= k holds ||. ( seq . k ) - ( seq . k ) .|| < r for X being set , P being a_partition of X , x , a , b being set st x in a & a in P & b in P & x in P & b in P holds a = b Z c= dom ( ( #Z 2 ) * ( ( #Z 2 ) * f ) ) \ ( ( #Z 2 ) * ( ( #Z 2 ) * f ) ) " { 0 } ) & Z c= dom ( ( #Z 2 ) * ( ( #Z 2 ) * f ) ) ; ex j being Nat st j in dom ( l ^ <* x *> ) & j = ( l ^ <* x *> ) . j & i = 1 + len l & z = 1 + len l & j = len l + 1 ; for u , v being VECTOR of V , r being Real st 0 < r & u in ex v being VECTOR of V st r * u + ( 1-r * v ) in N A , Int A , Cl ( A , B ) , Cl ( A , B ) , Cl ( A , B ) , Cl ( A , B ) , Cl ( A , B ) , Cl ( A , B ) , Cl ( A , B ) ) st A , B , Cl ( A , B ) , Cl ( A , B ) , Cl ( A , B ) , Cl ( A , B ) , Cl ( A , B ) , Cl ( A , B ) , Cl ( A , B ) , Cl ( A , B ) , Cl ( A , B ) , Cl ( A , B ) , Cl ( - Sum <* v , u , w *> = - ( v + u + w ) .= - ( v + u ) + w .= - ( v + u ) + w .= - ( v + u ) + w ; ( Exec ( a := b , s ) ) . IC SCM R = ( Exec ( a := b , s ) ) . IC SCM R .= succ IC s .= succ IC s .= succ IC s .= ( succ IC s ) .= ( succ IC s ) ; 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 ) . ( f . x ) ; for S1 , S2 being non empty reflexive RelStr , D being non empty directed Subset of [: S1 , S2 :] holds cos ( D ) is directed & cos ( D ) is directed & cos ( D ) is directed card X = 2 implies ex x , y st x in X & y in X & x <> y & x in X or x = y & x = y or x = y E-max L~ Cage ( C , n ) in rng ( Cage ( C , n ) :- W-min L~ Cage ( C , n ) ) & E-max L~ Cage ( C , n ) = ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ; for T being as Tree , p , T being Element of dom T st p in dom T holds ( T , p ) dom ( T , p ) = T . q [ i2 + 1 , j2 ] in Indices G & [ i2 + 1 , j2 ] in Indices G & f /. k = G * ( i2 + 1 , j2 ) & f /. k = G * ( i2 + 1 , j2 ) ; cluster ( k gcd n ) divides ( k gcd n ) & n divides ( k gcd n ) & ( k divides ( k gcd n ) ) & ( k divides ( k gcd n ) ) implies ( k divides ( k gcd n ) ) & ( k divides ( k gcd n ) ) dom F " = the carrier of X2 & rng F " = the carrier of X1 & F " = ( the carrier of X2 ) /\ ( the carrier of X1 ) & F " = ( the carrier of X2 ) /\ ( the carrier of X2 ) ; consider C being finite Subset of V such that C c= A and card C = $1 and the carrier of V = Lin ( B9 \/ C ) and C c= B and C c= B ; V is prime implies for X , Y being Element of [: the topology of T , the topology of T :] st X /\ Y c= V & X c= Y holds X c= Y or Y c= V 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 ( p , p3 , p2 ) .= angle ( p , p3 , p2 ) .= angle ( p , 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 . 1 = p2 & f . 1 = p3 & f . - 1 = p2 & f . - 1 = p4 ; attr f is partial differentiable on 2 means : Def1 : SVF1 ( 2 , pdiff1 ( f , 1 ) , u0 ) . u = ( proj ( 2 , 3 ) * ( pdiff1 ( f , 1 ) , u0 ) ) . u ; ex r , s st x = |[ r , s ]| & G * ( len G , 1 ) `1 < r & r < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < s & s < G * ( 1 , 1 ) `2 ; assume that f is_sequence_on G and 1 <= t & t <= len G and G * ( t , width G ) `2 >= N-bound L~ f and f /. t = ( GoB f ) * ( 1 , width G ) and f /. t = ( GoB f ) * ( 1 , width G ) and f /. t = ( GoB f ) * ( 1 , width G ) ; pred i in dom G means : Def1 : r (#) ( f * reproj ( i , x ) ) = r (#) ( reproj ( i , x ) ) & ( f * reproj ( i , x ) ) . x = r * ( reproj ( i , x ) ) . x ; consider c1 , c2 being bag of o1 + o2 such that ( decomp c ) /. k = <* c1 , c2 *> and c = ( decomp c1 ) /. ( k + 1 ) and ( decomp c ) /. ( k + 1 ) = c1 /. ( k + 1 ) ; u0 in { |[ r1 , s1 ]| : r1 < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < s1 & s1 < G * ( 1 , 1 ) `2 } ; ( X ^ Y ) . k = the carrier of X . k2 .= ( C ^ ( X ^ Y ) ) . k .= ( C ^ ( X ^ Y ) ) . k .= ( C ^ ( X ^ Y ) ) . k ; attr M1 = len M2 means : Def1 : width M1 = width M2 & for i st i in dom M1 holds M1 * ( i , j ) = M1 * ( i , j ) & M1 * ( i , j ) = M2 * ( i , j ) ; consider g2 be Real such that 0 < g2 and { y where y is Point of S : ||. ( - x0 ) * ( y - x0 ) .|| < g2 & y in dom ( ( - 1 ) (#) ( g - f ) ) & ( g - 1 ) (#) ( g - f ) ) < g2 } c= N2 ; assume x < ( - b + sqrt ( thesis ( a , b , c ) ) / ( 2 * a ) ) or x > ( - b - sqrt ( delta ( a , b , c ) ) / ( 2 * a ) ) ; ( G1 '&' G2 ) . i = ( <* 3 *> ^ G1 ) . i & ( H1 '&' H2 ) . i = ( <* 3 *> ^ H1 ) . i & ( H1 '&' H2 ) . i = ( <* 3 *> ^ H1 ) . i ; for i , j st [ i , j ] in Indices ( M3 + M1 ) holds ( ( M3 + M1 ) * ( i , j ) ) * ( i , j ) < M2 * ( i , j ) for f being FinSequence of NAT , i being Element of NAT st i in dom f & i divides len f holds i divides ( f /. j ) & ( i divides len f implies i divides len f ) assume F = { [ a , b ] where a , b is set , c is set : for a , b being set st a in B\mathopen { a , b } & b in B\mathopen { a , b } } holds a c= c ; b2 * q2 + ( b3 * q3 ) + ( - ( ( a - ( a - ( b - ( a - ( b - ( a - ( b - a ) ) ) ) ) ) ) ) * ( ( a - ( b - ( b - a ) ) ) ) ) = 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 & Cl ( B \/ C ) c= D } ; attr seq is summable means : : : seq is summable & seq is summable & Sum ( seq ) = Sum ( seq ) & Sum ( seq ) = Sum ( seq ) & Sum ( seq ) = Sum ( seq ) & Sum ( seq ) = Sum ( seq ) ; dom ( ( ( cn - cn ) | D ) | D ) = ( the carrier of ( ( TOP-REAL 2 ) | D ) ) /\ D .= the carrier of ( ( TOP-REAL 2 ) | D ) | D .= D ; X [ X \to Z ] means X is full non empty full SubRelStr of ( Omega Z ) |^ the carrier of Z & X [ X \to Y ] implies X = Y G * ( 1 , j ) `2 = G * ( i , j ) `2 & G * ( 1 , j ) `2 <= G * ( i , j ) `2 & G * ( 1 , j ) `2 <= G * ( i , j ) `2 ; synonym m1 c= m2 means : Def1 : for p being set st p in P holds the \HM { m1 where m1 is Nat : m1 in dom p & p . m1 = m2 } & for p being non empty set st p in P & p . m1 = m2 holds p . m1 = m2 . ( p . m1 ) ; 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 ] ; attr IT is \vert means : : : for a being multiplicative O , s being Element of IT holds the multF of IT is set of the carrier of IT & the multF of IT is Relation of the carrier of IT & the multF of IT is Relation of the carrier of IT ; sequence ( a , b ) + 1 + and sequence ( c , d ) . 1 = b + ( c , d ) .= b + ( c , d ) .= ( the non empty set ) --> ( c , d ) .= ( the carrier of a + c , d + 1 ) ; cluster + ( i , j ) -> Element of INT means : Def1 : for i1 , i2 being Element of INT holds it . ( i1 , i2 ) = + ( ( i , j ) --> ( i1 , i2 ) ) ; ( ( - s2 ) * p1 + ( s2 * p2 ) - ( s2 * p2 ) ) = ( ( - r2 ) * p1 + ( ( - s2 ) * p2 ) ) - ( ( ( - r2 ) * p1 ) + ( ( - s2 ) * p2 ) ) ; eval ( ( a | ( n , L ) ) *' , x ) = eval ( a | ( n , L ) ) * eval ( p , x ) .= a * eval ( p , x ) .= eval ( p , x ) ; assume that the TopStruct of S = the TopStruct of T and for D being non empty directed Subset of S , V being Subset of Omega S st V in V & V is open holds sup V in V & V is open & V is open ; assume that 1 <= k & k <= len w + 1 and T-7 . ( ( q , w ) U . k , w ) = ( ( T-7 . k , w ) U . ( k + 1 ) ) . ( ( q , w ) U . k , w ) ) . ( ( q , w ) U . k , w ) ; 2 * a |^ ( n + 1 ) + ( 2 * b |^ ( n + 1 ) ) >= a |^ ( n + 1 ) + ( ( a |^ n ) * b + ( b |^ n ) * a ) + ( ( a |^ n ) * b + ( b |^ n ) * a ) ; M , v / ( x. 3 , m ) / ( x. 0 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , m ) / ( x. 0 , m ) ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 0 , m ) ; assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f . x0 & for x1 st x1 in l holds f . x1 - f . x0 < f . x1 & f . x0 < f . x1 ; for G1 being _Graph , W being Walk of G1 , e being Vertex of G2 , e being Vertex of G2 st e in W & e in W holds not e in W & not e in W implies not e in V vs is non empty iff ( not .| is non empty or ' is non empty & not ( not .| is non empty & not ( not ' is non empty & not ' ( y ) is non empty ) & not ( not ( x is non empty & y is non empty & not x is non empty ) ) & not ( y is non empty & not x is non empty ) ) & not ( x is non empty & y is non empty ) & not c is non empty ; Indices GoB f = [: dom GoB f , Seg width GoB f :] & i1 + 1 in dom GoB f & i2 + 1 in dom GoB f & 1 <= i1 & i1 + 1 <= len GoB f & 1 <= i2 & i2 + 1 <= width GoB f & f /. ( 1 + 1 ) = ( GoB f ) * ( i1 , i2 + 1 ) ; for G1 , G2 , G3 being Group , O being strict Subgroup of G2 , A being set st G1 is stable & G2 is stable & A is stable holds ( G1 * A ) * ( A , B ) is stable UsedIntLoc ( in1 ( f , 1 ) ) = { intloc 0 ( 1 ) , intloc 1 ( ) , intloc 3 ( ) , intloc 4 ( ) , intloc 4 ( ) , intloc 4 ( ) , intloc 4 ( ) , intloc 0 ( ) , intloc 4 ( ) , intloc 0 ( ) , intloc 4 ( ) , intloc 0 ( ) , intloc 0 ( ) , intloc 0 ( ) , intloc 0 ( ) ) , intloc 4 ( ) ) } ; for f1 , f2 being 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 ) ) ^2 = ( q `1 ) ^2 / ( sqrt ( 1 + ( q `1 / q `2 ) ^2 ) ) ^2 ; for x1 , x2 , x3 being Element of REAL n holds |( x1 - x2 , x3 - x3 )| = |( x1 , x2 )| + |( x2 , x3 )| & |( x1 - x2 , x3 )| = |( x1 , x2 )| + |( x2 , x3 )| + |( x3 , x3 )| for x st x in dom ( ( F | A ) | A ) holds ( ( ( F | A ) | A ) . ( - x ) ) = - ( ( ( F | A ) | A ) . x ) for T being non empty TopSpace , P being Subset-Family of T st P c= the topology of T for B being Basis of T st B c= P & B c= P holds P is Basis of T ( a 'or' b 'imp' c ) . x = 'not' ( ( a 'or' b ) . x ) 'or' c . x .= 'not' ( ( a 'or' b ) . x ) 'or' c . x .= TRUE 'or' TRUE .= TRUE 'or' TRUE .= TRUE ; for e being set st e in [: A , Y1 :] ex X1 being Subset of Y , Y1 being Subset of X st e = [: X1 , Y1 :] & X1 is open & Y1 is open & Y1 c= Y1 & Y1 c= Y1 for i being set st i in the carrier of S for f being Function of [: S . i , S1 . i :] , S1 . i st f = H . i & F . i = f | ( the carrier of S1 ) 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 ) , J ) . v = Valid ( VERUM ( Al , J ) , J ) . w card D = card D1 + card D2 - card { i , j } .= ( c1 + 1 ) + ( 1 - 1 ) .= ( c1 + 1 ) + ( 1 - 1 ) .= ( c1 + 1 - 1 ) * ( c1 + 1 - 1 ) .= 2 * c1 + 1 - 1 .= 2 * c1 + 1 - 1 ; IC Exec ( i , s ) = ( s +* ( 0 .--> succ ( s . 0 ) ) ) . 0 .= ( 0 .--> succ ( s . 0 ) ) . 0 .= ( ( 0 .--> s . 0 ) ) . 0 .= ( ( 0 .--> s . 0 ) ) . 0 .= ( ( 0 .--> s ) . 0 ) . 0 .= ( ( 0 .--> s ) . 0 ) . 0 .= ( ( 0 .--> s ) . 0 ) .= ( 0 .--> s ) . 0 .= ( ( 0 .--> s ) . 0 .= ( 0 .--> s ) . 0 .= ( 0 .--> s ) . 0 .= ( 0 .--> s ) . 0 .= ( 0 .--> s ) . 0 .= ( 0 .--> s ) . 0 .= len f /. ( ( i -' 1 ) -' 1 + 1 ) = len f /. ( ( i -' 1 ) + 1 ) - 1 .= len f -' ( ( i -' 1 ) + 1 ) .= len f -' ( i -' 1 ) + 1 .= len f -' ( i -' 1 ) + 1 ; for a , b , c being Element of NAT st 1 <= a & 2 <= b & k <= a or a = b + b-2 or a = a + b-2 or b = b + b-2 or a = a + b-2 or b = b + 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 ) & i <= len f holds Index ( p , f ) <= i & Index ( p , f ) <= len f lim ( ( curry ( ( P , s ) . k + 1 ) ) # x ) = lim ( ( curry ( ( P , s ) . k ) ) # x ) + lim ( ( curry ( ( P , s ) . k + 1 ) ) # x ) ; z2 = g /. ( ( i -' n1 + 1 ) + 1 ) .= g . ( i -' n2 + 1 + 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 C6 & [ f . 0 , f . 2 ] in the InternalRel of C6 ; for G being Subset-Family of B st G = { R [ X ] where X is Subset of [: A , B :] , Y is Subset of [: A , B :] st X in F & Y in G holds ( Intersect ( F ) ) . X = Intersect ( G ) . Y holds ( Intersect ( F ) ) . Y = Intersect ( G ) CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) = CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m1 + m2 ) ) .= CurInstr ( P1 , Comput ( P1 , s1 , m1 ) ) .= CurInstr ( P1 , s1 ) .= ( CurInstr ( P1 , s1 ) ) ; assume that a on M and b on M and c on N and p on N and a on P and c on P and p on Q and a on P and c on Q and p on Q and p on Q and c on Q and p on Q and a on Q and c on Q and p on Q and a on Q and a on Q and a on Q and a on Q and a on Q and c on Q and a on Q ; assume that T is \hbox 4 -such that F is closed and F is closed and for n being Nat st n in dom F ex F being Subset-Family of T st F is finite-ind & ind F <= 0 & ind F <= 0 ; for g1 , g2 st g1 in ]. r - g2 , r .[ & g2 in ]. r - g2 , r + g2 .[ holds |. f . g1 - f . g2 .| <= ( ( g1 - f ) / ( r - g2 ) ) * ( ( g1 - f ) / ( r - g2 ) ) ( ( - ( x - z ) ) / ( ( x - z ) * ( y - z ) ) ) * ( ( x - z ) / ( ( x - z ) * ( y - z ) ) ) = ( ( - ( x - z ) ) / ( ( x - z ) * ( y - z ) ) ) * ( x - z ) ; F . i = F /. i .= 0. R + r2 .= ( b |^ ( n + 1 ) ) * ( ( n + 1 ) -tuples_on ( n + 1 ) ) .= <* ( n + 1 ) -tuples_on ( n + 1 ) -tuples_on ( n + 1 ) , ( n + 1 ) -tuples_on ( n + 1 ) ) ; ex y being set , f being Function st y = f . n & dom f = NAT & f . 0 = A ( ) & for n holds f . ( n + 1 ) = R ( n , f . n , f . n ) ; func f (#) F -> FinSequence of V means : Def1 : len it = len F & for i be Nat st i in dom it holds it . i = F /. i * f /. ( F /. i ) * F /. ( F /. i ) ; { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } = { x1 , x2 , x3 , x4 } \/ { x4 , x5 , x5 , x5 , x5 } \/ { x5 , x5 , x5 } \/ { x5 , x5 , x5 , x5 } ; for n being Nat for 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 & ( ( S , l ) . e = e ) & ( ( S , l ) . e = e ) & ( S , l ) . e = e ) & ( S , l ) . e = e ) ; consider P being FinSequence of GW2 such that pI = Product P and for i st i in dom P ex t being Element of the carrier of G st P . i = t & t . i = ( t . i ) * ( t . i ) ; for T1 , T2 being strict non empty TopSpace , P being Basis of T1 , Q being Basis of T2 st the carrier of T1 = the carrier of T2 & P = Q holds P is Basis of T1 & P is Basis of T2 & P c= Q assume that f is_is_is_len pdiff1 f and r (#) pdiff1 ( f , 3 ) is_partial_differentiable_in u0 , 2 and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 , 2 ) = r * pdiff1 ( f , u0 ) . u0 and partdiff ( r (#) pdiff1 ( f , 3 ) , u0 , 2 ) = r * pdiff1 ( f , u0 ) . u0 ; defpred P [ Nat ] means for F , G being FinSequence of ExtREAL for G being Permutation of ( Seg $1 ) st len F = $1 & G = F & not G = F * s holds Sum ( F ) = Sum ( G ) & Sum ( F ) = Sum ( G ) ; ex j st 1 <= j & j < width GoB f & ( GoB f ) * ( 1 , j ) `2 <= s & s <= ( GoB f ) * ( 1 , j + 1 ) `2 or s <= ( GoB f ) * ( 1 , j + 1 ) `2 & s <= ( GoB f ) * ( 1 , j + 1 ) `2 & s <= ( GoB f ) * ( 1 , j + 1 ) `2 ; defpred U [ set , set ] means ex FF be Subset-Family of T st $1 = FF & union FF is open & union FF is open & union FF c= $1 & union FF c= $1 & union FF c= union the topology of T & union FF c= $1 & union FF c= $1 & union FF c= $1 & union FF c= $1 & union FF c= $1 & union FF c= $1 ; for p4 being Point of TOP-REAL 2 st LE p4 , p , P & LE p1 , p2 , P & LE p2 , p , P holds LE p4 , p , P or LE p2 , p , P & LE p1 , p2 , P & LE p2 , p , P & LE p2 , p , P & LE p4 , p , P & LE p2 , p , P & LE p4 , p , P & LE p2 , p , P & LE p4 , p , P & LE p2 , p , P & LE p2 , p , P & LE p2 , p , P & LE p2 , p , P & LE p2 , p , P & LE p2 , p , P & LE p2 , p , P & LE p2 , p , P & LE p2 , p , P & LE p2 , p , P & LE p1 , p , P & LE p1 , p , P & LE p1 , p , P & LE p2 f in D & for y st g in S holds for x st g . y <> f . y holds x = y & g in D & f . x = f . ( All ( x , H ) ) implies f in D & f . x = f . ( All ( x , H ) ) ex 8 being Point of TOP-REAL 2 st x = 8 & ( ( ( - 1 ) * ( ( - 1 ) * ( - 1 ) ) / ( 1 + ( - 1 ) ) ) ) * ( ( - 1 ) * ( ( - 1 ) * ( - 1 ) ) ) & ( ( - 1 ) * ( ( - 1 ) * ( - 1 ) ) / ( 1 + ( - 1 ) ) ) ) * ( 1 + ( - 1 ) ) ) = ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( 1 + ( 1 + ( 1 + ( 1 + ( 1 + ( 1 + ( 1 + ( - 1 ) ) ) / ( 1 + ( 1 + ( ( - 1 ) * ( ( - 1 ) * ( 1 + assume for d7 being Element of NAT st d7 <= max ( ( n + 1 ) -d, ( n + 1 ) -d7 ) holds s1 . ( ( n + 1 ) -d7 ) = s2 . ( ( n + 1 ) -d7 ) . ( ( n + 1 ) -d7 ) ; assume that s <> t and s is Point of Sphere ( x , r ) and s is not Point of Sphere ( x , r ) and ex e being Point of E st { e } = Ball ( x , t ) /\ Ball ( s , r ) & Ball ( e , s ) c= Ball ( x , r ) ; given r such that 0 < r and for s st 0 < s ex x1 be Point of CNS st x1 in dom f & ||. x1 - x0 .|| < s & |. f /. x1 - f /. x0 .| < r & |. f /. x1 - f /. x0 .| < r ; ( p | x ) | ( ( 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 * cos . x ) / ( sin . x + cos . x ) ^2 and sin . x = - 1 & cos . x = 1 ; assume that i in dom A and len A > 1 and B c= the carrier of K and B c= the carrier of K and for i st i in dom A & B <> {} holds A * ( i , j ) = A * ( i , j ) and A * ( i , j ) = B * ( i , j ) ; for i being non zero Element of NAT st i in Seg n holds ( i divides n or i = n or i divides n ) & ( i divides n implies h . i = <* 1. F_Complex , 0. F_Complex *> or i divides n & h . i = 1. F_Complex ) ( ( b1 'imp' b2 ) '&' ( c1 'or' c2 ) '&' ( ( b1 'or' b2 ) '&' ( c1 'or' c2 ) '&' 'not' ( a2 '&' c2 ) '&' 'not' ( a1 '&' a2 '&' 'not' ( b1 '&' b2 ) ) ) ) '&' 'not' ( a2 '&' 'not' ( b1 '&' b2 ) '&' 'not' ( a2 '&' b2 ) '&' 'not' ( a1 '&' a2 '&' 'not' ( b1 '&' b2 ) ) ) '&' 'not' ( a2 '&' 'not' ( b1 '&' b2 ) '&' 'not' ( b2 '&' c2 ) ) ; assume that for x holds f . x = ( ( cot * ( sin - cos ) ) `| Z ) . x and for x st x in Z holds ( ( ( cot * ( sin - cos ) ) `| Z ) . x = - cos . ( x - h / ( sin . x ) ) ) and for x st x in Z holds ( ( ( cot * ( sin - cos ) ) `| Z ) . x = - cos . ( x - h / ( sin . x ) ) ; consider R8 , I-8 be Real such that R8 = Integral ( M , Re ( F . n ) ) and I-8 = Integral ( M , Im ( F . n ) ) and Integral ( M , Im ( F . n ) ) = Integral ( M , Im ( F . n ) ) and Integral ( M , Im ( F . n ) ) = Integral ( M , Im ( F . n ) ) ; ex k being Element of NAT st k = k & 0 < d & for q being Element of product G st q in X & ||. q- partdiff ( f , x , k ) .|| < r holds ||. partdiff ( f , x , k ) - partdiff ( f , x , k ) .|| < r ; x in { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } iff x in { x1 , x2 , x3 , x4 } \/ { x4 , x5 , x5 , x5 } or x in { x1 , x2 , x3 , x4 , x5 , x5 } G * ( j , i ) `2 = G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 .= G * ( 1 , i ) `2 ; f1 * p = p .= ( ( the Arity of S1 ) * ( the Arity of S2 ) ) . o .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) .= ( the Arity of S1 ) . ( ( the Arity of S1 ) . o ) ; func tree ( T , P , T1 , T1 ) -> FinSequence means : : : for q st q in it holds q in P & p = q or p = q or p = p & q = p or p = q & p = q ; F /. ( k + 1 ) = F . ( k + 1 -' 1 ) .= F^2 ( p . ( k + 1 -' 1 ) , k + 1 -' 1 ) .= F^2 ( p . ( k + 1 -' 1 ) , k + 1 -' 1 ) .= F^2 ( p . ( k + 1 -' 1 ) , k + 1 ) .= F^2 ( p . ( k + 1 -' 1 ) , k + 1 ) ; for A , B , C being Matrix of K st len B = len C & width B = width C & len B = width C & len A > 0 & width A > 0 & len B > 0 & width A > 0 & width A > 0 & width A > 0 & width A > 0 & width A > 0 & A + B = 0 holds A * ( B + C ) = B * ( A + B ) seq . ( k + 1 ) = 0. F_Complex + seq . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) .= ( Partial_Sums ( seq ) ) . ( k + 1 ) .= ( Partial_Sums ( 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 z = [ x , y ] ; defpred P [ Element of NAT ] means for f st len f = $1 holds ( VAL g ) . ( f . ( k + 1 ) ) = ( VAL g ) . ( f . ( k + 1 ) ) '&' ( VAL g ) . ( f . ( k + 1 ) ) ; assume that 1 <= k and k + 1 <= len f and f /. [ i , j ] in Indices G and [ i + 1 , 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 cn < 1 and ( q `1 / |. q .| - cn ) / ( 1 + cn ) >= 0 and ( p `1 / |. q .| - cn ) / ( 1 + cn ) >= 0 and ( p `1 / |. q .| - cn ) / ( 1 + cn ) >= 0 and ( p `1 / |. p .| - cn ) / ( 1 + cn ) >= 0 ; for M being non empty dist , x being Point of M , f being Function of M , M st x = x `1 holds ex x being Point of M st f . x = Ball ( x `1 , 1 / ( n + 1 ) ) & f . x = Ball ( x `1 / ( n + 1 ) ) defpred P [ Element of omega ] means f1 is_differentiable_on Z & f2 is_differentiable_on Z & ( for x st x in Z holds ( ( f1 - f2 ) `| Z ) . x = a * x + b * x ) & ( f1 - f2 ) `| Z = ( f1 - f2 ) `| Z ) . ( x + 1 ) ; defpred P1 [ Nat , Point of CNS ] means ( $1 in Y & ||. f /. $2 - f /. ( $1 + 1 ) .|| < r ) & ( ||. f /. $2 - f /. ( $1 + 1 ) .|| < r implies ||. f /. ( $1 + 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 * ( 2 * n0 + 2 * ( n + 2 ) ) ) * ( 2 * ( n + 2 ) ) = ( 1 - 2 * ( 2 * ( n + 2 ) ) ) * ( 2 * ( n + 2 ) ) .= 1 * ( 2 * ( n + 2 ) ) ; defpred P [ Nat ] means for G being non empty strict finite strict strict let G being non empty RelStr st G is non empty for H being strict strict strict strict non empty RelStr st H = ( the carrier of G ) \/ the carrier of H & the carrier of H = { H } holds H = G ; assume that not f /. 1 in Ball ( u , r ) and 1 <= m and m <= len ( f . ( i + 1 ) ) and LSeg ( f . ( i + 1 ) , r ) /\ Ball ( u , r ) <> {} and not m <= len ( f . ( i + 1 ) ) and not m <= len ( f . ( i + 1 ) ) ; defpred P [ Element of NAT ] means ( Partial_Sums ( cos * ( ]. - r , r .[ ) ) ) . $1 = ( Partial_Sums ( cos * ( ]. - r , r .[ ) ) ) . ( 2 * $1 ) ) . ( 2 * $1 ) ; for x being Element of product F holds x is FinSequence of G & ( for i being set st i in dom F holds x . i in I ) & for i being set st i in dom F holds x . i in ( the carrier of F ) . i ) & for i being set st i in dom F holds x . i in ( the carrier of F ) . i ( x " ) |^ ( n + 1 ) = ( x " ) * x " .= ( x * x ) " .= ( x * x ) " .= ( x * x ) " .= ( x * x ) " .= ( x * x ) " .= ( x * x ) " ; DataPart Comput ( P +* ( a , I ) , Initialized s ) = DataPart Comput ( P +* I , ( LifeSpan ( P +* I , P +* I ) ) , LifeSpan ( P +* I , s ) + 3 ) .= DataPart Comput ( P +* I , ( Initialize s ) , LifeSpan ( P +* I , P +* I ) ) ; given r such that 0 < r and ]. x0 , x0 + r .[ c= dom f1 /\ dom f2 and for g st g in ]. x0 , x0 + r .[ /\ dom f1 holds f1 . g <= f2 . g and for g st g in ]. x0 , x0 + r .[ /\ dom f2 & g <= x0 + r & g <= x0 + r & g in dom f1 /\ dom f2 ; assume that X c= dom f1 /\ dom f2 and f1 | X is continuous and f2 | X is continuous and ( for r st r in X /\ dom f2 holds ( f1 - f2 ) | X is continuous ) and ( f1 - f2 ) | X is continuous and ( f1 - f2 ) | X is continuous and ( f1 - f2 ) | X is continuous ; for L being continuous complete LATTICE for l being Element of L ex X being Subset of L st l = sup X & for x being Element of L st x in X holds x is finite & x is finite & x is finite & x is finite & x is finite Support ( e *' p ) in { Support ( m *' p ) where m is Nat : ex p being Polynomial of n , L st p in Support ( m *' p ) & ( p . ( i + 1 ) ) = p . ( i + 1 ) & ( p . ( i + 1 ) ) = p . ( i + 1 ) ) ; ( f1 - f2 ) /. ( lim s1 ) = lim ( f1 /* s1 ) - lim ( f2 /* s1 ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) .= lim ( ( f1 /* s1 ) - ( f2 /* s1 ) ) ; ex p1 being Element of CQC-WFF ( Al ) st F . p = g `1 & for g being Function of [: CQC-WFF ( Al ) , D ( ) :] , D ( ) st P [ g , f . ( len f ) ) holds P [ g , f . ( len f ) ] ; ( mid ( f , i , len f -' 1 ) ^ <* f /. j *> ) /. j = ( mid ( f , i , len f -' 1 ) ^ <* f /. j *> ) /. j .= ( mid ( f , i , len f -' 1 ) ^ <* f /. j *> ) /. j .= f /. ( j + 1 ) .= f /. ( j + 1 ) ; ( ( p ^ q ) ^ r ) . ( len p + k ) = ( ( p ^ q ) ^ r ) . ( len p + k ) .= ( ( p ^ q ) ^ r ) . ( len p + k ) .= ( ( p ^ q ) ^ r ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( k + 1 ) ; len mid ( D2 , D1 , indx ( D2 , D1 , j ) + 1 ) = indx ( D2 , D1 , j ) + 1 .= indx ( D2 , D1 , j ) + 1 .= indx ( D2 , D1 , j ) + 1 .= indx ( D2 , D1 , j ) + 1 ; x * y * z = Mz . ( x * y , z ) .= ( x * y ) * ( x * z ) .= ( x * y ) * ( x * z ) .= ( x * y ) * ( x * z ) .= ( x * y ) * ( x * z ) ; v . <* x , y *> - ( <* x0 , y0 *> - ( x - y ) * i ) = partdiff ( v , ( x - y ) * ( x - y ) + ( proj ( 1 , 1 ) * ( x - y ) ) ) + ( proj ( 1 , 1 ) * ( x - y ) ) ; i * i = <* 0 * ( - 1 ) * ( 0 - 1 ) , 0 * 0 + 0 * 0 , 0 * 0 + 0 * 0 + 0 * 0 + 0 * 0 , 0 * 1 + 0 * 0 + 0 * 0 + 0 * 1 + 0 * 0 , 0 * 1 + 0 * 0 + 0 * 1 + 0 * 0 + 0 * 1 + 0 * 0 + 0 * 0 , 0 * 1 + 0 * 0 + 0 * 1 + 0 * 0 , 0 * 1 * 0 + 0 * 0 + 0 * 0 + 0 * 1 + 0 * 1 + 0 * 0 + 0 * 0 * 1 * 0 + 0 * 1 + 0 * 1 + 0 * 1 + 0 * 1 + 0 * 1 + 0 * 1 + 0 * 1 + 0 * 1 , 1 * 1 + 0 * 1 + 0 * 1 + 0 * 1 + 0 * 1 + 0 * 1 * 1 + 0 * 1 * 1 + 0 * 1 + 0 * 1 + 0 * 1 + 0 * 1 * 1 + 0 * 1 * 1 + 0 * 1 + 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 ) + Sum ( L (#) F2 ) .= Sum ( L (#) F1 ) + Sum ( L (#) F2 ) ; ex r be Real st for e be Real st 0 < e ex Y1 be finite Subset of X st Y1 is non empty & for Y be finite Subset of X st Y c= Y & Y c= Y holds |. ( - lower_bound ( X , Y ) ) .| < r ; ( GoB f ) * ( i , j ) `1 = f /. ( k + 2 ) & ( GoB f ) * ( i , j + 1 ) `1 = f /. ( k + 2 ) or ( GoB f ) * ( i , j + 1 ) `1 = f /. ( k + 2 ) or ( GoB f ) * ( i , j + 1 ) `1 = f /. ( k + 2 ) ) `1 ; ( ( - cos - sin ) / ( sin - cos ) ^2 ) = ( ( - sin ) / ( sin - cos ) ^2 ) / ( sin - cos ^2 ) .= ( ( - 1 ) / ( sin - cos ^2 ) ) / ( sin - cos ^2 ) .= ( ( - 1 ) / ( sin - cos ^2 ) ) / ( sin - cos ^2 ) .= ( ( - 1 ) / ( sin - cos ^2 ) ) / ( sin - cos ^2 ) ; ( - sqrt ( b - sqrt ( b - a ) ^2 ) ) / 2 * a < 0 & ( - sqrt ( b - a ) ^2 + sqrt ( b - a ^2 ) ) / 2 * a < 0 or ( - sqrt ( b - a ) ^2 + sqrt ( b - a ^2 ) ) / 2 * a ) ; assume that ex_inf_of uparrow "\/" ( X /\ C ) , L and ex_sup_of X , L and sup ( ( uparrow x ) /\ C ) = "/\" ( ( uparrow x ) /\ C ) and not "\/" ( ( uparrow x ) /\ C ) in C and not "\/" ( ( uparrow x ) /\ C , L ) in C ; ( for j holds ( j = i implies j = i ) implies ( j = i or j = i ) ) & ( ( j = i implies j = i or j = i ) implies ( j = i or j = i ) ) & ( j = i implies j = i ) implies ( j = i implies j = i ) )