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 ; b ; X ; b in D ; x = e ; let m ; h is onto ; N in K ; let i ; j = 1 ; x = u ; let n ; let k ; y in A ; let x ; let x ; m c= y ; F is one-to-one ; let q ; m = 1 ; 1 < k ; G is prime ; b in A ; d divides a ; i < n ; s <= b ; b in B ; let r ; B is one-to-one ; R is total ; x = 2 ; d in D ; let c ; let c ; b = Y ; 0 < k ; let b ; let n ; r <= b ; x in X ; i >= 8 ; let n ; let n ; y in f ; let n ; 1 < j ; a in L ; C is boundary ; a in A ; 1 < x ; S is finite ; u in I ; z << z ; x in V ; r < t ; let t ; x c= y ; a <= b ; m in NAT ; assume f is prime ; not x in Y ; z = +infty ; let k be Nat ; K is being_line ; assume n >= N ; assume n >= N ; assume X is with with 0. X ; assume x in I ; q is \upharpoonright 0 ; assume c in x ; 'not' p > 0 ; assume x in Z ; assume x in Z ; 1 <= k12 ; assume m <= i ; assume G is prime ; assume a divides b ; assume P is closed ; O > 0 ; assume q in A ; W is not bounded ; f is means : Def1 : f is one-to-one ; assume A is 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 <= 10 ; let X be set ; let X be set ; let y be element ; let x be element ; P [ 0 ] let E be set , X be set ; let C be category ; let x be element ; let k be Nat ; let x be element ; let x be element ; let e be element ; let x be element ; P [ 0 ] let c be element ; let y be element ; let x be element ; let a be Real ; let x be element ; let X be element ; P [ 0 ] let x be element ; let x be element ; let y be element ; r in REAL ; let e be element ; n1 is invertible ; Q halts_on s ; x in \rbrack ; M < m + 1 ; T2 is open ; z in b *^ a ; R2 is well-ordering ; 1 <= k + 1 ; i > n + 1 ; q1 is one-to-one ; let x be trivial set ; P3 is one-to-one ; n <= n + 2 ; 1 <= k + 1 ; 1 <= k + 1 ; let e be Real ; i < i + 1 ; p3 in P ; p1 in K ; y in C1 ; k + 1 <= n ; let a be Real , b be Real ; X |- r => p ; x in { A } ; let n be Nat ; let k be Nat ; let k be Nat ; let m be Nat ; 0 < 0 + k ; f is_differentiable_in x ; let x0 ; let E be Ordinal ; o <=' o1 ; O <> O2 ; let r be Real ; let f be FinSeq-Location ; let i be Nat ; let n be Nat ; Cl A = A ; L c= Cl L ; A /\ M = B ; let V be RealUnitarySpace , X be Subset of V ; not s in Y |^ 0 ; rng f <= w ; b "/\" e = b ; m = m1 ; t in h . D ; P [ 0 ] ; assume z = x * y ; S . n is bounded ; let V be RealUnitarySpace , X 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 ] ; a! <= r ; R [ 0 ] ; b in f .: X ; assume q = q2 ; x in [#] V ; f . u = 0 ; assume e1 > 0 ; let V be RealUnitarySpace , X be Subset of V ; s is trivial & s is trivial ; dom c = Q ; P [ 0 ] ; f . n in T ; N . j in S ; let T be complete LATTICE , X be Subset 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 ; vu < n ; Spion1 is bounded ; assume p = p2 ; len f = n ; assume x in P1 ; i in dom q ; let U1 ; pp `2 = c ; j in dom h ; let k ; f | Z is continuous ; k in dom G ; UBD C = B ; 1 <= len M ; p in right_open_halfline x ; 1 <= jj & jj <= j ; set A = Triv, B = incid; card a [= c ; e in rng f ; cluster B \oplus A -> empty ; H is \rm has \textit { x } ; assume x0 <= m ; T is increasing implies T is increasing e1 <> e2 ; Z c= dom g ; dom p = X ; H is proper of G ; i + 1 <= n ; v <> 0. V ; A c= Affin A ; S c= dom F ; m in dom f ; let X0 be set ; c = sup N ; R is connected implies union M 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 , A :] ; C c= Cmax C ; m1 <> {} & m2 <> {} ; let x be Element of Y ; let f be \rbrack , g be Function ; not n in Seg 3 ; assume X in f .: A ; assume that p <= n and p <= m ; assume not u in { v } ; d is Element of A ; A |^ b misses B ; e in v `2 ; - y in I ; let A be non empty set , X be Subset of REAL ; P0 = 1 ; assume r in F . k ; assume f is simple ; let A be exists exists exists set ; rng f c= NAT ; assume P [ k ] ; f6 <> {} & f6 <> {} ; let o be Ordinal ; assume x is sum of squares ; assume not v in { 1 } ; let I1 , I2 ; assume that 1 <= j and j < l ; v = - u ; assume s . b > 0 ; d1 in X ; assume t . 1 in A ; let Y be non empty TopSpace , X be non empty Subset of Y ; assume a in ]. s , t .[ ; let S be non empty Poset ; a , b // b , a ; a * b = p * q ; assume x , y are_the space of X ; assume x in [#] ( f | X ) ; [ a , c ] in X ; m-14 <> {} ; M + N c= M + M ; assume M is \llangle Mhho , M ; assume f is with_inbriis 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 = k2 ; m + 1 < n + 1 ; s in S \/ { s } ; n + i >= n + 1 ; assume Re ( y ) = 0 ; k1 <= j1 & j1 <= j2 ; f | A is continuous ; f . x `1 <= 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 , n = max A ; [ x , x ] in R ; assume x in succ 0 ; a in sup phi ; CX ; q2 c= C1 & q2 c= C2 ; a2 < c2 & c2 < c2 ; s2 is 0 -started ; IC s = 0 ; 6 = s3 & 7 = s3 ; let V ; let x , y be element ; let x be Element of T ; assume a in rng F ; x in dom T ` ; let S be MSAlgebra over L ; y " <> 0 ; y " <> 0 ; 0. V = uw ; y2 , y , w , y is_collinear ; R8 ; let a , b be Real , r be Real ; let a be Object of C ; let x be Vertex of G ; let o be Object of C , a , b be Object of C ; r '&' q = P \lbrack l , l .] ; let i , j ; let s be State of A , t be Element of S ; 4 . n = N ; set y = ( x `1 ) / ( x `2 ) ; [: the carrier of S , the carrier of S :] c= dom g ; l . 2 = y1 ; |. g . y .| <= r ; f . x in Cx0 ; V 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 & f | Z is continuous ; G = 0 .--> goto 0 ; let A be Subset of X ; assume that x0 is dense and A is dense ; |. f . x .| <= r ; let x be Element of R ; let b be Element of L ; assume x in W-19 ; P [ k , a ] ; let X be Subset of L ; let b be Object of B ; let A , B be category ; set X = Vars C , Y = Vars C ; let o be OperSymbol of S ; let R be connected non empty Poset ; n + 1 = succ n ; x9 c= [: X1 , X2 :] ; dom f = C1 & rng f c= C2 ; assume [ a , y ] in X ; Re ( seq . n ) is convergent ; assume a1 = b1 & a2 = b2 ; A = sInt A ; a <= b or b <= a ; n + 1 in dom f ; let F be Instruction of S , s be Element of S ; assume r2 > x0 & x0 < r2 ; let Y be non empty set , X be non empty set ; 2 * x in dom W ; m in dom ( g2 | X ) ; n in dom g1 & g1 . n = g1 . n ; k + 1 in dom f ; not the still of { s } is finite ; assume x1 <> x2 & x1 <> x3 ; v2 in V1 & v2 in V1 ; not [ b `1 , b `2 ] in T ; ij + 1 = i ; T c= NetUniv ( T ) ; ( l - 1 ) * ( l - 1 ) = 0 ; let n be Nat ; ( t `2 ) ^2 = r ; Ax0 is_integrable_on M & Ax0 is_integrable_on M ; set t = Bottom t ; let A , B be real-membered set ; k <= len G + 1 ; [: C , D :] misses [: C , D :] ; Product ( seq ) is non empty ; e <= f or f <= e ; cluster -> non empty for normal Real_Sequence ; 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 v-4 is Cauchy and lim vK = 0 ; IC Comput ( P3 , s3 , k ) = 0 ; k in N or k in K ; F1 \/ F2 c= F ; Int G1 <> {} & Int G2 <> {} ; ( z `2 ) ^2 = 0 ; p1 <> p1 & p1 <> p2 & p1 <> p2 ; assume z in { y , w } ; MaxADSet ( a ) c= F ; ex_sup_of ]. s , t .[ , T ; f . x <= f . y ; let T be up-complete non empty reflexive RelStr ; q |^ m >= 1 ; a >= X & b >= Y ; assume <* a , c *> <> {} ; F . c = g . c ; G is one-to-one implies G is onto onto ; A \/ { a } c= B ; 0. V = 0. Y .= 0. Y ; let I be `| SCM , S be Instruction of S ; f-24 . x = 1 ; assume z \ x = 0. X ; C4 = 2 to_power n ; let B be sequence of Sigma ; assume X1 = p .: D & X2 = p .: D ; n + l in NAT ; f " P is compact ; assume x1 in REAL & x2 in REAL ; p1 = ( K + 1 ) * ( K + 1 ) ; M . k = <*> REAL ; phi . 0 in rng phi ; MMMA is closed ; assume x0 <> 0. L & y0 <> 0. L ; n < NN . k ; 0 <= seq . 0 & seq . 0 <= seq . 0 ; - q + p = v ; { v } is Subset of B ; set g = f /. 1 , h = f /. len f ; the carrier of R is stable of R ; set RR = Vertices R , S = the carrier of R ; pp c= P3 & q c= P3 ; x in [. 0 , 1 .] ; f . y in dom F ; let T be Scott Scott Scott Scott Scott Scott Scott from S ; inf the carrier of S in S ; \neg a = downarrow b & T = downarrow b ; P , C , K is_collinear ; assume x in LSeg ( s , r ) ; 2 |^ i < 2 |^ m ; x + z = x + z + q ; x \ ( a \ x ) = x ; ||. x-y - x .|| <= r ; assume that Y c= field Q and Y <> {} and Y <> {} ; a , b are_connected & b , a are_connected ; assume a in [: A . i , A . i :] ; k in dom ( q | k ) ; p is FinSequence of S ; i -' 1 = i-1 - 1 ; f | A is one-to-one ; assume x in f .: [: X , Y :] ; i2 - i1 = 0 & i2 - i1 = 0 ; j2 + 1 + 1 <= i2 ; g " * a in N ; K <> { [ {} , {} ] } ; cluster strict strict for commutative Ring ; |. q .| ^2 > 0 ; |. p3 .| = |. p .| ; s2 - s1 > 0 & s2 - s1 > 0 ; assume x in { Gik } ; W-min C in C & W-min C in C ; assume x in { Gik } ; assume i + 1 = len G ; assume i + 1 = len G ; dom I = Seg n & dom I = Seg n ; assume that k in dom C and k <> i ; 1 + 1-1 <= i + 1-1 ; dom S = dom F ; let s be Element of NAT ; let R be ManySortedSet of A ; let n be Element of NAT ; let S be non empty non void non empty non void void void void \mathbb of S ; let f be ManySortedSet of I ; let z be Element of COMPLEX , x be Element of COMPLEX ; u in { \hbox { \boldmath $ g } } ; 2 * n < ( 2 * n ) + 1 ; let x , y be set ; BW c= [: V , V :] ; assume I is_closed_on s , P ; U = U U & U is open ; M /. 1 = z /. 1 ; x9 = x9 & y9 = y9 & x9 = y9 ; i + 1 < n + 1 + 1 ; x in { {} , <* 0 *> } ; ( f | X ) . x <= ( f | X ) . x ; let l be Element of L ; x in dom ( F | d ) ; let i be Element of NAT ; r8 is ( len r ) -element ; assume <* o2 , o *> <> {} ; s . x |^ 0 = 1 ; card ( K + 1 ) in M ; assume that X in U and Y in U ; let D be Subset-Family of Omega ; set r = { q + 1 } ; y = W . ( 2 * x ) ; assume dom g = cod f & cod g = cod f ; let X , Y be non empty TopSpace , f be Function of X , Y ; x in A -- B ; |. <*> A .| . a = 0 ; cluster strict for non empty RelStr ; a1 in B . s1 & a2 in B . s1 ; let V be finite VectSp of F , v be Element of V ; A * B on B * A ; f-3 = ( NAT --> 0 ) +* ( NAT --> 0 ) ; let A , B be Subset of V ; z1 = P1 . j & z2 = P1 . j ; assume f " P is closed ; reconsider j = i as Element of M ; let a , b be Element of L ; assume q in A \/ ( B "\/" C ) ; dom ( F * C ) = o ; set S = ( ( id X ) | X ) | X ; z in dom ( A --> y ) ; P [ y , h . y ] ; { x0 } c= dom f & f | X is continuous ; let B be non-empty ManySortedSet of I , A be ManySortedSet of I ; sqrt 2 < Arg z & Im z < Im z ; reconsider z1 = 0 as Nat ; LIN a , d , c ; [ y , x ] in [: I , I :] ; ( Q ) * ( 1 , 3 ) = 0 ; set j = x0 div m , n = x0 mod m ; assume a in { x , y , c } ; j2 - ( j + 1 ) > 0 ; I \! \mathop { \rm \hbox { - } phi } = 1 ; [ y , d ] in F-8 ; let f be Function of X , Y ; set A2 = sqrt B , C = sqrt C , D = sqrt C ; s1 , s2 , s1 , s2 , s3 , s3 , s2 , s3 , s3 be Element of R ; j1 -' 1 = 0 & j1 - 1 = 0 ; set m2 = 2 * n + j ; reconsider t = t as bag of n ; I2 . j = m . j ; i |^ s , n are_relative_prime ; set g = f | [: D , D2 :] ; assume that X is lower and 0 <= r ; ( p1 `1 ) ^2 = 1 ^2 + ( p1 `2 ) ^2 ; a < ( p3 `1 ) ^2 + ( p3 `2 ) ^2 ; L \ { m } c= UBD C ; x in Ball ( x , 10 ) ; not a in LSeg ( c , m ) ; 1 <= i1 -' 1 & i1 -' 1 <= i1 -' 1 ; 1 <= i1 -' 1 & i1 -' 1 <= i1 -' 1 ; i + i2 <= len h & i + 1 <= len h ; x = W-min ( P ) & x = W-min ( P ) ; [ x , z ] in X \times Z ; assume y in [. x0 , x .] ; assume p = <* 1 , 2 , 3 , 4 *> ; len <* A1 *> = 1 & len <* A1 *> = 1 ; set H = h . ( g . ( h . ( g . ( h . ( h . ( g . ( h . ( h . ( g . ( h . ( h . card b * a = |. a .| ; Shift ( w , 0 ) |= v ; set h = h2 ** h1 , h1 = h2 ** h2 , h2 = h2 (#) h1 , h2 = h2 (#) h2 , h2 = h2 (#) h1 , h2 = h2 (#) h2 , h2 assume x in X1 /\ X2 & x in X1 /\ X2 ; ||. h .|| < d1 & ||. h .|| < d1 ; not x in the carrier of f & not x in the carrier of f ; f . y = F ( y ) ; for n holds X [ n ] ; k -' l = kl ; <* p , q *> /. 2 = q ; let S be Subset of the carrier of Y ; let P , Q be Initialize of s ; Q /\ M c= union ( F | M ) f = b * canFS ( S ) ; let a , b be Element of G ; f .: X <= f . sup X ; let L be non empty reflexive RelStr , X be Subset of L ; S-20 is x -to_power i let r be non positive Real ; M , v |= All ( x , y ) ; v + w = 0. ( Z ) ; P [ len F ] & P [ len F ] ; assume InsCode i = 8 & InsCode i = 8 ; the zero of M = 0 & the carrier of M = the carrier of M ; cluster z * seq -> summable ; let O be Subset of the carrier of C ; ||. f .|| | X is continuous ; x2 = g . ( j + 1 ) ; cluster -> non empty for Element of S ; reconsider l1 = ll as Nat ; v4 is Vertex of r2 & v4 is Vertex of G ; T3 is SubSpace of T2 & the TopStruct of T2 = T2 | A ; Q1 /\ Q1 <> {} & ( for x st x in Q holds Q [ x ] ) implies Q /\ Q <> {} let k be Nat ; q " is Element of X ; F . t is set of M , M ; assume that n <> 0 and n <> 1 ; set e1 = EmptyBag n , e2 = EmptyBag n , T = EmptyBag n ; let b be Element of Bags n ; assume for i holds b . i is commutative ; x is root of ( p `2 ) / ( p `2 ) ; not r in ]. p , q .[ ; let R be FinSequence of REAL ; S7 does not destroy b1 & not LIN b1 , b2 , b1 IC SCM R <> a & IC SCM R <> a ; |. |[ x , y ]| .| >= r ; 1 * seq = seq . ( n + 1 ) * seq . ( n + 1 ) ; let x be FinSequence of NAT ; let f be Function of C , D ; for a holds 0. L + a = a IC s = s . NAT .= n ; H + G = F- G ; Cx . x = x2 & Cx . x = x2 ; f1 = f . ( f . x ) .= f2 . ( f . x ) ; Sum <* p . 0 *> = p . 0 ; assume v + W = v + u + W ; { a1 } = { a2 } & { a2 } = { a2 } ; a1 , b1 _|_ b , a ; d2 , o _|_ o , a3 & a3 , o _|_ o , o ; I1 is_reflexive & I2 is_reflexive implies C is reflexive [: I1 , I2 :] is antisymmetric & [: I2 , I2 :] is antisymmetric ; sup rng ( H1 | ( i + 1 ) ) = e & e in rng ( H1 | ( i + 1 ) ) ; x = ( a * a9 ) * ( a * a9 ) ; |. p1 .| ^2 >= 1 ^2 + ( p1 `2 ) ^2 ; assume j2 -' 1 < j2 & j2 -' 1 < j2 -' 1 ; rng s c= dom f1 /\ dom f2 ; assume support a misses support b & not b in support b ; let L be associative non empty doubleLoopStr , F be FinSequence of L ; s " + 0 < n + 1 ; p . c = ( f . 1 ) . c ; R . n <= R . ( n + 1 ) ; Directed ( I1 , I2 ) = I1 +* I2 ; set f = + ( x , y , r ) ; cluster Ball ( x , r ) -> bounded ; consider r being Real such that r in A ; cluster -> non empty for Function of NAT , NAT ; let X be non empty directed Subset of S ; let S be non empty full SubRelStr of L ; cluster <* I1 . N , I2 . N *> -> complete non trivial ; sqrt 1 / a = a / a ^2 .= a ; ( q . {} ) `1 = o & ( q . {} ) `2 = o ; n - ( i -' 1 ) > 0 ; assume sqrt ( 1 ^2 ) <= t ^2 ; card B = k + 1 - 1 ; x in union rng ( f | ( X \/ Y ) ) ; assume x in the carrier of R & y in the carrier of R ; d ; f . 1 = L . F ; the carrier of G = { v } & { v } = { v } ; let G be finite connected _Graph ; e , v2 , v , w , y ; c . ( i1 + 1 ) in rng c ; f2 /* ( q ^\ k ) is divergent_to+infty ; set z1 = - ( z1 + z2 ) , z2 = - ( z1 + z2 ) , z2 = - ( z1 + z2 ) , z2 = - ( z2 + z2 ) , z1 = - ( z1 assume w is llof S , G ; set f = p \! \mathop { t } , g = p \! \mathop { t } , h = p \! \mathop { t } , n = t \! \mathop { t } , n = t \! \mathop let c be Object of C ; assume ex a st P [ a ] ; let x be Element of REAL m m m -tuples_on REAL ; let I1 be Subset-Family of X ; reconsider p = p as Element of NAT ; let v , w be Point of X ; let s be State of SCM+FSA , I be Program of SCM+FSA ; p is FinSequence of NAT , q be FinSequence of NAT ; stop I c= P & card I = card J + 2 ; set ci = f /. i , fi = f /. i , fi = f /. i , , fj = f /. i , , f = g /. i , i = g /. i w ^ t ^ t ^ <* s *> ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ W1 /\ W = W1 /\ W2 /\ W2 .= W1 /\ W2 ; f . j is Element of J . j ; let x , y be Element of T2 , T ; ex d st a , b // b , d ; a <> 0 & b <> 0 implies c <> 0 ord ( x ) = 1 & x is positive ; set g2 = lim ( seq , n ) , g1 = lim ( seq , n ) ; 2 * x >= 2 * sqrt ( 1 + x ^2 ) ; assume ( a 'or' c ) . z <> TRUE ; f (*) g in Hom ( c , c ) ; Hom ( c , c + d ) <> {} ; assume 2 * Sum ( q | m ) > m ; L1 . ( F1 . ( F . k ) ) = 0 ; ( R1 \/ R2 ) \/ R1 = R2 \/ R2 .= R2 ; ( ( sin * cos ) `| Z ) . x <> 0 ; ( ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( o1 in ( X1 /\ X2 ) /\ ( X2 /\ X2 ) ; e , v2 , v , w , y ; r3 > sqrt ( 1 - 0 ) * ( 1 - 0 ) ; x in P .: ( F -ideal ) ; let J be closed Ideal of R ; h . p1 = f2 . O & h . O = g2 . O ; Index ( p , f ) + 1 <= j ; len ( q | i ) = width M & width ( q | i ) = width M ; the carrier of \mathbb K c= A & A c= A ; dom f c= union rng ( F | X ) ; k + 1 in support ( support ( n ) ) ; let X be ManySortedSet of the carrier of S ; [ x `1 , y `2 ] in ( field R ) ; i = D1 or i = D2 or i = D2 ; assume a mod n = b mod n .= b mod n ; h . x2 = g . x1 & h . x2 = f . x2 ; F c= 2 -tuples_on the carrier of X reconsider w = |. s1 .| as Real_Sequence ; sqrt ( 1 - m * r + r * r ) < p ; dom f = dom ( ( I --> finite ) +* ( J --> v ) ) ; [#] ( ( TOP-REAL 2 ) | P ) = [#] ( ( TOP-REAL 2 ) | P ) ; cluster - x -> real for number ; then { d1 } c= A ; cluster [: TOP-REAL n , TOP-REAL n :] -> finite-ind ; let w1 be Element of M ; let x be Element of dyadic ( n ) ; u in W1 & v in W2 & u in W1 /\ W2 ; reconsider y = y as Element of L2 ; N is full SubRelStr of T & N is full SubRelStr of T ; sup { x , y } = c "\/" c ; g . n = n to_power 1 .= n to_power 1 .= n to_power 1 ; h . J = EqClass ( u , J ) ; let seq be summable sequence of X ; dist ( x `1 , y ) < sqrt ( r ^2 + r ^2 ) ; reconsider mm = m - 1 as Element of NAT ; - x0 < r1 - x0 & r1 < x0 + r2 ; reconsider P = P `1 as strict Subgroup of N ; set g1 = p * ( Seg ( q `2 ) ) , g2 = p * ( q `2 ) ; let n , m , k be non zero Nat ; assume that 0 < e and f | A is bounded ; D2 . ( I2 , I2 ) in { x } ; cluster -> subcondensed for Subset of T ; let P be compact non empty Subset of TOP-REAL 2 , p1 , p2 , p1 , p3 be Point of TOP-REAL 2 ; Gik in LSeg ( \pi , 1 ) ; let n be Element of NAT , x be Element of NAT ; reconsider ST = S as Subset of T ; dom ( i .--> X ' ) = { i } ; let X be non-empty ManySortedSet of S ; let X be non-empty ManySortedSet of S ; [: { {} } , { {} } :] c= { [ {} , {} ] } ; reconsider m = m2 as Element of NAT ; reconsider d = x as Element of COMPLEX n ; let s be 0 -started State of SCMPDS , P be Initialize s ; let t be 0 -started State of SCMPDS ; b , b , x , y is_collinear & a , b , x , y is_collinear ; assume that i = n \/ { n } and j = k \/ { n } ; let f be PartFunc of X , Y ; N1 >= sqrt ( c ^2 + d ^2 ) + sqrt ( c ^2 + d ^2 ) ; reconsider tT = TT as TopSpace , y be Point of TOP-REAL n ; set q = h * p ^ <* d *> ; z2 in U . ( y1 , y2 ) /\ Q . ( y2 , z2 ) ; A |^ 0 = { <* E *> } ; len W2 = len W + len W2 + 1 ; len ( h2 ^ g2 ) in dom h2 /\ dom g2 ; i + 1 in Seg len ( s2 | Seg len s2 ) ; z in dom g1 /\ dom ( f | X ) ; assume p2 = W-min ( K ) & p1 = W-min ( K ) ; len G + 1 + 1 <= i1 + 1 + 1 ; f1 * ( f2 * f1 ) is_differentiable_in x0 & f2 * ( f1 * f2 ) . x0 > 0 ; cluster seq + ( seq + seq2 ) -> summable ; assume j in dom ( M1 /. i ) ; let A , B , C be Subset of X ; let x , y , z be Point of X , p be Point of X ; b ^2 - ( 4 * a * c ) >= 0 ; <* xny *> ^ <* 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 & x in X ; len q = len ( K (#) G ) ; s1 = Initialize ( ( Initialized s ) +* ( k + 1 ) ) .= s1 ; consider w being Nat such that q = z + w ; x ` is Element of x ` & x ` is ` ; k = 0 & n <> k or k > n ; then X is discrete implies X is closed ; for x st x in L holds x is FinSequence of REAL ||. f /. c - f /. c .|| <= r1 & r1 - f /. c <= r2 + c ; c in ]. p , q .[ & not c in { p } ; reconsider V = V as Subset of the topology of TOP-REAL n ; let N , M be being being being being being being being being being being being being being being being being being being being being being net of L ; then z >= waybelow x & z >= compactbelow x ; M = f & M = f & M = g implies M = g ( ( #Z 1 ) * ( ( #Z 1 ) * ( x + 1 ) ) ) = TRUE ; dom g = dom f /\ X .= X ; mode ^ of G is mpcluster of G ; [ i , j ] in Indices M & M * ( i , j ) = M * ( i , j ) ; reconsider s = x " as Element of H ; let f be Element of ( dom p ) -tuples_on the carrier of ( dom q ) ; F1 . ( a1 , a2 - a1 ) = G1 . ( a1 , a2 - a2 ) ; cluster circle ( a , b , r ) -> compact ; let a , b , c be Real ; rng s c= dom ( 1 / 2 ) & rng s c= dom ( 1 / 2 ) ; curry ( F-19 , k ) is additive ; set k2 = card dom B , k1 = card C , k2 = card D ; set G = coprod ( X ) ; reconsider a = [ x , s ] as Object of G ; let a , b be Element of M , c be Element of M ; reconsider s1 = s as Element of S , s2 = s as Element of T ; rng p c= the carrier of L & p . n = 0. L ; let d be Subset of the Sorts of A ; ( x | x ) = 0 iff x = 0. W & x = 0. W ; I1 in dom stop I & I2 . x = stop I ; let g be continuous Function of X | B , Y ; reconsider D = Y as Subset of TOP-REAL n ; reconsider i0 = len p1 - 1 as Integer , i2 be Integer ; dom f = the carrier of S & rng f = the carrier of S ; rng h c= union ( ( the carrier of J ) --> the carrier of L ) cluster All ( x , H ) -> proper ; d * N1 > N1 * ( N1 * N2 ) ; ]. a , b .[ c= [. a , b .] ; set g = f " [: D1 , D2 :] , f = f " [: D2 , D1 :] ; dom ( p | ( m + 1 ) ) = ( m + 1 ) ; 3 + - 2 <= k + - 2 + - 2 ; tan * ( arctan * ( arctan + arccot ) ) is_differentiable_on Z ; x in rng ( f /^ ( p .. f ) ) ; let f , g be FinSequence of D ; [: p , q :] in the carrier of [: S1 , S2 :] ; rng f " ( dom f ) = dom f /\ ( dom g ) ; ( the Target of G ) . e = v ; width G -' 1 < width G -' 1 ; assume v in rng ( S | ( E , X ) ) ; assume x is root or x is root of h ; assume 0 in rng ( ( g2 | A ) | A ) ; let q be Point of TOP-REAL 2 , a , b , c be Real ; let p be Point of TOP-REAL 2 , q be Point of TOP-REAL 2 ; dist ( O , u ) <= |. p2 .| + 1 ; assume dist ( x , b ) < dist ( a , b ) ; <* S *> is Element of the carrier of C-20 & <* S *> is Element of C-20 ; i <= len ( G /. len G -' 1 ) ; let p be Point of TOP-REAL 2 , q be Point of TOP-REAL 2 ; x1 in the carrier of ( I[01] ) | P & x2 in P ; set p1 = f /. i , p2 = f /. j ; g in { g2 : r < g2 & g2 < x0 } ; Q = [: S , T :] .: ( Q , Q ) ; ( sqrt ( 1 - 2 * ( 1 / 2 ) ) ^2 ) is summable ; - p + I c= - p + A + - B ; n < LifeSpan ( P1 , s1 ) + 1 ; CurInstr ( p1 , s1 ) = i & CurInstr ( p1 , s1 ) = i ; A /\ Cl { x } <> {} ; rng f c= ]. r , r + 1 .[ ; let g be Function of S , V ; let f be Function of L1 , L2 ; reconsider z = z as Element of InclPoset ( L ) ; let f be Function of S , T ; reconsider g = g as Morphism of c opp , b opp ; [ s , I ] in S [: A , B :] ; len ( the connectives of C ) = 4 & len ( the connectives of C ) = 2 ; let C1 , C2 be Subsignature of C ; reconsider V1 = V as Subset of X | B ; pred p is valid means : Def1 : All ( x , p ) is valid ; assume that X c= dom f and f .: X c= dom g and g .: X c= dom f ; H |^ a is Subgroup of H & H |^ a is Subgroup of H ; let A1 be |^ O , A2 be Element of O ; p2 , q2 , r is_collinear & q2 , q2 , r is_collinear & r , q2 , r is_collinear ; consider x being element such that x in v ^ K and x in v ^ K ; not x in { 0. TOP-REAL 2 } & not x in { 0. TOP-REAL 2 } ; p in [#] ( ( TOP-REAL 2 ) | ( B ) ) ; 0 + ( M . E ) < M . E + M . E ; ^ ( c , c ) = c & c = ( c , c ) --> ( c , d ) ; consider c being element such that [ a , c ] in G ; a1 in dom ( F . ( s2 . m ) ) ; cluster -> -> Suba2 for LATTICE of L ; set i1 = the Nat of G , i2 = the Element of G . ( n + 1 ) ; let s be 0 -started State of SCM+FSA , P be s -started State of SCM+FSA ; assume y in ( f1 \/ f2 ) .: A & y in ( f1 \/ f2 ) .: 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 of Y , X be set ; cluster ( x `1 ) / ( x `2 ) -> -> -> -> -> -> -> -> -> -> -> of inBBBis ; set S = <* Bags n , sqrt n *> , i = <* i *> , j = <* i *> , i = j ; set T = [. 0 , 1 / 2 .] , G = [. 0 , 1 / 2 .] ; 1 in dom mid ( f , 1 , 1 ) ; sqrt 4 * PI < sqrt 2 * PI * PI ; x2 in dom f1 /\ dom ( f | X ) ; O c= dom I & { {} } = { {} } ; ( the Target of G ) . x = v ; { HT ( f , T ) } c= Support f ; reconsider h = R . k as Polynomial of n , L ; ex b being Element of G st y = b * H ; let x , y , z be Element of G ` ; h19 . i = f . ( h . i ) ; ( p `1 ) ^2 = ( ( p `1 ) ^2 + ( p `2 ) ^2 ) ; i + 1 <= len Cage ( C , n ) ; len <* P *> = len P & len <* P *> = len P ; set NN = the Element of the carrier of N , NN = the carrier of N ; len g\mathbin -' ( x + 1 ) <= x ; a on B & b on B implies b on B & a on C reconsider r-12 = r * I . v as FinSequence of REAL ; consider d such that x = d and a (*) d [= c and a [= d ; given u such that u in W and x = v + u ; len f /. ( \downharpoonright n ) = len TOP-REAL n + 1 .= len f ; set q2 = ( E-max C ) .. Cage ( C , n ) ; set S = that S = LSeg ( S1 , S2 ) , T = LSeg ( S2 , S2 ) ; MaxADSet ( b ) c= MaxADSet ( P /\ Q ) /\ Q ; Cl ( G . q1 ) c= F . r2 & G . q2 c= G . q2 ; f " D meets h " ( f .: V ) & f " ( f .: V ) /\ h .: V c= h .: V ; reconsider D = E as non empty directed Subset of L1 ; H = ( the_left_argument_of H ) '&' ( ( the_left_argument_of H ) '&' ( ( H ) '&' ( H ) '&' ( H ) ) ; assume t is Element of ( \mathfrak S ) . X ; rng f c= the carrier of S2 & f is one-to-one implies f is one-to-one consider y being Element of X such that x = { y } ; f1 . ( a1 , b1 ) = b1 . ( a1 , b1 ) ; the carrier of G = E \/ { E } .= { E } ; reconsider m = len ( q | k ) as Element of NAT ; set S1 = LSeg ( n , UMP C ) ; [ i , j ] in Indices M1 & M1 * ( i , j ) = M1 * ( i , j ) ; assume that P c= Seg m and M is Matrix of n , m , K ; 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 * ( 1 / p ) ; pp . i = p1 . i .= p1 . i .= p2 . i ; let PA , PA , G be a_partition of Y ; pred 0 < r & 1 < 1 & r < 1 implies r * ( 1 / r ) < r * ( 1 / r ) ; rng ( proj ( a , X ) | [#] 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 ( s | ( s | ( len s ) ) ) = card ( s | ( len s ) ) ; reconsider x2 = x1 , y2 = x2 as Element of L2 ; Q in FinMeetCl ( ( the topology of X ) \ B ) ; dom ( f | X ) c= dom ( u | X ) ; pred n divides m & m divides n & n = m ; reconsider x = x as Point of I[01] , a , b = b + a , c = b + a , d = d + c ; a in sequence the carrier of T2 ( T2 , T2 ) ; not y0 in the carrier of ( f | X ) & not ex g st g in the carrier of ( f | X ) ; Hom ( ( a \times b ) \times c , c ) <> {} ; consider k1 such that p " < k1 and k1 < len p and p . k1 < p . k1 ; consider c , d such that dom f = c \ d and c in X ; [ x , y ] in dom g & g . x = g . y ; set S1 = 1GateCircStr ( x , y , z ) ; l = m2 & l1 = m2 & l = m2 & l = m2 & l = m2 & l = m2 & l = m2 ; x0 in dom ( u | ( A /\ B ) ) /\ ( A /\ B ) ; reconsider p = x as Point of TOP-REAL 2 , q be Point of TOP-REAL 2 ; I[01] = ( ( 1 - B ) * ( B * ( 1 , j ) ) ) | ( B * ( 1 , j ) ) ; f . p3 <= f . p1 & f . p2 <= f . p1 ; ( ( F . x ) `1 ) ^2 <= ( F . x ) ^2 + ( F . x ) ^2 ; ( x `2 ) ^2 = ( ( W . n ) `1 ) ^2 + ( W . n ) ^2 ; for n being Element of NAT holds P [ n ] ; let J , K be non empty Subset of I ; assume 1 <= i & i <= len <* a " *> ; 0 |-> a = <*> ( the carrier of K ) .= <*> 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 ] ; reconsider s5 = seq . ( s + 1 ) as terminal of D ; ( - i -' 1 ) <= len ( - j ) ; [#] S c= [#] ( T | the carrier of S ) & the TopStruct of T = the TopStruct of T ; for V being strict Subspace of V holds V in W1 iff V is Subspace of W2 assume k in dom mid ( f , i , j ) ; let P be non empty Subset of TOP-REAL 2 , p1 , p2 , p3 be Point of TOP-REAL 2 ; let A , B be Matrix of n , K ; - a * ( - b ) = a * ( - b ) ; for A being being_line Subset of A9 holds A // A & A // A implies A // A ( id o2 ) . ( ( o , o2 ) --> ( o , o2 ) ) in <* ( o , o1 ) --> ( o , o2 ) *> ; then ||. x .|| = 0 & x = 0. X ; let N1 , N2 be strict Subgroup of G , N be normal Subgroup of G ; j >= len ( upper_volume ( g , D1 ) | divset ( D , i ) ) ; b = Q . ( len Q - 1 ) ; f2 * f1 /* s is divergent_to+infty & f2 * f1 is divergent_to+infty ; reconsider h = f * g as Function of N , G ; assume that a <> 0 and Polynom ( a , b , c ) >= 0 ; [ t , t ] in the InternalRel of A & t in the carrier of A ; ( v |-- E ) | n is Element of T | n ; {} = ( the carrier of L1 + L2 ) \/ { 0. ( L1 + L2 ) } .= { 0. ( L1 + L2 ) } ; Directed I is closed & Directed I is closed implies Directed I is closed Initialized p = Initialize ( ( p +* q ) +* q ) +* ( p +* q ) ; reconsider N2 = N1 , N2 = N2 as strict net of R , R2 ; reconsider Y = Y as Element of <* InclPoset ( Ids L ) , \subseteq \rangle ; "/\" ( { p } , { p } ) <> p ; consider j being Nat such that i2 = i1 + j and j in dom f and f /. ( j + 1 ) = f /. ( j + 1 ) ; not [ s , 0 ] in the carrier of S2 & not [ s , 0 ] in the carrier of S2 ; m-5 in ( B '&' C ) \ { {} } ; n <= len ( ( P +* ( i , n ) ) | i ) + len ( ( P +* ( i , n ) ) | i ) ; ( x1 - x2 ) `1 = ( x2 - y2 ) `1 .= ( x2 - y2 ) `1 ; InputVertices S = { x1 , x2 } & InputVertices S = { x1 , x2 } ; let x , y be Element of FT1 ( n , m ) ; p = |[ p `1 , p `2 ]| .= p `1 ; g * 1_ G = h " * g * h .= g * h .= g * h ; let p , q be Element of PFuncs ( V , C ) ; x0 in dom ( x1 | X ) /\ dom ( ( x1 | X ) ^ ) ; ( R qua Function ) " = R " ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R * ( R n in Seg len ( f /^ ( len f -' 1 ) ) ; for s being Real st s in R holds s <= s2 & s <= t implies t <= s rng s c= dom ( f2 * f1 ) /\ dom ( f2 * f1 ) ; synonym for for for for X is Subset of Fin ( X ) ; 1_ K * ( 1_ K ) = 1_ K * ( 1_ K ) .= 1_ K * ( 1_ K ) ; set S = Segm ( A , P1 , Q1 ) , Q1 = Segm ( A , P1 , Q1 ) ; ex w st e = sqrt w & w in F & w in G ; curry ( P , k ) # x is convergent & lim ( ( P # x ) # x ) is convergent ; cluster -> open for Subset of [: T , T :] ; len f1 = 1 .= len ( f1 | ( Seg len f1 ) ) .= len f1 ; sqrt ( i * p ) < sqrt ( 2 * p ) ; let x , y be Element of \rm SubSub ( U0 ) ; b1 , c1 // b9 , c9 & b , c // b9 , c9 ; consider p being element such that c1 . j = { p } ; assume that f " { 0 } = {} and f is total and f is total ; assume IC Comput ( F , s , k ) = n ; Reloc ( J , card I + 2 ) not LIN a , b , c ; Macro ( card I + 1 + 1 ) not LIN c , d , c set m1 = LifeSpan ( p3 , s3 ) , m2 = LifeSpan ( p2 , s2 ) ; IC Comput ( p , s , k ) in dom Initialize ( ( intloc 0 ) .--> 1 ) ; dom t = the carrier of ( R ) & dom t = the carrier of ( R ) ; ( ( E-max L~ f ) .. f ) .. f = 1 ; let a , b be Element of V ( ) ; Cl union ( union F ) c= Cl union F & Cl ( union F ) c= Cl ( union F ) ; the carrier of X1 union X2 misses ( A1 \/ A2 ) & the carrier of X1 union X2 misses A1 \/ A2 ; assume not LIN a , f . a , g . a , g . a ; consider i being Element of M such that i = d1 . i and i in d2 ; then Y c= { x } or Y = { x } or Y = { x } ; M , v / ( ( y , x ) / ( y , x ) ) / ( y , x ) / ( y , x ) / ( y , x ) / ( y , x ) / ( y , x ) / ( y , x ) consider m being element such that m in Intersect ( FF | m ) and m in meet ( FF | m ) ; reconsider A1 = ( support u1 ) \/ ( { x } ) as Subset of X ; card ( A \/ B ) = k-1 + ( 2 * 1 ) ; assume that a1 <> a3 and a2 <> a3 and a3 <> a4 and a1 <> a2 and a3 <> a4 and a1 <> a2 and a2 <> a4 and a3 <> a4 and a1 <> a4 and a1 <> a2 and a1 <> a3 and a2 <> a4 and a3 <> a4 ; cluster s \! \mathop { V } -> string of S ; LS /. ( n + 2 ) = LS /. ( n + 2 ) .= LS /. ( n + 2 ) ; let P be compact non empty Subset of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ; assume r2 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ; let A be non empty compact Subset of TOP-REAL n , B be Subset of TOP-REAL n ; assume [ k , m ] in Indices ( D * ) ; 0 <= ( sqrt ( 1 - ( 2 * p ) ^2 ) ) * p ; ( F . N ) | ( E . N ) = +infty ; pred X c= Y & Z c= V implies X \ V c= Y \ V ; ( y * z ) * ( ( z * y ) * ( y * z ) ) <> 0. I ; 1 + card ( X1 \/ X2 ) <= card u + card ( X1 \/ X2 ) ; set g = z \circlearrowleft ( ( L~ z ) /. 1 ) .. z , ( L~ z ) .. z ) ; then k = 1 & p . k = <* x , y *> ; cluster -> total for Element of C carrier ( X ) ; reconsider B = A as non empty Subset of TOP-REAL n , B be Subset of TOP-REAL n ; let a , b , c be Function of Y , BOOLEAN ; L1 . i = ( i .--> g ) . i .= g . i ; ( ( x1 , x2 , x3 ) --> ( x1 , x2 , x3 ) ) c= P ; n <= indx ( D2 , D1 , j1 ) + 1 ; ( ( g2 ) . O ) `1 = - 1 & ( g2 ) . O = 1 ; j + p .. f - len f <= len f - len f + 1 ; set W = W-bound C , S = N-bound C , E = E-bound C , N = E-bound C , N = E-bound C , S = E-bound C , N = E-bound C , S = E-bound C , N = E-bound C , S = E-bound C , S = E-bound S1 . ( a , e ) = a + e .= a + e ; 1 in Seg width ( M * ( ( Line ( M , 1 ) ) * ( ( M * ( 1 , width M ) ) ) ) ; dom ( i (#) Im ( f ) ) = dom ( Im ( f ) ) ; Dx . ( x `1 , p `2 ) = W . ( a , p `2 ) ; set Q = |= ( All ( g , f , h ) , R ) ; cluster -> Subrelation of U1 -> Subsqrt -> SubCl for Relation of U1 ; attr F = { A } means : Def1 : F is discrete ; reconsider z1 = .[ as Element of product G ; rng f c= rng f1 \/ rng f2 & f | X is one-to-one implies f | X is one-to-one consider x such that x in f .: A and x in f .: C ; f = <*> ( the carrier of V ) & f is FinSequence of the carrier of V ; E , j |= All ( j , H ) & E , j |= All ( x , H ) ; reconsider n1 = n as Morphism of o1 , o2 , o2 be Morphism of o2 , o2 ; assume that P is idempotent and R is idempotent and P = R and R = R and P = R ; card ( B2 \/ { x } ) = card ( B2 \/ { x } ) ; card ( x \ B1 ) /\ ( x \ B1 ) = 0 ; g + R in { s : g-r < s & s < g + r } ; set q1 = ( q , <* s *> ) | ( q , <* s *> ) ; for x being element st x in X holds x in rng f1 ; h1 /. ( i + 1 ) = h1 . ( i + 1 ) ; set mw = max ( B , min ( B , min ( A , m ) ) ) ; t in Seg width ( I ^ J ) & t in Seg width ( I ^ J ) ; reconsider X = dom f /\ C as Element of Fin NAT , X be Element of Fin NAT ; IncAddr ( i , k ) = <% l . k %> + k .= l . k ; ( ( E-max L~ f ) .. f ) <= ( ( E-max L~ f ) .. f ) .. f ; attr R is condensed means : Def1 : for X being Subset of R st X is condensed holds Cl R is condensed & R is condensed ; pred 0 <= a & b <= 1 & a * b <= 1 & a * b <= 1 * b ; u in ( ( c /\ ( ( d /\ ( ( ( ( ( ( ( ( ( ( ( ( ( d /\ b ) /\ b ) /\ b ) ) /\ e ) /\ f ) /\ f ) /\ j ) ) /\ j ) ) /\ j ) /\ j ) /\ j ) /\ j u in ( ( c /\ ( ( d /\ ( ( ( ( ( ( ( ( ( ( ( ( ( ( d /\ ( e /\ e ) ) ) /\ b ) ) /\ b ) /\ f ) /\ f ) /\ j ) ) /\ j ) ) /\ j ) /\ j ) /\ j len C + 2 - 2 >= 9 + ( - 2 ) + ( - 2 ) ; x , z , y is_collinear & x , y , z is_collinear & x , y , z is_collinear ; a |^ ( n1 + 1 ) = a |^ n1 * a |^ ( n1 + 1 ) ; <* \underbrace 0 , \dots , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 set y9 = <* y , c *> , f = <* y , c *> ; F2 /. 1 in rng Line ( D , 1 ) & F /. len F = Line ( D , 1 ) ; p . m Joins r /. m , r /. ( m + 1 ) , r /. ( m + 1 ) ; ( p `2 ) ^2 = ( f /. i1 ) ^2 + ( f /. i2 ) ^2 ; ( W-min X \/ Y ) = ( W-min X ) union ( Y \/ { p } ) .= ( E-max X ) \/ ( Y \/ { p } ) ; 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 & f2 /* ( seq ^\ k ) is divergent_to+infty ; reconsider u2 = u as VECTOR of \mathop { \rm Pmin ( X , Y ) , X ; p \! ( ( Sgm ( ( Seg len ( p | ( Seg len p ) ) ) ) ) ) = 0 ; len <* x *> + 1 < i + 1 + 1 ; assume that I is non empty and { x } /\ { y } = { 0. I } ; set ii = card I + 4 + ( card I + 4 ) , goto 0 = card I + 4 + 0 ; x in { x , y } & h . x = {} T & y in T ; consider y being Element of F such that y in B and y <= x `1 ; len S = len ( the charact of ( A ) ) .= len the charact of ( A ) ; reconsider m = M , n = I as Element of X ; A . ( j + 1 ) = B . ( j + 1 ) \/ A . j ; set Nmin = : : \leq Gmin ( G , n ) `1 , Nmin = G * ( 1 , n ) `1 ; rng F c= the carrier of gr ( { a } , { a } ) ; ( for K being Matrix of n , m , K holds P * ( K , n , K ) is Matrix of n , m , K f . k , f . ( mod n ) in rng f ; h " P /\ [#] ( ( TOP-REAL 2 ) | P ) = f " P /\ ( ( TOP-REAL 2 ) | P ) .= P ; g in dom ( f2 \ ( f2 " { 0 } ) ) /\ ( f2 " { 0 } ) ; g\times X /\ dom f1 = g1 " ( g1 " ( X /\ Y ) ) .= g1 " ( g1 " ( X /\ Y ) ) ; consider n being element such that n in NAT and Z = G . n ; set d1 = \bf \bf ( ( x1 , y1 ) , ( x2 , y2 ) ) / ( x1 , y1 ) , d2 = ( x1 , y2 ) / ( x2 , y2 ) / ( x2 , y2 ) / ( x2 , y2 ) = ( x1 , y1 ) / ( x2 , b `1 + sqrt ( 1 + ( b `1 / b `1 ) ^2 ) < sqrt ( 1 + ( b `1 / b `1 ) ^2 ) ; reconsider f1 = f as VECTOR of the carrier of X , Y ; pred i <> 0 implies i |^ ( i + 1 ) mod ( i + 1 ) = 1 mod ( i + 1 ) ; j2 in Seg len ( ( g2 . ( i2 + 1 ) ) | ( Seg ( i2 + 1 ) ) ) ; dom ( i - j ) = dom ( i - j ) .= a .= a . i ; cluster sec | ]. PI / 2 , PI / 2 .[ -> one-to-one ; Ball ( u , e ) = Ball ( f . p , e ) ; reconsider x1 = x0 as Function of S , T | [: S , T :] ; reconsider R1 = x , R2 = y as Relation of L ; consider a , b being Subset of A such that x = [ a , b ] ; ( <* 1 *> ^ p ) ^ <* n *> in ( R + S ) ; S1 +* S2 = S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 ( ( ( ( id Z ) (#) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 1 / / ) ) ) ) ) ) ) ) ) ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( cluster -> continuous for Function of C , REAL ; set C7 = 1GateCircStr ( <* z , x *> , f3 ) , C7 = 1GateCircStr ( <* z , x *> , f3 ) ; E8 . ( v2 , v2 ) = E8 . ( v2 , v2 ) -T . ( v2 , v2 ) ; ( ( arctan * ( arctan + arccot ) ) `| Z ) . x = ( arctan * ( arctan + arccot ) ) . x ; upper_bound A = sqrt ( 3 * 2 ) & lower_bound A = 0 ; F . ( dom f , - f ) is Functor of F . ( cod f , - f ) ; reconsider p9 = ( q `2 / |. q .| - sn ) / ( 1 - sn ) as Point of TOP-REAL 2 ; g . W in [#] ( Y | X ) & [#] ( Y | X ) c= [#] Y ; let C be compact non vertical connected Subset of TOP-REAL 2 , p , q be Point of TOP-REAL 2 ; LSeg ( f ^ g , j ) = LSeg ( f , j ) \/ LSeg ( g , j ) ; rng s c= dom f /\ ]. - ( r / 2 ) , - ( r / 2 ) .[ ; assume x in { ( idseq 2 ) . ( ( idseq 2 ) . ( i + 1 ) ) } ; reconsider n2 = n , m2 = m 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 + m2 .= m1 + m2 + m2 + m2 .= m1 + m2 + m2 + m2 + m2 + m2 + m2 ; assume for n holds H1 . n = G . n -H . n ; set Bf = f .: ( the carrier of X1 union X2 ) , Bf = f .: the carrier of X2 ; ex d being Element of L st d in D & x << d ; assume R " ( a /\ b ) c= R " ( b /\ c ) ; t in ]. r , s .[ or t = r or t = s or t = s ; z + v2 in W & x = u + ( z + ( v + v2 ) ) ; x2 |-- ( x2 , y2 ) iff P [ x2 , y2 ] ; pred x1 <> x2 means : Def1 : |. x1 - x2 .| > 0 & |. x1 - x2 .| > 0 ; assume p2 - p1 , p3 - p1 , p2 - p1 , p3 - p1 , p2 - p1 , p3 - p1 , p2 - p1 , p3 - p1 , p2 - p1 - p1 , p2 - p1 - p1 , p3 - p1 - p2 , p2 - p1 - p2 , p3 - p1 - p1 , p2 - p1 - p2 , p2 - p1 - p1 set q = ( ( f | 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' let f be PartFunc of REAL-NS 1 , REAL-NS 1 , REAL-NS 1 , i , j be Nat ; ( n mod ( 2 * k ) ) = n mod ( 2 * k ) ; dom ( T * ( that that T * ( that T * ( i , j ) ) ) ) = dom T ; consider x being element such that x in w and x in c and x in X ; assume ( F * G ) . ( v , x3 ) = v . ( x3 ) ; assume that the carrier of D1 c= the carrier of D2 and the carrier of D2 c= the carrier of D2 and the carrier of D2 = the carrier of D2 ; reconsider A1 = [. a , b .[ as Subset of REAL n , a , b be Real ; consider y being element such that y in dom F and F . y = x ; consider s being element such that s in dom o and a = o . s ; set p = ( W-min L~ Cage ( C , n ) ) .. Cage ( C , n ) ; n1 -' len f + 1 + 1 <= len f + ( len f -' 1 ) ; |. |. q , O1 , b , a , b , c , d , b , c , d , e , f , g , h , h , i ) = [ u , b , c , d ] ; set C-2 = ( .n] ) . ( k + 1 ) , CE = ( G . ( k + 1 ) ) ; Sum ( L * p ) = 0. R .= 0. V .= 0. V ; consider i being element such that i in dom p and t = p . i ; defpred Q [ Nat ] means 0 = Q ( $1 ) & $1 <> 0 implies $1 = ( $1 + 1 ) * ( $1 + 1 ) ; set s3 = Comput ( P1 , s1 , k ) , s2 = Comput ( P2 , s2 , k ) , s3 = Comput ( P2 , s2 , k ) ; let l be variable , A be Element of k -tuples_on NAT , B be Element of l -tuples_on A ; reconsider U = union ( G . n , G . n ) as Subset-Family of [: T , T :] ; consider r such that r > 0 and Ball ( p `1 , r ) c= Q ` ; ( h | ( n + 2 ) ) /. ( i + 2 ) = p1 & ( h | ( n + 2 ) ) /. ( i + 2 ) = p1 ; reconsider B = the carrier of X1 , C = the carrier of X2 as Subset of X ; pX1 = <* - ( c1 + c2 ) , 1 , - ( c1 + c2 ) *> ; synonym f is real-valued means : Def1 : rng f c= NAT & rng f c= NAT ; consider b being element such that b in dom F and a = F . b ; x0 < card ( X \/ Y ) + card Y + card X + card Y + card Y + card Y + card X + card Y + card Y + card Y + card Y + card Y + card X + card Y + card Y + card Y + card Y + card X + card Y + card Y + card Y pred X c= B1 means : Def1 : for B st B in B1 holds X c= B & B c= B1 ; then w in Ball ( x , r ) & dist ( x , w ) <= r ; angle ( x , y , z ) = angle ( x , y , z ) + angle ( y , z , w ) ; pred 1 <= len s means : Def1 : for i being Nat holds ( s . i ) . i = s . i ; fj c= f . ( k + ( n + 1 ) ) ; the carrier of { 1_ G } = { 1_ G , { 1_ G } } ; pred p '&' q in TAUT ( Al ) means : Def1 : q '&' p in TAUT ( Al ) & q '&' p in TAUT ( Al ) ; - ( t `2 / t `1 ) ^2 < ( ( t `2 / t `1 ) ^2 + ( t `2 / t `2 ) ^2 ; U . 1 = U /. 1 .= ( U /. 1 ) `1 .= ( U /. 1 ) `1 ; f .: ( the carrier of x ) = the carrier of x & f .: ( the carrier of x ) = the carrier of x ; Indices ( O * ( i , j ) ) = [: Seg n , Seg n :] ; for n being Element of NAT holds G . n c= G . ( n + 1 ) then V in M |^ M & ex x being Element of M st V = { x } ; ex f being Element of F-9 st f is U & f is U & f is U & f is U c= U ; [ h . 0 , h . 3 ] in the InternalRel of G ; s +* Initialize ( ( intloc 0 ) .--> 1 ) = s3 +* ( ( intloc 0 ) .--> 1 ) ; |[ w1 , v1 ]| `1 <> 0. TOP-REAL 2 & |[ w1 , v1 ]| `2 <> 0. TOP-REAL 2 ; reconsider t = t as Element of ( REAL n ) -tuples_on the carrier of X ; C \/ P c= [#] ( ( G | [#] ( ( G | ( [#] ( A \ A ) ) ) \ A ) ) ; f " V in ( ( ( X /\ Y ) /\ ( X , \alpha ) ) . X , ( X /\ Y ) . X ) ; x in [#] ( ( the carrier of Y ) /\ the carrier of Y ) ; g . x <= h1 . x & h1 . x <= h1 . x & h1 . x <= h1 . x ; InputVertices S = { xy , yz , yz , yz , yz , zx , a5 , a5 , a5 , cin } ; for n being Nat st P [ n ] holds P [ n + 1 ] set R = Line ( M , i ) * Line ( M , i ) ; assume that M1 is being_line and M2 is being_line and M2 is being_line and M2 is being_line and M2 is being_line and M2 is being_line ; reconsider a = ( f0 . i0 -' 1 ) * ( ( f . i0 ) + ( f . i0 ) + ( f . i1 ) * ( f . i1 ) ) as Element of K ; len ( B2 ^ ( F2 ^ ( F2 ^ ( F2 ^ ( F2 ^ ( F2 ^ ( F2 ^ ( F2 ^ ( F2 ^ ( F2 ^ ( F2 ^ ( F2 ^ ( F2 ^ ( F2 ^ ( F2 ^ ( F2 ^ ( F2 ^ ( F2 ^ ( F2 ^ ( F2 ^ ( F2 ^ ( F2 ^ ( len ( ( the FinSequence of TOP-REAL n ) | i ) = n & len ( ( the _ of TOP-REAL n ) | i ) = n ; dom ( f + g ) = dom ( f + g ) /\ dom g .= dom f /\ dom g ; ( ( for n holds seq . n = upper_bound Y1 ) implies for n holds seq . n = ( lim seq ) . n dom ( p1 ^ p2 ) = dom ( p1 ^ p2 ) .= dom p1 .= Seg len p1 .= Seg len p1 ; M . ( [ 1 , y ] , v1 ) = 1 * v1 .= ( 1 - 1 ) * v1 .= ( 1 - 1 ) * v1 .= ( 1 - 1 ) * v1 ; assume that W is non trivial and W c= the carrier of G2 and W is open ; C6 * ( i1 , i2 ) `2 = G1 * ( i1 , i2 ) `2 .= G1 * ( i1 , i2 ) `2 ; C8 |- 'not' All ( x , p ) 'or' 'not' All ( x , p ) .= 'not' All ( x , p ) ; for b st b in rng g holds lower_bound rng fD1 <= b - a - sqrt ( ( ( q `2 / |. q .| - sn ) / ( 1 - sn ) ) ^2 ) = 1 ; ( LSeg ( c , m ) \/ LSeg ( l , k ) ) \/ LSeg ( l , m ) c= R ; consider p being element such that p in LSeg ( x , p ) and p in L~ f and p = f /. ( len f ) ; Indices ( X @ ) = [: Seg n , Seg n :] ; cluster s => ( q => p ) -> valid ; Im ( ( Partial_Sums F ) . m ) , ( Partial_Sums F ) . m ) is_measurable_on E ; cluster f . ( x1 , x2 ) -> Element of D ( ) ; consider g being Function such that g = F . t & Q [ t , g ] ; p in LSeg ( ( Cage ( C , n ) /. ( i + 1 ) , ( E-max L~ Cage ( C , n ) ) /. ( i + 1 ) ) ; set RX = R |^ ]. 1 , + \infty .[ , RX = ]. 1 , 1 .[ ; IncAddr ( I , k ) = IncAddr ( da , k ) .= IncAddr ( da , k ) .= CurInstr ( P1 , Comput ( P2 , s2 , k ) ) ; seq . m <= ( ( the Sorts of A ) * ( ( the Sorts of A ) * ( ( the Sorts of A ) * ( ( the Sorts of A ) * ( ( the Arity of S ) * ( ( the Arity of S ) * ( ( the Arity of S ) * ( ( the Arity of S ) * a + b = ( a ` ) ` + ( b ` ` ` ` ` ) ` .= ( a ` + b ` ) ` + ( a ` + b ` ) ` ; id X /\ ( id X ) = id X /\ ( id X ) .= id X /\ id X .= id X ; for x being element st x in dom h holds h . x = f . x ; reconsider H = U1 \/ U2 as non empty Subset of U0 ; u in ( ( ( c /\ ( ( ( ( d /\ ( ( ( ( ( ( ( ( d /\ ( e /\ e ) /\ b ) ) /\ b ) /\ f ) /\ f ) /\ j ) ) /\ m ) ) /\ m ) ) /\ m ) /\ m ) /\ m ) /\ m ; consider y being element such that y in Y and P [ y , ( inf B ) . y ] ; consider A being finite stable Subset of R such that card A = card ( R * ) ; p2 in rng ( f |-- p1 ) \ rng <* p1 *> & p1 in rng ( f | p1 ) ; len s1 > 0 & len s2 > 0 & len s2 > 0 implies len s1 = len s2 ( ( E-max P ) `2 ) `2 = ( E-max P ) `2 .= ( E-max P ) `2 ; Ball ( e , r ) c= LeftComp ( Cage ( C , k + 1 ) ) \/ L~ Cage ( C , k + 1 ) ; f . a1 = f . ( a1 ` * a2 ` * x ` + f . ( a2 * x ` + c ) ) ; ( seq ^\ k ) . n in ]. - ( x0 - r ) , x0 .[ ; gg . x0 = g . ( ( f . x0 ) | G ) . ( ( f . x0 ) | G ) . ( ( f . x0 ) | G ) . ( ( f . x0 ) | G ) . ( ( f . x0 ) | G ) . ( ( f . x0 ) | G ) the InternalRel of S is RelStr of ( the InternalRel of S ) \/ ( the InternalRel of T ) ; deffunc F ( Ordinal , Ordinal ) = phi . ( $2 + $2 ) ; F . ( s1 . a1 ) = F . ( s2 . a1 ) .= F . ( s2 . a1 ) ; x `1 = A . o .= Den ( o , A ) . a .= Den ( o , A ) . a ; Cl ( f " P1 ) c= f " ( ( f " P1 ) " P1 ) & f " P1 c= f " ( ( f " P1 ) " P1 ) ; FinMeetCl ( ( the topology of S ) | the topology of T ) c= the topology of T ; synonym o is constructor means : Def1 : o <> \ast & o <> {} & o <> {} & o <> {} & o <> {} & o <> {} ; assume that X = Y + X and card Y <> card Y and card X <> card Y ; the carrier of s <= 1 + ( the carrier of S ) * ( ( the carrier of S ) * ( the carrier of S ) ) ; LIN a , a1 , d or b , c // b1 , d or b , c // d , c ; e . ( 2 + 1 ) = 0 & e . ( 2 * 2 ) = 1 & e . ( 2 * 2 ) = 0 ; EE in [: S , { 1 } :] & EE in { N } & EE in { N } ; set J = ( l , u ) transitive I ; set A1 = 1GateCircStr ( a9 , b9 , c9 ) , A2 = Following ( b9 , c9 ) ; set c9 = [ <* c9 , A1 *> , and2a ] , A2 = [ <* A1 , cin *> , and2a ] , it = [ <* cin , c9 *> , and2 ] ; x * z `1 * x `1 in x * ( z * N ) `1 * ( x * N ) `1 * ( x * N ) `1 in x * ( z * N ) `1 * ( x * N ) `1 ) ; for x being element st x in dom f holds f . x = g2 . x & f . x = g2 . x ; Int cell ( f , 1 , G ) c= RightComp f \/ RightComp f \/ RightComp f \/ RightComp f \/ RightComp f \/ RightComp f \/ RightComp f ; U is_an_arc_of W-min C , W-min C & E-max C in L~ Cage ( C , n ) implies W-min C in L~ Cage ( C , n ) \/ L~ Cage ( C , n ) set f9 = f .: @ g , f9 = f .: @ g ; attr S1 is convergent means : Def1 : for n , m st n >= m holds S1 - S2 is convergent & lim S1 = x0 ; f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a ; cluster -> Subsymmetric non empty for RelStr of the carrier of G , the carrier of G ; consider d being element such that R reduces b , d and R reduces c , d ; not b in dom Start-At ( ( card I + 2 ) + 2 + 2 ) ; ( z + a ) + x = z + ( a + y ) .= z + ( a + y ) .= z + ( a + y ) ; len ( l | [. 0 , PI / 2 .] ) = len l ; t4 is ( {} \/ rng ( t ^ <* n *> ) ) -valued FinSequence of ( X \/ rng ( t ^ <* n *> ) ) -valued FinSequence ; t = <* F . t *> ^ ( C ^ ( p ^ q ) ) *> ^ ( C ^ q ) ; set pp = W-min L~ Cage ( C , n ) , p = W-min L~ Cage ( C , n ) ; k9 -' ( i + 1 ) = ( i + 1 ) - ( i + 1 ) .= i - ( i + 1 ) ; consider u being Element of L such that u = u ` ` ` and u in D and u in D ; len ( ( width G ) |-> a ) = width ( ( width G ) |-> a ) .= width ( ( width G ) |-> a ) ; F3 . x in dom ( ( G * the_arity_of o ) . x ) ; set H2 = the carrier of H2 , Z = the carrier of H2 , I = the carrier of H , J = the carrier of H , H = the carrier of H , I = the carrier of H , J = the carrier of H , H = the carrier of H , I = the carrier of H , H = I ; set H1 = the carrier of H1 , H2 = the carrier of H2 , Z = the carrier of H2 , i = the carrier of H1 , j = the carrier of H2 ; ( Comput ( P , s , 6 ) ) . intpos ( m + 6 ) = s . intpos ( m + 6 ) ; IC Comput ( Q , t , k ) = l + 1 .= ( card I + 1 ) + 1 ; dom ( ( ( ( cos * ( cos ) ) * ( ( cos * ( ( cos * ( ( cos * ( ( cos * ( ( cos * ( ( cos * ( ( ( ( ( ( f * ( f ) ( f ) ) ) ) ) ) ) ) ) ) ) ) ) `| REAL ) ) ) = REAL ; cluster <* l *> ^ phi -> ( 1 + 1 ) -element string of S ; set b5 = [ <* A1 , cin *> , and2a ] , b5 = [ <* A1 , cin *> , and2 ] , b5 = [ <* A1 , cin *> , and2 ] , b5 = [ <* A1 , cin *> , and2 ] , b5 = [ <* A1 , cin *> , and2 ] , b5 = [ <* A1 , cin *> , and2 Line ( Segm ( M , P , Q ) , x ) = L * ( Sgm Q ) ; n in dom ( ( the Sorts of A ) * ( the_arity_of o ) ) ; cluster f1 + f2 -> continuous for PartFunc of REAL , REAL ; consider y be Point of X such that a = y and ||. y - x .|| <= r ; set x3 = t . DataLoc ( s . a , i ) , x4 = t . DataLoc ( s . a , i ) , x4 = t . DataLoc ( s . a , i ) , P4 = t . DataLoc ( s . a , i ) , P4 = t . intpos ( i + 3 ) , P4 = t . intpos set pp = stop I , p = stop I , s = stop I ; consider a being Point of D2 such that a in W1 and b = g . a and a in the carrier of D2 ; { A , B , C , D , E , F , J , M , N , M , N , N , N , M , N , N , N , M , N , N , N , M , N , N , N , N , M , N , N , N , N , M , N , N , let A , B , C , D , E , F , J , M , N , N , M , N , N , N , M , N , N , N , M , N , N , N , M , N , N , N , M , N , N , N , N , M , N , N , |. p2 .| ^2 - ( p2 `2 / p1 `1 ) ^2 >= 0 ; l -' 1 + 1 = l * ( l1 + 1 ) + ( l1 + 1 ) ; x = v + ( a * ( w1 + ( b * w2 ) ) + ( c * w2 ) ) + ( c * w2 ) ; the TopStruct of L = ( the TopStruct of L ) | ( the topology of L ) .= the TopStruct of L ; consider y being element such that y in dom H1 and x = H1 . y and y in H1 . x ; f9 \ { n } = ( ( Free ( 'not' ( 'not' ( 'not' ( 'not' ( 'not' ( 'not' ( 'not' ( 'not' ( 'not' ( 'not' 'not' 'not' 'not' ( 'not' ( 'not' ( 'not' ( 'not' ( 'not' ( 'not' ( 'not' ( 'not' ( 'not' ( 'not' ( 'not' ( 'not' ( 'not' ( 'not' ( 'not' ( 'not' ( 'not' ( 'not' ( 'not' ( 'not' ( 'not' ( 'not' ( 'not' for Y being Subset of X st Y is summable & Y is summable holds Y is not summable 2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ; for s being FinSequence holds len ( ( s * ( len s ) ) ) = len s & len ( s * ( len s ) ) = len s for x st x in Z holds ( ( exp_R * f ) `| Z ) . x = exp_R . x / ( cos . x ) ^2 rng ( h2 * ( f2 * ( f1 * ( f2 * ( f1 * ( f2 * ( f1 * ( f2 * ( f1 * ( f2 * ( f1 * ( f1 * ( f2 * ( f1 * ( f1 * ( f2 * ( f1 * ( f2 * ( f1 * ( f2 * ( f1 * ( f2 * ( f1 * ( f2 * ( f2 j + ( 1 + len f -' 1 ) <= len f + ( len f -' 1 ) - 1 ; reconsider R1 = R * I as PartFunc of REAL , REAL n , Y , Z be Subset of REAL n ; C8 . x = s1 . a .= C8 . x .= C8 . x .= C8 . x .= C8 . x ; power ( n , z ) . ( ( n , 1 ) * ( x , y ) ) = 1 .= ( x |^ n ) |^ ( ( n + 1 ) * ( x , y ) ) ; t at ( C , s ) = f . ( ( the connectives of S ) . o ) . t ; support ( f + g ) c= ( support f \/ C ) \/ ( { x } \/ { x } ) ; ex N st N = j1 & 2 * Sum ( ( r | N ) | N ) > 0 ; for y , p st P [ p ] holds P [ All ( y , p ) ] { x1 , x2 } is Subset of X1 union X2 & { x1 , x2 } c= the carrier of X1 union X2 ; h = ( i .--> j ) +* ( id B , id B ) .= H . i ; ex x1 being Element of G st x1 = x & x1 * N c= A & x1 * N c= A * N ; set X = ( ( |. q .| , O1 ) , ( |. q .| , O1 ) , m ) , Y = ( |. q .| , O1 ) , X = { m } , Y = { m , n } , Y = { m , n } , X = { m , n } , Y = { m , n } , Y = { m , n } , Y = n , m , m b . n in { g1 : x0 < g1 & g1 < x0 } & ( g1 . n ) . x0 < g1 . ( g1 . n ) ; f /* s1 is convergent & lim ( f /* s1 ) = lim ( f /* s1 ) ; the lattice of the lattice of Y = the lattice of Y & the carrier of X = the carrier of Y & the carrier of Y = the carrier of X ; 'not' ( a . x ) '&' b . x '&' a . x = TRUE ; 2 = len ( ( ( q ^ <* r *> ) | ( len q ) ) ) + len ( ( q ^ <* r *> ) | ( len q ) ) .= len ( ( q ^ <* r *> ) | ( len q ) ) ; sqrt ( 1 - a * ( ( ( sec * ( f1 + f2 ) ) `| Z ) ) ^2 - ( ( ( sec * ( f1 + f2 ) ) `| Z ) ) ^2 / ( 1 - ( ( sec * ( f1 + f2 ) ) ^2 ) ) ^2 ) * ( ( ( sec * ( f1 + f2 ) ) ^2 ) ) ^2 ) ) ^2 set K1 = ( ( lim ( ( lim ( H ) ) (#) ( ( lim ( H ) ) (#) ( lim ( H ) ) ) ) ) `| REAL ) ; assume that e in { |[ w1 , w2 ]| : w1 in F & w2 in G } and w1 in G and w2 in G and w1 in F and w2 in G and w1 in G and w2 in G ; reconsider d1 = dom a `1 , d2 = dom F , d2 = F . ( a , b ) as finite Subset of NAT ; LSeg ( f , len f -' 1 ) = LSeg ( f , j ) \/ LSeg ( f , j ) ; assume X in { T . ( N2 , K1 ) : h . ( N2 , K1 ) = T . ( N2 , K1 ) } ; assume that Hom ( d , c ) <> {} and <* f , g *> * f = <* f , g *> * f ; dom ( S | ( Seg n ) ) = dom S /\ ( Seg n ) .= Seg n .= Seg n ; x in H |^ a implies ex g st x = g |^ a & g in H |^ a & g in H |^ a ; a * ( a , 1 ) = a `1 - ( 0 * n ) .= a `1 - ( 0 * n ) ; D2 . j in { r : lower_bound A <= r & r <= upper_bound A } ; ex p being Point of TOP-REAL 2 st p = x & P [ p ] & p <> 0. TOP-REAL 2 ; for c holds f . c <= g . c implies f ^ <* c *> t t ^ <* c *> ^ <* d *> ^ t ^ <* d *> ^ t ^ <* d *> ^ t ^ <* c *> ^ t ^ <* d *> ^ t ^ t ^ <* d *> ^ t ^ t ^ t ^ <* d *> ^ t ^ t ^ t ^ t ^ t ^ t dom ( f1 (#) ( f2 (#) ( f1 + f2 (#) ( f1 + f2 (#) ( f2 + f3 (#) ( f1 + f2 ) ) ) ) ) /\ X c= dom ( f1 (#) ( f2 + ( f2 (#) ( f1 + f3 (#) ( f2 + f3 (#) ( f2 + f3 ) ) ) ) ; 1 = sqrt ( p * p ) .= p * p .= p * p .= p * p .= p * p .= p * p ; len g = len f + len <* x + y *> .= len f + len <* y *> .= len f + len <* y *> ; dom ( F | [: N1 , N2 :] ) = dom ( F | [: N1 , N2 :] ) .= [: N1 , N2 :] ; dom ( f . t * I ) = dom ( f . t * I . t ) .= dom ( f . t * I . t ) ; assume a in ( "\/" ( ( T |^ the carrier of S ) , T |^ the carrier of S ) ) .: D ; assume that g is one-to-one and ( the carrier of S ) /\ ( the carrier of S ) c= dom g and g is one-to-one and rng g c= the carrier of S ; ( ( x \ y ) \ ( ( x \ z ) \ ( y \ z ) ) ) \ ( ( x \ y ) \ ( x \ z ) ) = 0. X ; consider f such that f * f = id b & f * f = id b & f * f = id b & f * f = id b ; ( ( cos | [. - PI / 2 , 0 .[ ) | [. - PI / 2 , 0 .[ ) is differentiable ; Index ( p , co ) <= len LS - Index ( p , co ) + 1 + Index ( p , co ) ; t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , ( ( ( Frege ( Frege ( J , K ) ) ) . h ) . h <= ( ( Frege ( J , K ) ) . h ) . ( j , k ) ; then P [ f . ( i0 + 1 ) , f . ( i0 + 1 ) ] & F ( f . ( i0 + 1 ) , f . ( i0 + 1 ) ) < j ; Q [ ( D . ( [ D , x ] ) , F . ( D . ( D . ( D , x ) ) ) , F . ( D . ( D , x ) ) , F . ( D . ( D . ( D , x ) ) ) ] ; 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 Element of G . i ; the Sorts of A2 = ( ( the Sorts of A2 ) --> TRUE ) +* ( ( the Sorts of A2 ) --> TRUE ) +* ( ( the Sorts of A1 ) --> TRUE ) ; consider s being Function such that s is one-to-one & dom s = NAT & rng s = { 0 } and rng s c= { 0 } ; dist ( b1 , b2 ) <= dist ( b1 , a ) + dist ( a , b ) + dist ( a , b ) ; ( for n holds ( for C being Element of C holds ( Cage ( C , n ) ) /. ( len C ) ) `2 = ( /. n ) `2 q <= ( ( UMP L~ Cage ( C , n ) ) .. Cage ( C , n ) ) .. Cage ( C , n ) ; LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} ; given a being ExtReal such that a <= [: I , I :] and A = ]. a , I .] and a < b ; consider a , b being Complex such that z = a & y = b + a and a + b = a + b + c ; set X = { b |^ n where b is Element of NAT : b in X & b in X } , Y = { b where b is Element of NAT : b in X } ; ( ( x * y ) * z ) \ ( x * y ) \ ( x * z ) = 0. X ; set xy = [ <* xy , yz , yz , zx *> , [ <* xy , yz , yz *> , [ ] , [ ] , [ <* xy , yz , yz ] ] , [ <* xy , yz , yz *> , [ ] , [ ] , [ ] , [ ] , [ ] ] , [ <* xy , yz , yz ] ] ] ; ll /. len l = ll . ( len l ) .= l /. len l .= l /. len l .= l /. len l ; sqrt ( ( ( q `2 / |. q .| - sn ) / ( 1 - sn ) ) ^2 ) = 1 ^2 + ( q `2 / |. q .| - sn ) ^2 ; sqrt ( ( p `1 ) ^2 - ( p `2 ) ^2 + ( p `2 ) ^2 ) < 1 - ( p `2 ) ^2 + ( p `2 ) ^2 ; ( ( ( ( ( X \/ Y ) \/ Y ) \/ Y ) \/ X ) \/ Y ) = ( ( X \/ Y ) \/ Y ) \/ ( X \/ Y ) ) \/ ( X \/ Y ) .= ( X \/ Y ) \/ Y ) \/ ( Y \/ X ) ; ( seq - seq ) . k = seq . ( seq . k - seq . ( k + 1 ) ) .= seq . ( ( seq . k ) - seq . ( k + 1 ) ) ; rng ( ( h + c ) ^\ n ) c= dom ( SVF1 ( 1 , f , u0 ) ) ; the carrier of X = the carrier of X & the carrier of X = the carrier of X & the carrier of X = the carrier of X ; ex p3 st p3 = p3 & |. p3 - |[ a , b ]| .| = r & |. p3 - |[ a , b ]| .| = r ; set h = [: X , Y :] , i = [: X , Y :] , j = [: X , Y :] , f = [: X , Y :] , g = [: X , Y :] , h = [: Y , X :] , f = g ; R |^ ( 0 * n ) = ( Ireal ( X , X ) ) |^ ( 0 * n ) .= R |^ ( n + 1 ) ; ( Partial_Sums ( ( ( F1 + F2 ) / ( F2 + G2 ) ) ) | ( ( F1 + F2 ) / ( F2 + G2 ) ) | ( ( F1 + F2 ) / ( F2 + G2 ) ) ) is nonnegative ; f2 = C7 . ( ( E7 ) | ( ( V , the carrier of K ) | ( V , ( V , C ) | ( V , C ) ) ) ) ; S1 . b = s1 . b .= S2 . b .= S2 . b .= S2 . b .= S2 . b ; p2 in LSeg ( p2 , p1 ) /\ LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) .= { p2 } ; dom ( f . t ) = Seg n & dom ( I . t ) = Seg n & dom ( I . t ) = Seg n ; assume o = ( the connectives of S ) . 11 & ( the connectives of S ) . 11 = ( the connectives of S ) . 11 ; set phi = ( l1 , l2 ) \kern1pt , phi = ( l , X ) \kern1pt , phi = ( l , X ) , phi = ( l , X ) , phi = ( l , X ) , phi = ( l , X ) , phi = ( X , Y ) , 1 = ( X , Y ) , 1 = ( X , Y ) , 1 = X , 1 = X , 1 = X , synonym p is invertible means : Def1 : HT ( p , T ) = 1 & HT ( p , T ) = 0. L ; ( Y1 `2 ) ^2 = ( 1 - ( Y1 `2 ) ) ^2 & ( ( Y1 `2 ) ^2 + ( X1 `2 ) ^2 ) = ( Y1 `2 ) ^2 + ( X1 `2 ) ^2 + ( X1 `2 ) ^2 + ( X1 `2 ) ^2 = ( X1 union X2 ) ^2 + ( X2 union X2 ) ^2 ; defpred X [ Nat , set , set ] means P [ $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 , $2 ) ; consider k be Nat such that for n be Nat st k <= n holds s . n < x0 + g ; Det ( I |^ ( m -' n ) ) = ( I |^ ( m -' n ) ) * ( I |^ ( m -' n ) ) .= 0. K ; sqrt ( b - sqrt ( b ^2 - c ^2 ) ) * ( sqrt ( b ^2 - c ^2 ) ) < 0 ; CC . d = CC . ( d1 , d2 ) mod C . ( d2 , d2 ) mod C . ( d2 , d2 ) mod C . ( d2 , d2 ) mod C . ( d2 , d2 ) mod C . ( d2 , d2 ) mod C . ( d2 , d2 ) mod C . ( d2 , d2 ) mod C . ( d2 , d2 ) mod C . ( d2 , d2 ) mod C attr X1 is dense means : Def1 : X1 is dense implies X1 meet X2 is dense implies X1 meet X2 is dense SubSpace of X1 meet X2 & X1 meet X2 = X1 meet X2 ; deffunc F ( Element of E , Element of I ( ) , Element of I ( ) ) = $1 * $2 ; t ^ <* n *> in { t ^ <* i *> : Q [ i , T . t ] } ; ( x \ y ) \ x = ( x \ x ) \ y .= 0. X .= 0. X ; for X being non empty set holds X is Basis of <* X , Y *> synonym A , B , C , D , E , F , G , G , G , H , G , H , G , H , G , H , G , G , H , G , F , G , G , G , H , G , G , H , G , H , G , H , G , H , G , H , G , H implies H , G , G , H , G len ( M1 ^ M2 ) = len p & width ( M1 ^ M2 ) = width M1 & width ( M1 ^ M2 ) = width M1 & width ( M1 ^ M2 ) = width M1 ; J = { x where x is Element of K : 0 < v & v < x & x < 0 } ; ( ( Sgm ( Seg m ) ) . d - ( Sgm ( Seg m ) ) . e ) <> 0 ; lower_bound divset ( D2 , k + 1 ) = D2 . ( k + 1 ) - D2 . ( k + 1 ) ; g . r1 = - 2 * r1 + 1 & dom h = [. 0 , 1 .] & rng h = [. 0 , 1 .] & rng h c= [. 0 , 1 .] ; |. a .| * ||. f .|| = 0 * ||. f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| .= ||. a * f .|| ; f . x = ( h . x ) `1 & g . x = ( h . x ) `1 & g . x = ( h . x ) `1 ; ex w st w in dom B1 & <* 1 *> ^ s = <* 1 *> ^ w & <* 1 *> ^ s = <* 1 *> ^ w ; [ 1 , {} , {} , {} ] in ( { [ 0 , {} , {} ] } \/ [: { 0 , {} } , { 0 } :] \/ [: { 0 , {} } , { 0 } :] ) \/ [: { 0 , {} } , { 0 , {} } :] ; IC Exec ( i , s1 ) + n = IC Exec ( i , s2 ) + n .= IC Exec ( i , s2 ) + n .= ( n + 1 ) + n ; IC Comput ( P , s , 1 ) = IC Comput ( P , s , 1 ) .= 5 + 1 .= 5 + 1 ; ( IExec ( W6 , Q , t ) ) . intpos ( i + 1 ) = t . intpos ( i + 1 ) .= t . intpos ( i + 1 ) ; LSeg ( f , i -' j ) misses LSeg ( f , i -' j ) \/ LSeg ( f , i -' j ) ; assume for x , y being Element of L st x , y ] in C holds x <= y or y <= x or y <= x ; integral ( f , C ) . x = f . ( ( sup C ) . x ) - ( lower_bound C ) . x ; for F , G being FinSequence of NAT st rng F misses rng G & G is one-to-one holds F ^ G is one-to-one ||. R /. ( L . h ) - R /. ( K + 1 ) .|| < e1 * ( K + 1 ) + K * ( L + 1 ) ; assume a in { q where q is Element of M : dist ( z , q ) <= r } ; set p3 = [ 2 , 1 , 0 ] .--> [ 2 , 0 , 1 ] ; consider x , y being Subset of X such that [ x , y ] in F and x c= d and y in d and x in d and y in d ; for y , x being Element of REAL m st y in Y & x in X & y in Y holds x <= y + x ; func |. p ^ <* p *> -> variable of A means : Def1 : for n holds it . n = ( p ^ <* p *> ) . n ; consider t being Element of S such that x , y , z , t is_collinear and x , y , t , z is_collinear and x , z , t is_collinear and x , y , t is_collinear ; dom x1 = Seg len ( x1 ^ y1 ) & len ( x1 ^ y1 ) = len ( x1 ^ y1 ) & len ( x1 ^ y1 ) = len ( x1 ^ y1 ) ; consider y2 being Real such that x2 = y2 & 0 <= y2 and y2 <= 1 and y2 <= 1 and y2 <= 1 and y2 <= 1 and y2 <= 1 / 2 * ( y2 - y2 ) ; ||. f | X , f | X .|| = ||. f | X .|| & ||. f | X .|| = ||. f .|| | X ; ( the InternalRel of A ) | ( the carrier of A ) /\ ( the carrier of A ) = {} .= {} .= {} ; assume that i in dom p and for j being Nat st j in dom p holds P [ i , p . j ] and P [ i , j ] ; reconsider h = f | [: X , Y :] as Function of [: X , Y :] , Y ; u1 in the carrier of W1 & u2 in the carrier of W2 & v2 in the carrier of W1 + W2 implies ( the carrier of W1 ) /\ ( the carrier of W2 ) = { 0. V } defpred P [ Element of L ] means M <= f . $1 & $1 <= $1 implies f . ( $1 + 1 ) <= f . ( $1 + 1 ) ; ( u , a , v , w ) = s * x + ( - ( s * x ) + ( s * x ) + ( s * x ) ) .= b ; - ( R1 + R2 ) = - x + ( - x ) .= - x + ( - x ) .= x + ( - x ) .= x + ( - x ) ; given a being Point of GX such that for x being Point of X holds a , x ] in the topology of X ; fthesis = [ [ dom ( f , g ) , cod ( f , g ) ] , [ cod ( f , g ) , cod ( f , g ) ] ; for k , n being Nat st k <> 0 & k < n & n < m holds ( ( m + n ) * ( k + 1 ) ) |^ ( n + 1 ) = ( m + n ) |^ ( k + 1 ) for x being element holds x in A |^ d iff x in ( ( A ` ) ` ) ` & ( A ` ) ` = ( ( A ` ) ` ) ` consider u , v being Element of R , a being Element of A such that l /. i = u * a * v and a in A * v ; ( ( - ( p `2 / |. p .| - sn ) ) / ( 1 - sn ) ) ^2 > 0 ; LS . k = LS . ( F . k ) & F . ( F . k ) in dom LS & F . ( F . k ) in dom LS ; set i2 = AddTo ( a , i , - n ) , i2 = AddTo ( a , i , - n ) , C = D2 ; pred B is atomic means : Def1 : for S being non empty ManySortedSign holds -sqrt ( ( ( B \/ ( B \/ S ) ) * ( S \/ S ) ) ) = ( B \/ S ) * ( S * ( S * ( S \/ T ) ) ) ; a = { a "/\" D where d is Element of N : d in D & a in D } ; ( ( ( ( ( q `2 / |. q .| - sn ) / ( 1 - sn ) ) / ( 1 - sn ) ) ) ^2 >= ( ( q `2 / |. q .| - sn ) ) ^2 / ( 1 - sn ) ) ^2 ; ( - f ) . ( upper_bound A ) = ( ( - f ) | A ) . ( upper_bound A ) .= ( - f ) . ( upper_bound A ) ; ( G * ( len G , k ) , G * ( len G , k ) ) `1 = ( G * ( 1 , k ) ) `1 .= G * ( 1 , k ) `1 ; ( Proj ( i , n ) ) . ( ( proj ( i , n ) * ( ( proj ( i , n ) * ( ( Proj ( i , n ) * ( ( Proj ( i , n ) * ( ( Proj ( i , n ) * ( ( Proj ( i , n ) * ( ( Proj ( i , n ) * ( ( n + 1 ) * ( ( n + 1 ) * ( ( n + 1 ) * f1 + f2 * reproj ( i , x ) * ( reproj ( i , x ) ) is_differentiable_in ( ( reproj ( i , x ) * ( reproj ( i , x ) ) ) . ( x + h ) . x ) ; pred ( ( ( cos * ( cos * ( cos ) ) ) `| Z ) . x = ( cos * ( cos * ( cos ( ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( ex t being SortSymbol of S st t = s & h1 . t = ( h . t ) . x & t . x = ( h . t ) . x ; defpred C [ Nat ] means ( P [ $1 ] implies ( $1 in A ) & ( $1 in A ) & ( $1 in A ) & ( m in A ) & ( m in A ) & n in A ) ; consider y being element such that y in dom ( ( p | i ) | j ) and ( ( p | i ) | j ) . y = ( p | i ) . y ; reconsider L = product ( { x1 } +* ( index B , l ) ) as Subset of product A ; for c being Element of C holds T . ( id c ) = id c & T . ( id c ) = id c & T . ( id c ) = id c Rotate ( f , n , p ) = ( f | n ) /^ ( p -' 1 ) .= f /^ ( p -' 1 ) .= f /^ ( p -' 1 ) ; ( f * g ) . x = f . ( g . x ) & ( f * h ) . x = f . ( h . x ) ; p in { |[ 1 , 0 ]| * ( G * ( i + 1 , j ) + G * ( i + 1 , j ) ) `1 } ; f `2 - p = ( ( c | ( n , L ) ) *' ) *' ( ( c | ( n , L ) ) *' ) .= ( ( c (#) ( f *' ) ) *' ) *' ; consider r being Real such that r in rng ( f | divset ( D , j ) ) and r < m + r ; f1 . ( |[ r2 , r1 ]| ) in ( f1 .: ( W1 /\ W2 ) ) .: ( W1 /\ W2 ) ; eval ( a | ( n , L ) , x ) = ( a | ( n , L ) ) . x .= a . ( x , x ) ; z = DigA ( t , x ) .= DigA ( t , x ) .= ( DigA ( t , x ) ) . ( ( DigA ( t , x ) ) . ( ( k + 1 ) ) . ( ( k + 1 ) ) . ( ( k + 1 ) ) . ( ( k + 1 ) + 1 ) ) ; set H = { Intersect S where S is Subset-Family of X : S c= G & S c= G } , H = { {} , { {} } where S is Subset-Family of X : S is open } ; consider S19 being Element of D such that S = [: S , <* d *> :] and [: S , <* d *> :] = [: S , <* 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 ) ) / ( 1 - sn ) ; ( for x being VECTOR of V , a , b being Element of A holds Sum ( a * b ) = 0. V ) & Sum ( a * b ) = 0. V & Sum ( b * a ) = 0. V let k1 , k2 , k1 , k2 , k2 be Element of NAT , I , J be Program of SCM+FSA , a , b , c , d be Int-Location , a , b , c , d be Int-Location ; consider j being element such that j in dom a and j in g . ( k + 1 ) and x = a . j and x = a . j ; H1 . ( x1 , x2 ) c= H1 . ( x2 , x1 ) or H1 . ( x2 , x1 ) c= H1 . ( x2 , x1 ) & H1 . ( x1 , x2 ) c= H1 . ( x2 , x1 ) ; consider a being Real such that p = tree * p1 + ( a * p2 ) and 0 <= a and a <= 1 and a <= 1 and a <= 1 and a <= 1 ; assume that a <= c & c <= d and [ a , b ] c= dom f and f . a = g . b ; cell ( Gauge ( C , m ) , 1 , width Gauge ( C , m ) -' 1 , width Gauge ( C , m ) -' 1 , width Gauge ( C , m ) -' 1 ) is non empty ; A5 in { ( S . i ) `1 where i is Element of NAT : P [ i ] } ; ( T * b1 ) . y = L * ( b * ( b * ( b * ( b * ( b * ( b * ( b * ( b * ( b * ( b * ( b * ( b * ( b * ( b * ( b * ( b * ( b * ( b * ( b * ( b * ( b * ( b * ( b * ( b * ( b * ( b * ( b * ( b * ( b * ( g . ( s , I ) . ( s , I ) . y = |. s . ( x , I ) .| & g . ( s , I ) . y = |. s . ( x , I ) . ( s , I ) .| ; ( ( log ( 2 , k ) ) / ( 2 * ( ( 2 * k ) + 1 ) ) ^2 ) >= ( ( 2 * ( ( 2 * k ) + 1 ) ) ^2 ; then p => q in the carrier of ( ( the carrier of A ) \/ { p } ) \/ { q } & q => p in the carrier of ( A ) ; dom ( ( the ^ of r-10 ) | ( dom ( the ^ of r-10 ) ) ) = ( the ^ of ( the ^ of r-10 ) ) | ( dom ( the ^ of r-10 ) ) ; synonym f is extended real-valued means : Def1 : for x being set st x in rng f holds f . x is integer ; assume for a being Element of D holds f . { a } = a & f . ( f .: { a } ) = f . ( union X ) ; i = len p1 .= len p1 + len <* x *> .= len p1 + len <* x *> .= len p1 + len <* x *> .= len p1 + len <* x *> .= len p1 + len <* x *> ; ( l , 3 ) `1 = ( g . ( k , 3 ) ) `1 + ( g . ( k , 3 ) ) `1 .= ( g . ( k , 3 ) ) `1 + ( g . ( k , 3 ) ) `1 ; CurInstr ( P2 , Comput ( P2 , s2 , l ) ) = halt SCM+FSA .= CurInstr ( P2 , Comput ( P2 , s2 , l ) ) .= halt SCM+FSA ; assume for n be Nat holds ||. ( seq . n ) - ( seq . n ) .|| <= ( ( seq . n ) - ( seq . n ) ) & ( ( seq . n ) - ( seq . n ) ) is summable ; sin ( st st Y = sin ( r ) * cos ( ( cos ( r ) * cos ( s ) ) ) .= 0 ; set q = |[ g1 `1 , g2 `2 ]| , g1 = |[ g1 , g2 ]| , g1 = |[ g1 , g2 ]| , g2 = |[ g2 , g1 ]| , g1 = |[ g2 , g2 ]| , g2 = |[ g2 , g1 ]| , g2 = |[ g2 , g1 ]| , g1 = |[ g2 , g2 ]| , g2 = |[ g2 , g1 ]| , g2 = |[ g2 , g2 ]| , g2 = |[ g2 , g1 ]| , g2 ]| ; consider G being sequence of S such that for n being Element of NAT holds G . n in SubSet ( F . n ) ; consider G such that F = G and ex G1 , G2 st G1 in [: X1 , X2 :] & G2 in [: X2 , X1 :] & G2 in [: X1 , X2 :] & G1 = G * ( G1 , G2 ) ; the root of ( ( the Sorts of C ) . s ) . ( ( the Sorts of C ) . s ) . ( ( the Sorts of C ) . s ) . ( ( the_arity_of o ) . s ) . ( ( the_arity_of o ) . s ) . ( ( the_arity_of o ) . s ) . ( ( the_arity_of o ) . s ) . ( ( the_arity_of o ) . s ) . ( ( the_arity_of o ) . s ) = ( the_arity_of o Z c= dom ( ( ( exp_R + ( exp_R * ( ( exp_R + ( exp_R * ( exp_R * ( exp_R + ( exp_R * ( exp_R + ( exp_R * ( exp_R * ( exp_R + ( exp_R * ( exp_R * ( exp_R + ( exp_R * ( exp_R * ( exp_R + ( exp_R * ( exp_R * ( f1 + ( exp_R * ( f1 + ( exp_R * ( f1 + ( exp_R * ( f1 + ( exp_R * ( f1 + ( f1 * for k being Element of NAT holds ( r (#) ( Im ( f ) ) ) . k = ( ( Im ( f ) ) . k ) * ( ( Im ( f ) ) . k ) ; assume that - 1 < sn and sn < 1 and q `2 / |. q .| - sn ) < 0 and q `2 / |. q .| = sn and q `1 / |. q .| = sn ; assume that f is continuous and a < b and c < d and f . a = c and f . b = d and f . a = c ; consider r being Element of NAT such that seq = Comput ( P1 , s1 , r ) and r <= len ( P1 +* I ) and r <= len ( P2 +* I ) ; LE f /. ( i + 1 ) , f /. ( j + 1 ) , f /. ( j + 1 ) , f /. ( j + 1 ) , f /. ( j + 1 ) ; assume that x in the carrier of K and y in the carrier of K and inf { x , y } in the carrier of K and x in the carrier of K ; assume f +* ( i1 , \xi ) in ( ( proj ( F , i2 ) * ( ( proj ( F , i2 ) * ( ( proj ( F , i2 ) * ( f | ( A * ( i , j ) ) ) ) ) ) ) " ( ( proj ( F , i2 ) * ( ( f | ( A * ( i , j ) ) ) ) ) ; rng ( ( Flow M ) | ( the carrier of M ) ) c= the carrier of M & ( ( 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 { t where t is Element of T : t in X } ; consider l be Nat such that for m be Nat st l <= m holds ||. s1 . m - g .|| < s / 2 ; consider t being VECTOR of product G such that [: the carrier of G , the carrier of G :] = ||. ( t + s ) * ( t + s ) .|| and ||. t + s * ( t + t ) .|| <= 1 ; assume that the carrier of v = 2 and v ^ <* 0 *> in dom p and v ^ <* 1 *> in dom p and p ^ <* 0 *> in dom p ; consider a being Element of the points of [ X1 , X2 ] , A being Element of the topology of X2 such that a on A and a on A and b on A ; ( - x ) |^ ( k + 1 ) * ( ( - x ) |^ ( k + 1 ) ) = 1 * ( ( - x ) |^ ( k + 1 ) ) " .= 1 * ( ( - x ) |^ ( k + 1 ) ) " ; for D being set st i in dom p holds p . i in D & p . i in D & p . i in D defpred R [ element ] means ex x , y being element st [ x , y ] = $1 & y in X & P [ x , y ] ; L~ f2 = union { LSeg ( p1 , p2 ) where p1 , p2 is Point of TOP-REAL 2 : p1 `1 = p2 & p2 `1 <= p1 `1 } \/ { p1 , p2 } ; i -' len ( h | ( len h -' 1 ) ) + 1 < i -' len ( h | ( len h -' 1 ) ) + 1 ; for n being Element of NAT st n in dom F holds F . n = |. ( F /. n ) .| + 1 - F /. n .| for r , s1 , s2 , s2 , s3 , s3 , r1 , r2 , r2 , r3 st r1 in [. r , r2 .] & r2 in [. r , s .] & s1 <= s2 holds s1 <= s2 & s2 <= s3 * ( r2 , s2 ) assume v in { G where G is Subset of T2 : G in B & G c= B1 & G c= B2 } ; let g be C.Funcs of A , ( X , Y ) --> 0 & ( X , Y ) --> 0 <> ( X --> 0 ) --> 0 & ( X --> 0 ) --> 0 = ( X --> 0 ) --> 0 ) min ( g . [ x , y ] , k ) = ( min ( g , k ) ) . [ y , z ] ; consider q1 being sequence of CC such that for n holds P [ n , q1 . n ] & P [ n , q1 . ( n + 1 ) ] ; consider f being Function such that dom f = NAT & for n being Element of NAT holds f . n = F ( n ) & f . n = F ( n ) ; reconsider B-6 = B /\ B , BU = O /\ ( Z \/ { Z } ) as Subset of B ; consider j being Element of NAT such that x = ( the thesis of n ) --> ( j , n ) and 1 <= j and j <= n and n <= len f and f /. j = f /. ( n + 1 ) ; consider x such that z = x and card ( x * ( O * ( O * ( O * ( O * ( O * ( O * ( O * ( O * ( O * ( O * ( O * ( O * ( O * ( O * ( O * ( O * ( O * ( O * ( O * ( O * ( O * ( O * ( O * ( O * ( O * ( O * ( O * ( O * ( C * ( [: k , n2 :] ) ) . 0 = C . ( ( ( ( ( ( ( ( ( n , n2 ) --> ( k , n2 ) ) ) . 0 ) ) . 0 ) .= C . ( ( ( ( ( n + 1 ) --> ( k + 1 ) ) ) . 0 ) ; dom ( X --> rng f ) = X & dom ( X --> f ) = X --> dom ( X --> f ) ; ( ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ) <= ( ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ) ; synonym x , y means : Def1 : { x , y } = y or { x , y } c= l ; consider X being element such that X in dom ( f | ( n + 1 ) ) and ( f | ( n + 1 ) ) . X = Y ; assume that \mathop { k } is continuous and for x , y being Element of L st a = x & b = y holds a << b & b << a & a << b ; sqrt ( 1 - ( ( cos * ( arctan * ( arctan ) ) ) ^2 ) * ( ( ( ( sin * ( ( cos * ( arctan ) ) ) ^2 ) ) ) ^2 ) ) * ( ( ( ( cos * ( ( cos * ( arctan ) ) ^2 ) ) ^2 ) ) * ( ( ( cos * ( ( cos * ( arctan ) ) ^2 ) ) ^2 ) ) ) * ( ( cos * ( ( cos * ( cos * ( ( cos * ( ( cos * ( ( cos * ( ( cos * ( ( sin * ( ( sin * defpred P [ Element of omega \omega ( A1 , B1 ) , set ] means ( $1 in A1 & $2 = A1 . $1 ) & $2 = ( $1 + 1 ) * ( $1 + 1 ) ; IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 2 ) .= 6 + 1 .= 6 + 1 .= 6 + 1 ; f . x = f . ( g1 * f . ( g1 * f . ( g1 * f . ( g1 * f . ( g1 * f . ( g1 * f . ( g1 * f . ( g1 * f . ( g1 * f . ( g1 * f . ( g1 * f . ( g1 * f . ( g1 * f ) ) ) ) ) ) ) ) .= f . ( g1 * f . ( g1 * f . ( g1 * f . ( g1 * f . ( g1 * f . ( g1 * f ( M * ( F-4 ) ) . n = M . ( ( F * ( ( id [#] ( Omega ) ) ) . 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 L2 ) .= the carrier of L1 + ( the carrier of L2 ) .= the carrier of L1 + ( the carrier of L2 ) .= the carrier of L1 + ( the carrier of L2 ) ; pred a , b , c , x , y , z is_collinear means : Def1 : a , b , c , x , y is_collinear & a , b , z is_collinear & a , c , x is_collinear & b , c , y is_collinear & a , c , x , y is_collinear & a , b , z is_collinear & b , c , x , y is_collinear & a , c , x , y is_collinear & b , c , x is_collinear & a , b , y is_collinear & b , c , z is_collinear & a , c , y , ( ( the PartFunc of S , T ) . n ) . n <= ( ( the Sorts of A ) * ( ( the Sorts of A ) . n ) ) . ( ( the Sorts of A ) . ( ( the Sorts of A ) . n ) ; pred - 1 <= r & r <= 1 implies ( ( ( - 1 ) (#) ( ( r - 1 ) * ( ( r - 1 ) * ( ( r - 1 ) * ( ( r - 1 ) * ( ( r - 1 ) * ( ( r - 1 ) * ( ( r - 1 ) * ( ( r - 1 ) * ( ( r - 1 ) * ( ( r - 1 ) * ( ( r - 1 ) * ( ( r - 1 ) ) ) ) ) ) ) ) ) ^2 = s8 in { p ^ <* n *> where n is Nat : p ^ <* n *> in T } ; |[ x1 , x2 , x3 ]| . 2 - |[ y1 , y2 ]| . 2 = ( x2 - y2 ) . 2 - ( y2 - y1 ) . 2 .= ( x2 - y2 ) . 2 - ( y2 - y1 ) . 2 ; attr F . m is nonnegative means : Def1 : ( Partial_Sums F ) . m is nonnegative & ( Partial_Sums F ) . m is nonnegative ; len ( ( G . ( ( y - z ) ) * ( G . ( y - z ) ) ) ) = len ( G . ( ( y - z ) * ( G . ( y - z ) ) ) ) ; consider u , v being VECTOR of V such that x = u + v and u in W1 /\ W2 and v in W2 /\ W1 and u in W2 /\ W1 and v in W2 /\ W1 and v in W1 /\ W2 and u in W2 /\ W2 and v in W1 /\ W2 ; given F being FinSequence of NAT such that F = x & dom F = n & rng F c= { 0 , 1 } and F is one-to-one & F is one-to-one & rng F c= { 0 , 1 } and F is one-to-one ; 0 = @ @ @ 1- @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ consider n be Nat such that for m be Nat st n <= m holds |. ( f # x ) . m - ( f # x ) . n .| < e ; cluster non empty Boolean for RelStr (# carrier of L , ( ( ML ) | ( D ) ) , ( ( ML ) | ( D ) ) , ( ( ML ) | D ) #) -> Boolean ; "/\" ( B , L ) = "\/" ( B , L ) .= "\/" ( [#] ( S , L ) , L ) .= "\/" ( [#] ( S , L ) , L ) .= "\/" ( [#] ( S , L ) , L ) .= "\/" ( ( S , L ) , L ) .= "\/" ( ( S , L ) , L ) ; sqrt ( r ^2 + ( r ^2 + ( r ^2 + ( r ^2 + ( r ^2 + 1 ) ) ^2 ) ) ^2 <= sqrt ( r ^2 + ( r ^2 + ( r ^2 + 1 ) ^2 ) ) ^2 + ( r ^2 + ( r ^2 + 1 ) ^2 ) ; for x being element st x in A /\ dom ( f `| X ) holds ( ( f `| X ) `| X ) . x >= r2 2 * r1 - ( 2 * |[ a , c ]| - ( 2 * |[ b , c ]| ) ) = 0. TOP-REAL 2 + ( 2 * |[ a , c ]| ) .= 0. TOP-REAL 2 ; reconsider p = P /. ( \square , 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 be Nat st 1 <= n & n <= len q1 holds q1 . n = ( ( upper_bound rng g ) | ( len q1 ) ) . n consider y , z being element such that y in the carrier of A & z in the carrier of A and y = [ y , z ] and i = [ y , z ] ; given H1 , H2 being strict Subgroup of G such that x = H1 & y = H2 & H1 = H2 and H2 = H1 & H1 = H2 and H2 = H2 and H1 = H2 and H2 = H1 & H1 = H2 ; for S , T being non empty RelStr for d being Function of T , S , e being Function of T , S st d is directed-sups-preserving & e is monotone holds d is monotone & e is monotone [ a + 0 , i + ( b + ( b + a ) ) ] in ( the carrier of V ) & [ a , b ] in ( the carrier of V ) ; reconsider m5 = max ( len F1 , len <* x *> ) * ( <* x *> |^ n ) as Element of NAT ; I <= width GoB ( GoB ( GoB ( h , n ) , 1 ) , ( GoB ( h , n ) ) * ( 1 , 1 ) ) ; f2 /* q = ( f2 /* ( f1 /* ( s ^\ k ) ) ) ^\ k .= ( f2 /* ( s ^\ k ) ) ^\ k .= ( f2 /* ( s ^\ k ) ) ^\ k ; attr A1 : : Def1 : A1 : A1 : A1 : A1 : A1 : A1 : A1 : A1 : A1 : A1 : A1 : A1 : A1 : A1 : A1 : A1 : A1 : A1 : A1 : A1 : A1 : A1 : A1 : A1 : A1 : A1 : A1 is linearly-independent & A2 c= A1 & A1 is linearly-independent & A1 is linearly-independent & A2 c= A1 & A1 c= A2 & A1 c= A1 & A2 c= A1 & A1 c= A2 & A1 is linearly-independent & A1 is linearly-independent & A2 c= A1 & A2 c= A1 & A1 is linearly-independent & A1 c= A1 & A2 func A -carrier C -> set equals union { A where A is Element of R : A in C } ; dom ( Line ( v , i + 1 ) (#) ( ( Line ( v , i ) ) * ( ( Line ( v , i ) ) * ( ( Line ( v , i ) ) * ( ( Line ( v , i ) ) * ( ( Line ( v , i ) ) * ( ( i , j ) * ( ( i , j ) * ( ( i , j ) * ( ( i , j ) * ( i , j ) ) ) ) ) ) ) = dom ( ( ( i , j ) * ( ( i , j ) * ( ( i cluster [ x , y ] -> ( x , y ) `1 & ( x , y ) `1 = ( x , y ) `1 & ( x , y ) `2 = ( x , y ) `2 & ( x , y ) `2 = ( x , y ) `2 ; E , ( All ( x2 , x1 ) '&' ( ( x2 '&' ( x2 '&' ( x3 '&' x4 ) ) '&' ( x3 '&' x4 ) '&' ( x4 '&' x4 ) '&' ( x4 '&' x4 ) '&' ( x4 '&' x4 ) '&' ( x4 '&' x4 ) '&' ( x4 '&' x4 ) '&' ( x4 '&' x4 ) '&' ( x4 '&' x4 ) '&' ( x4 '&' x4 ) '&' ( x4 '&' x4 ) '&' ( x4 '&' x4 ) '&' E '&' E '&' E '&' ( E '&' x4 ) '&' ( E '&' x4 ) '&' ( x4 '&' x4 ) '&' ( x4 '&' x4 ) '&' F .: ( id X , g ) . x = F . ( id X , g ) .= F . ( id X , g ) .= F . ( F . ( g , f ) ) .= F . ( F . ( g , f ) ) ; R . ( h . m ) = F . x0 + ( h . m ) + ( h . m ) + ( h . m ) + ( h . m ) + ( h . m ) ) ; cell ( G , ( X1 -' 1 , Y + 1 ) , Y ) \ ( ( the carrier of X + Y ) \/ ( the carrier of Y ) \/ ( the carrier of Y ) ) meets ( the carrier of Y ) \/ ( ( the carrier of Y ) \/ ( the carrier of Y ) ) ; IC Comput ( P2 , s2 , LifeSpan ( P2 , s2 ) + 1 ) = IC Comput ( P2 , s2 , LifeSpan ( P2 , s2 ) + 1 ) .= card I + card J + 2 .= card I + 2 + 2 .= card I + 2 + 2 + 1 .= card I + 2 + 2 + 1 + 1 .= card I + 2 + 2 + 1 ; sqrt ( ( ( ( ( ( ( ( ( q `2 / |. q .| - sn ) / ( 1 - sn ) ) / ( 1 - sn ) ) ) / ( 1 - sn ) ) ^2 ) ) ^2 ) > 0 ; consider x0 being element such that x0 in dom a and x0 in dom a and a . x0 = ( g . x0 ) . x0 and ( g . x0 ) . x0 = a . x0 ; dom ( r1 (#) ( f | A ) ) = dom ( ( f | A ) | A ) .= A /\ ( ( f | A ) | A ) .= C /\ ( A /\ A ) .= C /\ ( A /\ A ) .= C /\ ( A /\ A ) .= C /\ ( A /\ A ) .= C /\ ( A /\ A ) ; d1 . ( y , z ) = ( ( ( y , z ) `1 ) * ( ( y , z ) `2 ) * ( ( y , z ) `2 ) * ( ( y , z ) `2 ) * ( ( y , z ) `2 ) * ( ( y , z ) `2 ) * ( ( y , z ) `2 ) * ( ( y , z ) `2 ) * ( ( y , z ) `2 ) ; pred for i being Nat holds C . i = A . i /\ B . i & C . i c= C . i /\ B . i ; assume that x0 in dom f and f | X is continuous and for x st x in X holds ( f | X ) . x = ( f | X ) . x and ( f | X ) . x = ( f | X ) . x ; p in Cl A implies for Q being Basis of p , A being Subset of T st Q in A & Q in A holds A meets Q for x being Element of REAL n st x in Line ( x1 , x2 ) holds |. ( y1 - y2 ) . x - ( y2 - x2 ) . x .| <= |. y1 - y2 .| func <* a *> -> Ordinal of a means : Def1 : a in it & for b being Ordinal st a in it holds it . b c= a & it . b = b ; [ a1 , a2 , a3 ] in ( the carrier of A ) & ( the carrier of A ) c= ( the carrier of A ) \/ ( the carrier of A ) ; ex a , b being element st a in the carrier of S1 & b in the carrier of S2 & x = [ a , b ] & x = [ a , b ] ; ||. ( v - ( n + 1 ) ) * ( ( v - ( n + 1 ) ) * ( x - ( v + ( n + 1 ) ) * ( x - ( v + ( n + 1 ) ) * ( x - ( v + ( n + 1 ) ) * ( x - ( v + ( n + 1 ) ) * ( x - ( v + ( n + 1 ) ) ) ) ) .|| < e ; then for Z being set st Z in { Y where Y is Element of [: I , I :] , I is finite Subset of I : z in Z & Y in Z } holds z in Z ; sup ( compactbelow ( s , t ) ) = [ sup ( { s } , t ) , t ] .= sup ( { s } , t ) .= t ; consider i , j being Element of NAT such that i < j and [ y , f . i ] in [: I , J :] and [ f . i , f . j ] in [: I , J :] and f . i = f . j ; for D being non empty set , p , q being FinSequence of D st p c= q & q in D holds p ^ q = q ^ p consider e1 being Element of the carrier of X such that c9 , a9 // a9 , b9 and a , c // a9 , b9 and a9 , b9 // b9 , c9 and a , c // a9 , b9 and a , c // a9 , b9 and a , c // a9 , b9 and a , c // a9 , b9 and a , c // a9 , b9 // b9 , c9 and a , c // b9 , c9 ; set U2 = I \! \mathop { + } ( { {} } , {} ) ; |. q1 .| ^2 = ( ( ( q1 `2 / |. q1 .| - sn ) / ( 1 - sn ) ) ^2 + ( ( ( q1 `2 / |. q1 .| - sn ) / ( 1 - sn ) ) ^2 ) ^2 + ( ( q `2 / |. q .| - sn ) / ( 1 - sn ) ) ^2 .= ( q `2 / |. q .| ) ^2 ; for T being non empty TopSpace , x , y being Element of [: the topology of T , the topology of T :] holds x "\/" y = x "\/" y & x "/\" y = x "\/" y & x "\/" y = x "\/" y dom ( ( the charact of U1 ) * the charact of U2 ) = dom ( ( the charact of U1 ) * the Arity of U2 ) & rng ( ( the charact of U1 ) * the Arity of U2 ) = ( the charact of U1 ) * the Arity of U2 ) & ( the charact of U1 ) * the Arity of U2 = ( the charact of U1 ) * the Arity of U2 ; dom ( h | X ) = dom h /\ X .= dom ( ( h | X ) | X ) .= X /\ ( ( h | X ) | X ) .= X /\ ( ( h | X ) | X ) .= X /\ ( ( h | X ) | X ) .= X /\ ( ( h | X ) | X ) .= X /\ ( ( h | X ) | X ) ; for N1 , N1 , N2 being Element of [: N1 , N2 :] holds ( h . N1 ) . N1 = N1 & ( h . N1 ) . N1 = ( h . N1 ) . N1 & ( h . N1 ) . N1 = ( h . N1 ) . N1 ) . N1 & ( h . N1 ) . N1 = ( h . N1 ) . N1 ) . N1 ( mod ( u , m ) + mod ( v , m ) ) . i = ( mod ( u , m ) ) . i + ( mod ( v , m ) ) . i .= ( mod ( v , m ) ) . i + ( mod ( v , m ) ) . i ; - ( q `2 / |. q .| - sn ) ^2 < - ( q `2 / |. q .| - sn ) & ( q `2 / |. q .| - sn ) ^2 >= ( q `2 / |. q .| - sn ) ^2 & ( q `2 / |. q .| - sn ) ^2 >= ( q `2 / |. q .| - sn ) ^2 ; pred r1 = f9 & r2 = ( f | X ) | X & ( f | X ) | X = ( f | X ) | X & ( f | X ) | X = f | X & ( f | X ) | X = f | X ) ; ( for m be bounded Function of X , Y holds ( for m be Nat holds ||. ( vseq ( vseq ( X , Y ) ) ) . m - ( vseq ( X , Y ) ) . x ) = ( seq_id ( vseq ( X , Y ) ) ) . x ) * ( seq_id ( vseq ( Y , X ) ) . x ) pred a <> b & b <> c & angle ( a , b , c ) = PI & angle ( b , c , a ) = PI & angle ( a , c , b ) = PI & angle ( b , c , a ) = PI & angle ( a , c , b ) = PI & angle ( a , b , c , a ) = PI ; consider i , j being Nat , r being Real such that p1 = |[ i , r ]| and r < j and r < 1 and r < 1 and r < 1 and r < 1 / 2 * ( i , j ) ; |. p .| ^2 - ( 2 * ( p `2 / |. p .| - sn ) ) ^2 = |. p .| ^2 + ( p `2 / |. p .| - sn ) ^2 ; consider p1 , q1 being Element of [: X , Y :] such that y = p1 ^ q1 and q1 in X and p1 ^ q1 = p1 ^ q1 and q1 = p1 ^ q2 and q1 = p1 ^ q2 and q2 = p2 ^ q2 ; |[ ( 1 / ( r1 + r2 ) ) / ( s1 + s2 ) , ( 1 / ( s1 + s2 ) ) / ( s1 + s2 ) ]| = sqrt ( s2 + ( s1 + s2 ) / ( s1 + s2 ) ) .= sqrt ( s2 + ( s1 + s2 ) ) ; ( ( LMP A ) | ( ( proj2 .: ( ( TOP-REAL n ) /\ ( TOP-REAL n ) ) /\ ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( TOP-REAL n ) ) ) ) ) ) ) & ( proj2 .: ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( TOP-REAL n ) ) ) ) ) is non empty ; s , ( H / ( H1 , H2 ) ) |= H2 iff s , ( H / ( H1 , H2 ) ) |= H2 '&' ( H1 '&' H2 ) ; len ( s + 1 ) = card ( ( support ( b ) ) + ( support ( b ) ) ) .= card ( ( support ( b ) ) + ( { b } ) ) .= card ( ( support ( b ) ) + { b } ) .= card ( ( support ( b ) ) + { b } ) .= card ( { b } ) ; 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 >= y ; LSeg ( ( UMP D ) . ( ( ( sup D ) `2 ) / 2 ) * ( ( sup D ) / 2 ) * ( ( sup D ) / 2 ) * ( ( sup D ) / 2 ) * ( ( sup D ) / 2 ) ) / 2 ) = { ( sup D ) / 2 * ( ( sup D ) / 2 ) * ( ( sup D ) / 2 ) / 2 * ( ( sup D ) / 2 ) * ( ( sup D ) / 2 ) ) ; lim ( ( ( f `| N ) /* ( g `| N ) ) /* b ) = ( ( f `| N ) /* ( g `| N ) ) . b .= ( f `| N ) . b .= ( f `| N ) . b ; P [ i , pr1 ( f , i ) . ( pr1 ( f , i ) . ( pr1 ( f , i ) . ( pr1 ( f , i ) . ( pr1 ( f , i ) . ( pr1 ( f , i ) . ( pr1 ( f , i ) . ( pr1 ( f , i ) . ( i + 1 ) ) ) ) ] ; for r be Real st 0 < r ex m be Nat st for k be Nat st m <= k holds ||. ( seq . m ) - ( lim ( seq . m ) ) .|| < r for P being set , P being a_partition of X , a , b being Element of P , x being set st x in P & P in P & a in P & b in P & x in P & a in P & b in P & x in P & a in P & b in P & x in P & b in P & a in P & x in P holds a = b Z c= dom ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ex j being Nat st j in dom ( l ^ <* x *> ) & j < i & i = ( l ^ <* x *> ) . j & j = ( l ^ <* x *> ) . j & j = ( l ^ <* x *> ) . j ; for u , v being VECTOR of V , r being Real st 0 < r & u in N holds r * u + ( r * v ) + ( r * ( v + ( r * u ) ) ) in N * N A , B , C , D , E , F , G , G , G , H , G , H , G , H , G implies A , B , C , G , H , G , G , H , G , G , H , G , G , H , G , H , G , H , G , H , G , H , G , H , G , H , G , H , H , G ) , H , H , G , G , H , G , G , G , H , G , G , G , G , H , G , H , G , H , G , G , - Sum <* v , u , w *> = - ( v + u + w ) .= - ( v + u ) + ( w + u ) .= ( v + u ) + ( w + u ) .= ( v + u ) + ( w + u ) .= ( v + u ) + ( w + u ) ; ( Exec ( a := b , s ) ) . IC SCM R = ( Exec ( a := b , s ) ) . IC SCM R .= Exec ( a := b , s ) . IC SCM R .= Exec ( a := b , s ) . IC SCM R .= Exec ( a := b , s ) . IC SCM R .= succ IC s ; consider h being Function such that f . a = h & dom h = I & for x being element st x in I holds h . x = ( the carrier of J ) . x ; for D being non empty reflexive RelStr for S1 , S2 being non empty Subset of [: S1 , S2 :] , S1 , S2 being non empty Subset of S2 holds cos ( S1 ) is directed & cos ( S2 ) is directed & cos ( S1 ) is directed & cos ( S2 ) is directed card X = 2 implies ex x , y st x in X & y in X & x in Y & y in Y & x <> y & x in Y & y in Y & x = y or x = y ; ( E-max L~ Cage ( C , n ) ) .. ( Cage ( C , n ) ) in rng ( Cage ( C , n ) -: Cage ( C , n ) ) ; for T , T being tree of T , p , q being Element of dom T holds ( T -tree ( p , q ) ) . ( p , q ) = T . ( q , p ) [ i2 + 1 , j2 ] in Indices G & f /. ( i2 + 1 ) = G * ( i2 , j2 ) & f /. ( k + 1 ) = G * ( i2 , j2 ) ; cluster ( k , n ) divides ( ( k , n ) div ( m , n ) ) & ( k , n ) divides ( ( m , n ) div ( m , n ) ) & ( n , m ) div ( m , n ) ) divides ( ( m , n ) div ( m , n ) ) ; dom F = the carrier of X1 & rng F = the carrier of X2 & F is one-to-one & rng F = the carrier of X2 & F is one-to-one & rng F = the carrier of X2 & F is one-to-one & rng F = the carrier of X2 & F is one-to-one & rng F = the carrier of X2 & F is one-to-one & rng F = the carrier of X2 & F is one-to-one & rng F = the carrier of X2 & F is one-to-one & F is one-to-one implies F is one-to-one consider C being finite Subset of V such that C c= A and card C = n and the carrier of V = Lin ( B \/ C ) and C = Lin ( B \/ C ) ; V is prime implies for X , Y being Subset of [: the topology of T , the topology of T :] st X /\ Y c= V & Y c= V holds X c= Y or Y c= V set X = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } , Y = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } , Z = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } , Z = { F ( v1 ) where v1 is Element of B ( ) : P [ v1 ] } , Z ( ) = { F ( v1 ) : P [ v1 ] } ; angle ( p1 , p3 , p2 ) = 0 .= angle ( p2 , p3 , p2 ) .= angle ( p2 , p3 , p2 ) .= angle ( p2 , p3 , p2 ) .= angle ( p2 , p2 , p3 ) .= angle ( p2 , p3 , p2 ) .= angle ( p2 , p2 , p4 ) ; - sqrt ( ( ( q `2 / |. q .| - sn ) / ( 1 - sn ) ) ^2 ) = - ( q `1 / |. q .| - sn ) ^2 .= - ( q `1 / |. q .| - sn ) ^2 / ( 1 - sn ) ^2 ; ex f being Function of I[01] , TOP-REAL 2 st f is continuous one-to-one & rng f = P & f . 0 = p1 & f . 1 = p2 & f . 0 = p1 & f . 1 = p2 & f . 1 = p2 & f . 1 = p3 & f . 0 = p4 ; pred f is_differentiable on One means : Def1 : ( SVF1 ( 2 , f , 3 ) ) . y0 - ( SVF1 ( 2 , f , 3 ) ) . y0 = ( proj ( 2 , 3 ) ) . y0 - ( proj ( 2 , 3 ) ) . y0 ; ex r , s st x = |[ r , s ]| & ( G * ( 1 , 1 ) `1 < r & r < G * ( 1 , 1 ) `2 & s < G * ( 1 , 1 ) `2 & r < G * ( 1 , 1 ) `2 & s < G * ( 1 , 1 ) `2 ; assume that f is special and 1 <= t and t <= len G and t <= width G and G * ( t , width G ) `2 >= G * ( t , width G ) `2 ; pred i in dom G means : Def1 : r * ( ( f (#) reproj ( i , x ) ) * ( reproj ( i , x ) ) ) = r * ( reproj ( i , x ) ) ; consider c1 , c2 being bag of o1 + o2 such that ( O /. k ) = <* c1 , c2 *> and ( O /. k ) = ( ( O /. k ) + ( O /. k ) ) + ( ( O /. k ) + ( O /. k ) ) + ( ( O /. k ) + ( O /. k ) ) ; x0 in { |[ r1 , s1 ]| : r1 < r1 & r1 < G * ( 1 , 1 ) `1 & G * ( 1 , 1 ) `2 < G * ( 1 , 1 ) `2 } ; Cl ( X ^ Y ) = the carrier of X .= C4 ( ( Y + X ) \/ Y ) .= C4 ( Y ) .= C4 ( Y ) .= C4 ( Y ) ; pred len M1 = len M2 & width M1 = width M2 & width M2 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M1 = width M2 & width M2 = width M2 ; consider g2 be Real such that 0 < g2 and { y where y is Point of S : ||. y - x0 .|| < g2 & g2 < x0 } c= N2 & ( for y be Point of S st y in N holds ||. ( y - x0 ) .|| < g2 . y ) * ( ||. x0 - x0 .|| ) ; assume x < sqrt ( - ( b ^2 + c ^2 ) ) * ( 2 * a ) or x > - ( b ^2 + c ^2 ) * ( 2 * a ) + ( c ^2 + c ^2 ) * ( 2 * a ) + ( c ^2 + c ^2 ) * ( 2 * a ) + ( c ^2 + c ^2 ) * ( 2 * a ) + ( c ^2 + c ^2 ) * ( 2 * a ) + assume assume x < c ; ( G1 '&' G2 ) . i = ( <* 3 *> ^ ( G1 ^ G2 ) ) . i & ( G1 '&' G2 ) . i = ( <* 3 *> ^ ( G1 ^ G2 ) ) . i & ( G1 ^ G2 ) . i = ( <* 3 *> ^ ( G1 ^ G2 ) ) . i ; for i , j st [ i , j ] in Indices ( M2 + M1 ) & j in dom M2 holds ( M2 + M2 ) * ( i , j ) < M2 * ( i , j ) for f being FinSequence of NAT , j being Element of NAT st j in dom f & j in dom f holds f . j divides f /. j & f /. j = f /. j assume F = { [ a , b ] where a , b is Element of X : for c being set st c in B holds a c= c & b c= c & c c= c & a c= b } & b c= c & c c= c & c c= d & d c= b & b c= c & d c= c ; b2 * q2 + ( b2 * q2 ) + ( ( b2 * q1 ) + ( ( b1 * q2 ) + ( b2 * q2 ) ) + ( ( b1 * q2 ) + ( ( b2 * q2 ) + ( ( b1 * q2 ) + ( b2 * q2 ) + ( ( b1 * q2 ) + ( b2 * q2 ) ) ) = ( ( b1 * q2 ) + ( b2 * q2 ) + ( ( b2 * q2 ) + ( b1 * q2 ) ) + ( ( b1 * q2 ) + ( b2 * q2 ) + ( ( b2 * q2 ) + ( ( b1 * q2 ) + ( b2 * q2 ) + ( b1 * q2 ) + ( b2 * q2 ) + ( card F = card D where D is Subset of T : ex B being Subset of T st B = union F & B in F & B c= F & F is closed & B c= F & F is closed & F is closed & F is closed & F is closed & F is closed & F is closed & F is closed & F is closed & F is closed & F is closed & for B being Subset of T st B in F holds F is closed & F is closed & F is closed & F is closed & F is closed & F is closed & F is closed & F is closed & F is closed & F is closed & F is closed & F is closed & F is closed & F is closed attr seq is summable means : Def1 : seq is summable & seq is summable & lim seq = Sum ( seq ) implies Sum ( seq ) = Sum ( seq ) + Sum ( seq ) + Sum ( seq ) ; dom ( ( ( cn ) | D ) | D ) = ( ( ( cn ) | D ) | D ) .= ( ( ( cn ) | D ) | D ) | D .= ( ( cn ) | D ) | D ) | D .= ( ( cn ) | D ) | D .= ( ( cn ) | D ) | D ) | D .= ( ( cn ) | D ) | D ) | D ; [ X \to Z , Y ] is full SubRelStr of ( ( Omega X ) |^ ( [#] Y ) ) |^ ( [#] Y ) , ( id ( ( Omega Y ) |^ ( [#] Y ) ) ) |^ ( [#] Y ) ) ; ( G * ( 1 , j ) ) `2 = ( G * ( 1 , j ) ) `2 & ( G * ( 1 , j ) ) `2 <= ( G * ( 1 , j ) ) `2 ; pred m1 c= m2 means : Def1 : for p being set st p in P holds the carrier of ( m1 , m2 ) <= ( the carrier of ( m1 , m2 ) ) | ( ( m1 , m2 ) | ( ( m2 , m2 ) | ( ( m2 , m2 ) | ( m2 , m1 ) ) ) ; consider a being Element of [: B ( ) , C ( ) :] such that x = F ( a ) & a in { G ( ) where b is Element of B ( ) : P [ b ] } and P [ b ] ; func multiplicative M -> Function of the carrier of M , the carrier of M , the carrier of M means : Def1 : the carrier of it = [: the carrier of M , the carrier of M :] & the carrier of it = [: the carrier of M , the carrier of M :] ; l ( a , b , c , d ) + l ( c , d , b ) = b + c .= b + c + d .= b + d + c .= b + d + d .= b + d + c + d .= b + d + d + c .= b + d + d + d .= b + d + d + d .= b + d + d + d + c ; cluster -> real for Element of REAL n , i1 , i2 , j2 be Element of REAL n n -tuples_on REAL n , i2 , j2 be Element of REAL n ; ( ( s2 * p1 + ( s1 * p2 ) - ( s2 * p2 ) ) ) * ( ( s1 * p2 ) + ( s2 * p2 ) ) * ( ( s2 * p2 ) + ( s2 * p2 ) ) * ( ( s2 * p2 ) + ( s2 * p2 ) ) * ( ( s2 * p2 ) + ( s2 * p2 ) ) * ( ( s2 * p2 ) + ( s2 * p2 ) ) * ( ( s2 * p2 ) ) * ( ( s2 * p2 ) ) ) = ( ( s2 * p2 ) ) * ( ( s2 * p2 ) ) * ( ( s2 * p2 ) ) ; eval ( ( a | ( n , L ) *' p ) , x ) = eval ( a | ( n , L ) , x ) .= a * eval ( p , x ) .= a * eval ( p , x ) ; assume that the TopStruct of S = the TopStruct of T and for D being non empty Subset of S , f being Function of Omega S , Omega T holds f .: D meets f .: ( [#] S ) & f .: D = f .: ( [#] S ) ; assume that 1 <= k and k <= len w + 1 and T . ( ( ( q , w ) . k ) , w ) = ( ( T . ( k + 1 ) ) , w . ( k + 1 ) ) , w . ( ( T . ( k + 1 ) ) , w . ( k + 1 ) ) ] ; 2 * a + ( 2 * ( b + 1 ) ) * ( b + 1 ) >= ( a + b ) * ( a + b ) + ( b + 1 ) * ( a + b ) + ( b + 1 ) * ( a + b ) + ( b + 1 ) * ( a + b ) + ( b + 1 ) * ( a + b ) ) + ( b + 1 ) * ( a + b ) >= ( a + b ) * ( a + b ) + ( b + 1 ) * ( a + b ) * ( a + b ) * ( a + b ) * ( a + 1 ) * ( a + b ) * ( a + b ) * ( a + b ) * ( a + b ) * ( a M , v / ( All ( x , H ) '&' ( All ( x , H ) '&' ( All ( x , H ) '&' ( All ( x , H ) '&' ( All ( x , H ) '&' ( All ( x , H ) '&' ( All ( x , H ) '&' ( ( All ( x , H ) '&' ( ( All ( x , H ) '&' ( ( All ( x , H ) '&' ( ( All ( x , H ) '&' ( ( ( x , H ) '&' ( ( ( x , H ) '&' ( ( ( ( x , H ) '&' ( ( ( x , H ) '&' ( ( ( x , H ) '&' ( ( ( x , H ) '&' ( ( x , H ) '&' ( ( x , H ) '&' ( ( assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < f /. x0 and for x1 st x1 in l holds f /. x1 - f /. x0 < f /. x1 + f /. x0 ; for G1 being _Graph , W being Walk of G1 , e being Vertex of G2 , x being set st e in W holds not W is walk of G2 & e in W c9 is not empty iff not ( ( ( ex y1 , y2 st y1 in \mathbin { 0 } & y2 in \mathbin { 0 } & not ( ( not ( ex y st y in \mathbin { 0 } & y in \mathbin { 0 } ) ) & not ( ( not y in \mathbin { 0 } ) & ( not ( ( y in Y ) ) & ( ( y in Y ) \ ( y in Y ) ) & ( y in Y ) \ ( ( y in Y ) ) & ( ( y in Y ) \ ( y in Y ) ) & ( y in Y ) \ ( y in Y ) ) & ( y in Y ) \ ( y in Y ) ) & ( y in Y ) \ ( y in Y ) \ ( y in Y ) \ ( Indices GoB f = [: dom f , Seg width GoB f :] & width GoB f = width GoB f & width GoB f = width GoB f & width GoB f = width GoB f & width GoB f = width GoB f & width GoB f = width GoB f & width GoB f = width GoB f & width GoB f = width GoB f ; for G1 , G2 being Subgroup of O , G2 being Subgroup of O , G1 , G2 being Subgroup of O , G2 , G2 being Subgroup of G1 , G2 being Subgroup of G2 , G1 , G2 being Subgroup of G2 , G2 , G1 being Subgroup of G1 holds G1 is Subgroup of G1 & G2 = G2 & G1 = G2 & G2 = G2 & G1 = G2 & G2 = H & G1 = H & G2 = H & G1 = H & G2 = H & G1 = H & G2 = H & G2 = H & G2 = H & G2 = H & G2 = H & G2 = H = H & H = H implies G1 = H & G2 = H & G2 = H & H = H & H = H & H = H & H = H & H = H & UsedIntLoc ( ( f . ( intloc 0 ) ) , ( f . ( intloc 0 ) ) , ( f . ( intloc 0 ) ) , ( f . ( intloc 0 ) ) ) = { f . ( ( intloc 0 ) .--> ( 1 ) ) , ( f . ( intloc 0 ) ) , ( f . ( intloc 0 ) ) , ( f . ( intloc 0 ) ) , ( f . ( intloc 0 ) ) , ( f . ( intloc 0 ) ) } ; for f1 , f2 being FinSequence of F st f1 ^ f2 is p -element & f2 is p & ( for n being Nat holds f1 . n = p ^ f2 ) & ( for n being Nat holds f2 . n = p . ( n + 1 ) ) holds f2 . ( n + 1 ) = p . ( n + 1 ) sqrt ( ( p `1 ) ^2 + ( p `2 ) ^2 + ( p `2 ) ^2 + ( p `2 ) ^2 ) = sqrt ( ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 ) .= ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 ; for x1 , x2 , x3 , x4 , x5 , N , M , N , N , M , N , N , N , N , M , N , N , N , M , N , N , N , N , M , N , N , N , N , M , N , N , N , N , M , N , N , N , N , N , N , N , M , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , M , N , N , M , N , N , N , N , N , N , N , N , N for x st x in dom ( ( ( \rbrack ) | A ) | A ) holds ( ( ( \rbrack ) | A ) | A ) . x = - ( ( ( \rbrack ) | A ) | A ) . x ) for T being non empty TopSpace , P being Subset-Family of T , B being Basis of T , x being Point of T st P c= the topology of T & B c= P & P c= P holds P is Basis of T ( a 'or' b ) . x = 'not' ( ( a 'or' b ) . x ) 'or' ( b 'or' c ) . x .= ( a 'or' b ) . x .= TRUE .= TRUE ; for e being set st e in AX ex X1 being Subset of Y st e = X1 & X1 is open & X1 is open & Y1 is open & X1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y2 is open & Y2 is open & Y1 is open & Y2 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & for i being set st i in the carrier of S for f being Function of [: S , T :] , the carrier of T for F being Function of S , T st F = H & F is one-to-one & F is one-to-one & F is one-to-one & F is one-to-one holds F is one-to-one for v , w st for y st x <> y holds w . y = v . y holds Valid ( All ( x , y ) , J ) . w = Valid ( x , y ) . w card D = card ( D1 + D2 ) - card { i + 1 } .= card ( D1 + D2 ) - card { i + 1 } .= ( D1 + D2 ) - ( D2 + D1 ) + ( D2 + D2 ) - ( D1 + D2 ) - ( D2 + D1 ) = ( D1 + D2 ) + ( D2 + D1 ) - ( D2 + D1 ) .= ( D1 + D2 ) + ( D2 + D1 ) - ( D2 + D1 ) - ( D2 + D2 ) - ( D2 + D2 ) - ( D2 + D2 ) - ( D1 + D2 ) - ( D2 + D2 ) - ( D2 + D2 ) - ( D1 - D2 ) - ( D2 + D2 ) - ( D1 - D2 ) - ( D1 IC Exec ( i , s ) = ( s +* ( 0 .--> ( s . 0 ) ) ) . 0 .= s . 0 .= s . 0 .= s . 0 .= s . 0 .= s . 0 .= s . 0 .= s . 0 .= s . 0 .= s . 0 .= s . 0 .= s . 0 ; len f -' ( i1 -' 1 ) + 1 = len f -' ( i1 -' 1 ) + ( i1 -' 1 ) .= len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 ; for a , b , c being Element of NAT st 1 <= a & 2 < b & a < b holds a + c < a + b or a = b + c for f being FinSequence of TOP-REAL 2 , p being Point of TOP-REAL 2 , i being Nat st p in LSeg ( f , i ) & p <> q & q <> p & p <> q holds Index ( p , f ) <= Index ( p , f ) lim ( ( curry ( P , k + 1 ) # x ) ) = lim ( ( curry ( P , k ) ) # x ) + ( ( curry ( P , k + 1 ) # x ) # x ) .= lim ( ( curry ( P , k ) ) # x ) + ( ( curry ( P , k ) # x ) # x ) ; z2 = g /. ( i -' ( n + 1 ) ) .= g /. ( i -' ( n + 1 ) ) .= g /. ( i -' ( n + 1 ) ) .= g /. ( i -' ( n + 1 ) ) .= g /. ( i -' ( n + 1 ) ) ; [ f . 0 , f . 3 ] in id ( the carrier of C ) \/ ( the InternalRel of C ) or [ f . 0 , f . 3 ] in the InternalRel of C & [ f . 0 , f . 3 ] in the InternalRel of C ; for G being Subset-Family of B for R being Subset of A st G = { R where R is Subset of A , Y is Subset of B holds ( Intersect F ) . ( G . Y ) = Intersect ( G , Y ) & ( Intersect F ) . ( G . Y ) = Intersect ( G , Y ) CurInstr ( P1 , Comput ( P1 , s1 , m1 + 1 ) ) = CurInstr ( P1 , Comput ( P2 , s2 , m1 ) ) .= CurInstr ( P2 , Comput ( P2 , s2 , m1 ) ) .= CurInstr ( P2 , Comput ( P2 , s2 , m1 ) ) .= CurInstr ( P2 , Comput ( P2 , s2 , m1 ) ) .= i ; assume that a on M and b on M and c on N and a on M and b on N and a on M and c on N and a on M and b on N and a on M and b on N and c on N and a on M and b on N and a on M and c on N and a on M and b on N and a on M and b on N and a on M and b on N and c on N and a on M and b on N and a on M and b on N and a on M and b on N and a on M and a on M and b on N and a on M and b on N and a on M and b on N and c on N and a on N and a on M and b on N and c , b on N and a on N and a on M and a on M and a , b on N and a on M assume that T is \hbox { T _ 4 } and F is closed and for F being Subset-Family of T st F is closed & F is finite-ind holds F is finite-ind & F is finite-ind & F is finite-ind & F is finite-ind & F is finite-ind & F is finite-ind ; for g1 , g2 , r , g st g1 in ]. r1 , r2 .[ & g1 in ]. r1 , r2 .[ & g2 in ]. r1 , r2 .[ & g1 < g2 & g2 < g1 holds |. ( f - g ) . g1 - ( f - g ) . g2 .| <= ( f - g ) . g1 - ( f - g ) . g2 .| <= ( f - g ) . g1 - ( f - g ) . g2 ( ( Re z ) + ( Im z ) * ( ( Re z ) + ( Im z ) * ( ( Re z ) + ( Im z ) * ( ( Re z ) + ( Im z ) * ( ( Im z ) + ( Im z ) * ( ( Re z ) + ( ( Re z ) + ( Im z ) * ( ( Re z ) + ( ( Im z ) * ( ( Re z ) + ( ( Re z ) * ( ( Re z ) + ( ( Im z ) * ( ( Re z ) + ( ( ( ( Re z ) + ( ( Re z ) * ( ( Re z ) * ( ( Re z ) * ( ( Re z ) * ( ( Re z ) * ( ( Re z ) * ( ( ( ( ( ( ( ( ( z ) ) * ( ( Re z ) * ( ( ( ( z ) ) ) ) ) ) ) F . i = F /. i + 0. R .= F /. ( n + 1 ) .= F /. ( n + 1 ) .= F /. ( n + 1 ) .= F /. ( n + 1 ) .= F /. ( n + 1 ) .= F /. ( n + 1 ) ; ex y being set , f being Function st y = f . n & dom f = NAT & rng f = { y } & f . ( n + 1 ) = R ( n ) & f . ( n + 1 ) = R ( n ) ; func f * F -> FinSequence of V means : Def1 : len it = len F & for i be Nat st i in dom F holds F . ( i , j ) = F . ( F /. i ) * F /. ( F /. j ) ; { x1 , x2 , x3 , x4 , x5 , x5 , 7 , 8 , 8 , 7 , 8 , 7 , 8 , 7 , 8 , 7 , 8 , 8 , 7 } = { x1 , x2 , x3 , x4 , x4 , x5 , 7 , 8 , 7 , 8 , 7 } \/ { x2 , x3 , x4 , 7 , 8 , 8 , 7 , 8 , 7 } .= { x1 , x2 , x3 , x4 } \/ { x2 , x4 } .= { x1 , x2 , x4 } \/ { x2 , x4 , 7 } .= { x1 , x2 , x4 } \/ { x2 , x4 } \/ { x2 , x4 } \/ { x2 , x4 , x4 } .= { x1 , x2 } \/ { x2 , x4 , x4 } \/ { x2 , x4 } .= { x2 , x2 } .= { x1 , x2 for n being Nat , x being set st x = h . n & x in InputVertices ( S ( ) ) & x in InputVertices ( S ( ) ) & x in InputVertices ( S ( ) ) & y in InputVertices ( S ( ) ) & x in InputVertices ( S ( ) ) & y in InputVertices ( S ( ) ) & z in InputVertices ( S ( ) ) & x in InputVertices ( S ( ) ) & y in InputVertices ( S ( ) ex S1 being Element of CQC-WFF ( Al ) st ( for e being Element of Al ( ) ) , e being Element of Al ( ) st e = S1 & e = S2 ( e ) holds ( e = e ) & ( e = e ) & ( e = e ) & ( e = e ) & ( e = e ) & ( e = e ) & ( e = e ) implies ( e = e ) & ( e = e ) = e ) & ( e = e ) & ( e = e ) . ( e = e ) ; consider P being FinSequence of ( the carrier of G ) * such that p = Product P and for i being Element of NAT st i in dom P ex t being Element of ( the carrier of G ) * st t = ( P * p ) . i & t . i = ( P * p ) . t ; for T1 , T2 being strict non empty TopSpace for P being Subset-Family of T1 , T2 being Subset of T2 , Q being Subset of T2 , P being Subset of T2 , Q being Subset of T2 , P being Subset of T2 , Q being Subset of T2 st P = the topology of T2 & Q = the topology of T2 & P is Basis of T2 holds P is Basis of T1 assume that f is_partial u0 to u and r (#) ( pdiff1 ( f , 3 ) ) is_differentiable_in x0 and ( r (#) ( pdiff1 ( f , 3 ) ) . y0 = r * ( ( pdiff1 ( f , 3 ) ) . y0 ) + ( ( r (#) ( pdiff1 ( f , 3 ) ) . y0 ) ; defpred P [ Nat ] means for F being FinSequence of REAL , G being FinSequence of REAL st len F = $1 & G = F * ( $1 , 1 ) & G = F * ( $1 , 1 ) holds Sum F = Sum G ; ex j st 1 <= j & j < width GoB f & ( GoB f ) * ( 1 , j ) `2 <= s & s * ( 1 , j ) `2 <= ( GoB f ) * ( 1 , j ) `2 & s * ( 1 , j ) `2 <= ( GoB f ) * ( 1 , j ) `2 ; defpred U [ set , set ] means ex Fi1 being Subset-Family of T st ( $1 = F & ( $1 = F . $1 ) & ( $1 = F . $1 implies $2 = union ( F . $1 ) ) & ( F . $1 ) is open ) & ( F is open implies F is open ) ; for p2 being Point of TOP-REAL 2 st LE p1 , p2 , P , p1 , p2 & LE p2 , p1 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 holds LE p1 , p2 , P , p1 , p2 & LE p2 , p1 , P , p1 , p2 f in St ( E , H ) & for y st y <> f . y holds ( for y st y in E holds f . y = ( \hbox { x } ) . y ) & ( for y st y in E holds f . y = ( \hbox { x } , H ) . y ) & ( for y st y in E holds f . y = ( \hbox { x } ) . y ) ; ex p2 being Point of TOP-REAL 2 st x = p2 & p2 `1 <= 0 & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p1 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p1 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p1 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p1 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p1 <> 0. TOP-REAL 2 & p1 <> 0. TOP-REAL 2 & p1 <> 0. TOP-REAL 2 & p2 <> 0. TOP-REAL 2 & p1 <> 0. TOP-REAL 2 & p1 <> 0. TOP-REAL 2 & p1 <> 0. TOP-REAL 2 & p1 <> 0. TOP-REAL 2 & p1 <> 0. TOP-REAL 2 & p1 <> 0. TOP-REAL 2 & p1 <> 0. TOP-REAL 2 & p1 <> 0. TOP-REAL 2 & p1 <> 0. TOP-REAL 2 & p1 <> 0. TOP-REAL 2 & p1 <> 0. TOP-REAL 2 & p1 <> 0. TOP-REAL 2 & p1 <> 0. TOP-REAL 2 & p1 <> 0. TOP-REAL 2 & p1 assume for d1 being Element of NAT st d1 <= ( s . ( n + 1 ) ) & ( for t being Element of NAT holds t . ( n + 1 ) = ( s . ( n + 1 ) ) * ( t . ( n + 1 ) ) & ( t . ( n + 1 ) ) * ( t . ( n + 1 ) ) = t . ( n + 1 ) ; assume that s <> t and s is Point of Sphere ( x , r ) and s is Point of Ball ( x , r ) and ex e being Point of TOP-REAL n st e = Ball ( x , r ) & e <> 0. TOP-REAL n ; given r such that 0 < r and for s being Point of C st 0 < s ex x1 , x2 being Point of C st x1 in dom f & x2 in dom f & f /. x1 - f /. x2 < r & f /. x2 - f /. ( x1 - x2 ) ) < r ; ( p | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x assume that x , x + h in dom ( ( cos * ( ( cos * ( cos * ( ( cos * ( ( cos * ( ( cos * ( ( cos * ( ( cos * ( ( cos * ( ( ( ( ( x ) / ( cos * ( ( cos * ( ( ( x ) / ( cos * ( ( cos * ( ( ( ( x ) / ( cos * ( ( ( x ) / ( cos * ( ( x ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) and ( ( ( cos * ( cos * ( cos * ( cos * ( ( cos * ( cos * ( ( cos * ( ( cos * ( cos * ( cos * ( ( cos * ( cos * ( ( cos * ( ( cos * ( ( ( ( ( ( ( ( cos * ( ( cos * ( ( cos * ( ( assume that i in dom A and len A > 1 and B > 0 and B is Matrix of n , m , m , K and B * ( i , j ) = A * ( i , j ) and B * ( i , j ) = B * ( i , j ) ; for i being non zero Element of NAT st i in Seg n holds ( i divides n implies ( i divides n ) |^ ( i -' 1 ) ) = <* ( n -' 1 ) |^ ( i -' 1 ) *> & ( i divides n ) |^ ( i -' 1 ) = ( n -' 1 ) |^ ( i -' 1 ) ) |^ ( i -' 1 ) ( ( ( ( b1 '&' b2 ) '&' ( ( b2 '&' c2 ) '&' ( ( b1 '&' b2 ) '&' ( ( b2 '&' c2 ) '&' ( ( b1 '&' b2 ) '&' ( ( b1 '&' b2 ) '&' ( ( b2 '&' c2 ) '&' ( ( b1 '&' b2 ) '&' ( ( b2 '&' c2 ) '&' ( ( b1 '&' b2 ) '&' ( ( b1 '&' b2 ) '&' ( ( b1 '&' b2 ) '&' ( b2 '&' c2 ) '&' ( ( b1 '&' b2 ) '&' ( ( b1 '&' b2 ) '&' ( b2 '&' b2 ) '&' ( b2 '&' b2 ) '&' ( ( b2 '&' b2 ) '&' ( ( b2 '&' c2 ) '&' ( b1 '&' b2 ) '&' ( b2 '&' b2 ) '&' ( b2 '&' c2 ) '&' ( b2 '&' c2 ) '&' ( ( b2 '&' c2 ) '&' ( ( b2 '&' c2 ) '&' ( b1 '&' c2 ) '&' ( b2 ) '&' ( b2 ) '&' ( b2 '&' c2 ) '&' ( b2 ) '&' assume that f . x = ( ( ( - 1 ) (#) ( ( sin * cos ) `| Z ) ) `| Z ) and ( ( ( sin * cos ) `| Z ) . x = ( ( sin * cos ) `| Z ) . x and ( ( sin * cos ) `| Z ) . x = ( sin . x ) / ( cos . x ) ^2 ) ; consider R8 , R8 be Real such that R8 = Integral ( M , F . n ) and I = Integral ( M , F . n ) and I = Integral ( M , F . n ) and I = Integral ( M , F . n ) and I = Integral ( M , F . n ) ; ex k being Element of NAT st x0 = k & 0 < d & d < 1 & for q be Element of product G st q in X & ||. ( f , q ) . ( n + 1 ) - f /. x0 .|| < r holds ||. ( f , q ) . ( n + 1 ) - f /. x0 .|| < r ; x in { x1 , x2 , x3 , x4 , x5 , x5 , 7 , 8 , 8 , 7 , 8 , 7 , 8 , 7 , 8 , 7 } iff x in { x1 , x2 , x3 , x4 , x5 , 7 , 8 , 7 , 8 , 8 } \/ { x2 , x3 , x4 , 7 , 8 , 7 , 8 , 8 , 7 } \/ { x1 , x2 , x4 } & x in { x2 , x3 , x4 } & x in { x2 , x4 , 7 , 8 , 8 , 8 , 7 , 8 , 7 , 8 , 8 , 7 , 8 , 7 , 8 , 7 , 7 , 8 , 7 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 7 , 7 , 7 , 8 , 7 , 7 , 8 , 7 , 7 , 8 , 8 , 8 , 7 , 7 , 8 , ( G * ( j , 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 ) ) .= ( ( the Arity of S2 ) +* ( the Arity of S2 ) ) +* ( o , ( the Arity of S1 ) +* ( o , ( the Arity of S2 ) +* ( o , ( the Arity of S1 ) +* ( o , the Arity of S2 ) ) ) ; func tree ( T , P , T ) -> tree of T means : Def1 : for p , q st p in P & q in P & p in T holds it . p = T ( p , q ) or it . p = T ( q , p ) ; F /. ( k + 1 ) = F . ( p . ( k + 1 ) ) .= F . ( p . ( k + 1 ) ) .= F . ( p . ( k + 1 ) ) .= F /. ( p /. ( k + 1 ) ) .= F /. ( p /. ( k + 1 ) ) ; for A , B , C , D being Matrix of n , K st len B = len C & len B = width C & width B > 0 & width B > 0 & width B > 0 & width B > 0 & width B > 0 holds B * ( BC ) = B * ( C * ( i , j ) ) seq . ( k + 1 ) = 0. ( ( seq . ( k + 1 ) ) + ( seq . ( k + 1 ) ) .= ( seq . ( k + 1 ) ) + ( seq . ( k + 1 ) ) .= ( seq . ( k + 1 ) ) + ( seq . ( k + 1 ) ) .= ( seq . ( k + 1 ) ) + ( seq . ( k + 1 ) ) .= ( seq . ( k + 1 ) + ( seq . ( k + 1 ) ) + ( seq . ( k + 1 ) + ( seq . ( k + 1 ) + ( seq . ( k + 1 ) + ( seq . ( k + 1 ) ) + ( seq . ( k + 1 ) ) + ( seq . ( k + 1 ) + ( seq . ( k + 1 ) ) .= ( seq . ( k + 1 ) ) + ( seq . ( k + 1 ) + ( seq . ( assume that x in ( the carrier of CC ) & y in ( the carrier of CC ) /\ the carrier of C and z = ( the carrier of C ) /\ the carrier of C and x = ( the carrier of C ) /\ the carrier of C and y = ( the carrier of C ) /\ the carrier of C ; defpred P [ Element of NAT ] means for f being FinSequence 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 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) assume that 1 <= k and k + 1 <= len f and f /. k = G * ( i , j ) and f /. k = G * ( i , j ) and f /. k = G * ( i , j ) and f /. ( k + 1 ) = G * ( i , j ) ; assume that seq < 1 and ( ( q `2 / |. q .| - sn ) / ( 1 - sn ) ) ^2 >= 0 and ( q `2 / |. q .| - sn ) ^2 >= 0 and ( q `2 / |. q .| - sn ) ^2 >= 0 ; for M being non empty MetrSpace , x being Point of M , n being Nat holds x = dist ( x , n ) iff ex f being Function of M , TOP-REAL n st for n being Nat holds f . n = dist ( x , n ) & f . n = dist ( x , n ) defpred P [ Element of omega \omega \omega ] means f1 | Z is continuous & ( f1 - f2 ) | Z is continuous implies ( f1 - f2 ) | Z is continuous & ( f1 - f2 ) | Z = ( f1 - f2 ) | Z ) | Z & ( f1 - f2 ) | Z = ( f1 - f2 ) | Z ) | Z ; defpred P1 [ Nat , Point of Cmin ( $1 , $1 ) - ( $2 + 1 ) ) / ( $2 + 1 ) < r / ( $2 + 1 ) & ( ||. ( f + g ) . ( $2 + 1 ) - ( f + g ) . ( $2 + 1 ) ) < r / ( ( f + g ) . ( $2 + 1 ) ) ; ( f ^ mid ( g , 2 , len g ) ) . i = ( mid ( g , 2 , len g ) ) . i .= ( g | ( i -' 1 ) ) . i .= ( g | ( i -' 1 ) ) . i .= ( g | ( i -' 1 ) ) . i .= ( g | ( i -' 1 ) ) . i ; sqrt ( 1 - 2 * ( ( n + 2 ) * ( n + 1 ) ) * ( ( n + 2 ) * ( ( n + 2 ) * ( n + 1 ) ) ) / ( n + 2 ) ) = ( n + 2 ) * ( ( n + 2 ) * ( n + 2 ) ) / ( n + 2 ) ) .= ( n + 2 ) / ( n + 2 ) ; defpred P [ Nat ] means ( G . $1 ) is finite or ( the carrier of G = the carrier of G ) & ( the carrier of G ) \/ the carrier of G = the carrier of G & the carrier of G = the carrier of G & the carrier of G = the carrier of G ; assume that f /. 1 in Ball ( u , r ) and 1 <= m and m <= len f and for i st 1 <= i & i <= len f holds not LSeg ( f /. i , f /. ( i + 1 ) ) /\ LSeg ( f /. ( i + 1 ) ) c= { f /. ( i + 1 ) } and not LSeg ( f /. ( i + 1 ) , f /. ( i + 1 ) ) /\ LSeg ( f /. ( i + 1 ) ) c= { f /. ( i + 1 ) /\ LSeg ( f /. ( i + 1 ) /\ LSeg ( f /. ( i + 1 ) ) /\ LSeg ( f /. ( i + 1 ) /\ LSeg ( f /. ( i + 1 ) ) /\ LSeg ( f /. ( i + 1 ) = { f /. ( i + 1 ) /\ LSeg ( f /. ( i + 1 ) /\ LSeg ( f /. ( i + 1 ) /\ LSeg ( f /. ( i + 1 ) /\ LSeg ( f /. ( i + 1 ) /\ defpred P [ Element of REAL ] means ( Sum ( cos ( $1 ) ) (#) sin ( $1 ) ) = ( Sum ( cos ( $1 ) ) ) * sin ( $1 ) * sin ( $1 ) * sin ( $1 ) * sin ( $1 ) * sin ( $1 ) * sin ( $1 ) * sin ( $1 ) * sin ( $1 ) * sin ( $1 ) * sin ( $1 ) * sin ( $1 ) * sin ( $1 ) * sin ( $1 ) ) ; for x being Element of product F holds x in ( the carrier of product F ) & x in ( the carrier of F ) & x in ( the carrier of F ) & x in ( the carrier of F ) & x in ( the carrier of F ) & x in ( the carrier of F ) & x in ( the carrier of F ) ; ( x " ) |^ ( n + 1 ) = ( x " ) * ( x |^ ( n + 1 ) ) .= ( x |^ ( n + 1 ) ) * x .= ( x |^ ( n + 1 ) ) * x .= ( x |^ ( n + 1 ) ) * x .= x |^ ( n + 1 ) * x .= x |^ ( n + 1 ) * x ; DataPart Comput ( P +* I , s , LifeSpan ( P +* I , s ) + 3 ) = DataPart Comput ( P +* I , s , LifeSpan ( P +* I , s ) + 3 ) .= DataPart Comput ( P +* I , s , LifeSpan ( P +* I , s ) + 3 ) ; given r such that 0 < r and ]. x0 - r , x0 .[ c= dom ( f1 + f2 ) /\ ]. x0 , x0 + r .[ and for g st g in ]. x0 - r , x0 .[ /\ ]. x0 - r , x0 .[ holds ( f1 + f2 ) . g <= ( f1 + f2 ) . g ; assume that X c= dom f1 /\ ( X /\ dom f2 ) and ( f1 | X ) | X is continuous and ( f1 | X ) | X is continuous and ( f1 | X ) | X is continuous ; for l being continuous continuous LATTICE , X being Subset of L st l = "\/" ( ( waybelow X ) , L ) & for x being Element of L st x in ( downarrow X ) /\ ( downarrow X ) holds x is prime Support ( A *' p ) = { m *' p where m is Element of NAT : p in Support ( m *' q ) & p is Polynomial of n , L : ex m being Element of NAT st m in Support ( m *' q ) & p . m = ( m *' q ) . m ; ( f1 - f2 ) /* ( ( f1 - f2 ) /* ( h + c ) ) = ( f1 - f2 ) /* ( h + c ) .= ( f1 - f2 ) /* ( h + c ) - f2 /* ( h + c ) .= ( f1 - f2 ) /* ( h + c ) - f2 /* ( h + c ) ; ex p1 being Element of CQC-WFF ( Al ) st F . p = g . ( p `1 ) & for g being Function of D ( ) , D ( ) st g is p & g is Function of D ( ) , D ( ) st P [ g , p , g ] holds P [ g ] ( mid ( f , i , len f -' 1 ) ) /. j = ( mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) ) /. j .= ( mid ( f , i , len f -' 1 ) ) /. j ; ( ( p ^ q ) | ( len q ) ) . ( len p + 1 ) = ( ( p ^ q ) | ( len q ) ) . ( len p + 1 ) .= ( ( p ^ q ) | ( len q ) ) . ( len q + 1 ) .= ( p ^ q ) . ( len p + 1 ) .= ( p ^ q ) . ( len p + 1 ) ; len mid ( ( ( indx ( D2 , D1 , j1 ) + 1 ) , indx ( D2 , D1 , j1 ) + 1 ) , indx ( D2 , D1 , j1 ) + 1 ) = indx ( D2 , D1 , j1 ) + 1 .= indx ( D2 , D1 , j1 ) + 1 ; x * y * z = ( x * ( y * z ) ) * ( ( y * z ) * ( y * z ) ) .= ( x * ( y * z ) ) * ( y * z ) .= ( x * ( y * z ) ) * ( y * z ) .= x * ( y * z ) ; v . ( <* x , y *> ) = ( ( <* x , y *> - ( y - x0 ) ) * ( ( ( ( ( 1 , y ) * ( ( i - x0 ) * ( ( i - x0 ) * ( ( i - x0 ) * ( ( i - x0 ) * ( ( i - x0 ) * ( ( i - x0 ) * ( ( i - x0 ) * ( ( i - x0 ) * ( ( i - x0 ) ) * ( ( i - x0 ) * ( ( i - x0 ) ) ) ) ) ) ) ) + ( ( i - x0 ) ) ) ) ) ) + ( ( i - x0 ) ) ) ) + ( ( ( ( i - x0 ) ) ) ) ) + ( ( i - x0 ) ) ) * ( ( ( ( ( ( ( i - x0 ) ) ) * ( ( ( ( i - x0 ) ) + ( ( ( ( i - x0 ) * ( ( ( i - x0 ) ) ) ) ) + ( ( i - x0 ) ) ) ) * ( ( i - x0 ) ) ) + ( ( ( ( ( i * i = <* 0 * ( 1 / 2 ) - ( 1 / 2 ) * ( 1 / 2 ) + ( 1 / 2 ) * ( 1 / 2 ) .= ( 1 / 2 ) * ( 1 / 2 ) + ( 1 / 2 ) * ( 1 / 2 ) .= ( 1 / 2 ) * ( 1 / 2 ) ; Sum ( L (#) F ) = Sum ( ( L (#) F ) ^ ( F (#) G ) ) .= Sum ( ( L (#) F ) ^ ( F (#) G ) ) .= Sum ( ( L (#) F ) (#) ( F (#) G ) ) .= Sum ( ( L (#) F ) (#) ( F (#) G ) ) .= Sum ( ( L (#) F ) (#) ( F (#) G ) ) ; ex e be Real st for Y be Subset of X , Y be Subset of REAL st Y in e & Y c= X & Y c= Y holds ( for Y be finite Subset of X st Y c= X & Y c= Y holds r * Y = r * Y ) & for Y be Subset of X st Y c= X holds r * Y = r * Y ) implies r * Y = r * Y ( GoB f ) * ( i , j + 1 ) = f /. ( k + 1 ) & ( GoB f ) * ( i , j + 1 ) = f /. ( k + 1 ) & ( GoB f ) * ( i , j + 1 ) = f /. ( k + 1 ) & ( GoB f ) * ( i , j + 1 ) = f /. ( k + 1 ) ; ( ( ( - 1 ) / ( r / ( r * ( 1 + r ) ) ) ) / ( 1 - ( r * ( 1 / ( r * ( 1 + r ) ) ) ) ) ) / ( 1 - r * ( 1 / ( r * ( 1 + r ) ) ) ) / ( 1 - r * ( 1 / ( 1 + r ) ) ) = ( ( r / ( 1 / ( r * ( 1 + r ) ) ) ) / ( 1 - r ) ) ) / ( 1 - r ) ) / ( 1 - r ) ) / ( 1 - r ) ) / ( 1 - r ) ) / ( 1 - r ) ) / ( 1 - r ) ) / ( 1 - r ) ) / ( 1 - r ) ) / ( 1 - r ) / ( 1 - r ) ) = ( 1 - r ) / ( 1 - r ) / ( 1 - r ) ) / ( 1 - r ) ) / ( 1 - r ) .= ( ( 1 - r ) ) / ( 1 - r ) / ( 1 - r ) ) / ( 1 - ( - b ) + sqrt ( integral ( a , b , c , d ) ) > 0 & sqrt ( - sqrt ( b , c , d ) ) > 0 & sqrt ( - sqrt ( a , b , c ) ) > 0 ; assume that inf ( { "\/" ( X , L ) : for X st X in X holds "\/" ( X , L ) in X } , L ) = "/\" ( X , L ) and "\/" ( X , L ) = "\/" ( X , L ) ; ( ( B , i ) --> ( j , i ) ) . j = ( j .--> ( i , j ) ) . j .= ( j .--> ( i , j ) ) . j .= ( j .--> ( i , j ) ) . j .= ( j .--> ( i , j ) ) . j .= ( j .--> ( i , j ) ) . j ;