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 c= A ; a <> c ; T c= S D c= B c ; b in X ; X is finite ; 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 continuous ; let q ; m = 1 ; 1 < k ; G is finite ; b in A ; d divides a ; i < n ; s <= b ; b in B ; let r ; B is one-to-one ; R is total ; x = 2 ; d in D ; let c ; let c ; b = Y ; 0 < k ; let b ; let n ; r <= b ; x in X ; i >= 8 ; let n ; let n ; y in dom f ; let n ; 1 < j ; a in L ; C is unital ; 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 < n ; assume f is one-to-one ; x in Y ; z = + \infty ; let k be Nat ; K is being_line ; assume n >= N ; assume n >= N ; assume X is \setminus ; assume x in I ; q is 0 iff q is 0 ; assume c in x ; p > 0 ; assume x in Z ; assume x in Z ; 1 <= k12 ; assume m <= i ; assume G is finite ; assume a divides b ; assume P is closed ; `2 > 0 ; assume q in A ; W is bounded ; f is one-to-one ; assume A is condensed ; g is special ; assume i > j ; assume t in X ; assume n <= m ; assume x in W ; assume r in X ; assume x in A ; assume b is even ; assume i in I ; assume 1 <= k ; X is non empty ; assume x in X ; assume n in M ; assume b in X ; assume x in A ; assume T c= W ; assume s is negative ; b `2 <= c `2 ; A meets W ; i `2 <= j `2 ; assume H is universal ; assume x in X ; let X be set ; let T be DecoratedTree of X ; let d be element ; let t be element ; let x be element ; let x be element ; let s be element ; k <= len f-2 ; 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 , y be Element of E ; let C be Category ; let x be element ; let k be Nat ; let x be element ; let x be element ; let e be element ; let x be element ; P [ 0 ] let c be element ; let y be element ; let x be element ; let a be Real ; let x be element ; let X be element ; P [ 0 ] let x be element ; let x be element ; let y be element ; r in REAL ; let e be element ; n1 is non-zero ; Q halts_on s ; x in hP ; M < m + 1 ; T2 is open ; z in b ; R 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 + 1 ; p1 in P ; p1 in K ; y in A1 ; k + 1 <= n ; let a be Real , x 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 + 1 ; f is_differentiable_on Z ; let x0 ; let E be Ordinal ; o indx ( o , 4 ) = o ; O <> 0. TOP-REAL 2 ; let r be Real ; let f be FinSequence of INT ; let i be Nat ; let n be Nat ; Cl A = A ; L c= Cl L ; A /\ M = B ; let V be complex linear space , f be Function of V , W ; s in Y |^ 0 ; rng f <= w ; b "/\" e = b ; m = m1 ; t in h . D ; P [ 0 ] ; assume z = x * y ; S . n is bounded ; let V be RealLinearSpace , W be Subspace of V ; P [ 1 ] ; P [ {} ] ; B1 is connected ; H = G . i ; 1 <= i + 1 + 1 ; F . m in A ; f . o = o ; P [ 0 ] ; aaomega <= b-a ; R [ 0 ] ; b in f .: X ; assume q = q2 ; x in [#] V ; f . u = 0 ; assume e > 0 ; let V be RealUnitarySpace , W be Subspace 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 Element of T ; the object of F is one-to-one ; sgn x = 1 ; k in support a ; 1 in Seg 1 ; rng f = X ; len T in X ; v1 < n ; SK1 is bounded ; assume p = p2 ; len f = n ; assume x in P1 ; i in dom q ; let U1 ; p1 = 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 <= j1 ; set A = Cl Cl A ; card a [= c ; e in rng f ; cluster B \oplus A -> non empty ; H has has has has \textit { - H } ; assume n1 <= m ; T is increasing ; e <> e1 ; Z c= dom g ; dom p = X ; H is proper implies H is proper or H is Subgroup of G i + 1 <= n ; v <> 0. V ; A c= Affin A ; S c= dom F ; m in dom f ; let X be set ; c = sup N ; R is connected ; assume x in REAL ; Im f is complete ; x in Cl y ; dom F = M ; a in On W ; assume e in { \cal A } ; C c= { C } ; m <> {} ; x be Element of Y ; let f be Chain of X , x be Element of X ; n in Seg 3 ; assume X in f .: A ; assume p <= n & p <= m ; assume u in { v } ; d is Element of A ; A |^ b misses B ; e in v v v ; - y in I ; let A be non empty set , x be set ; P1 = 1 ; assume r in F . k ; assume f is simple function ; let A be countable countable countable countable countable set ; rng f c= NAT ; assume P [ k ] ; f3 <> {} ; let o be Ordinal ; assume x is sum of squares ; assume v in { 1 } ; let I ; assume 1 <= j & j < l ; v = - u ; assume s . b > 0 ; d = q1 ; assume t . 1 in A ; let Y be non empty TopSpace , X be Subset of Y ; assume a in ]. s , t .[ ; let S be non empty RelStr ; a , b // b , a ; a * b = p * q ; assume x , y // the carrier of V ; assume x in \Omega ( X ) ; [ a , c ] in X ; n1 <> {} ; M + N c= M + N ; assume M is connected ; assume f is multiplicative w.r.t. B\dot -stable ; let x , y be element ; let T be non empty TopSpace ; b , a // b , c ; k in dom p ; let v be Element of V ; [ x , y ] in T ; assume len p = 0 ; assume C in rng f ; k1 = k2 ; m + 1 < n + 1 + 1 ; s in S \/ { s } ; n + i >= n + 1 ; assume Re ( y ) = 0 ; k1 <= j1 ; f | A is continuous ; f . x < b ; assume y in dom h ; x * y in { B } ; set X = Seg n ; 1 <= i2 + 1 ; k + 0 <= k + 1 + 1 ; p ^ q = p ^ q ; j |^ y divides m ; set m = max A ; [ x , x ] in R ; assume x in succ 0 ; a in sup \varphi ; B1 in C ; q2 c= C ; a2 < c2 ; s2 is 0 -started ; IC s = 0 ; s2 = s1 ; let V ; let x , y be element ; x is Element of T ; assume a in rng F ; x in dom T `2 ; let S be non-empty Subgroup of L ; y " <> 0 ; y " <> 0 ; 0. V = uV --16 ; y , u , v is_collinear ; R1 is total ; let a , b be Real ; let a be object of C ; let x be Vertex of G ; let o be object of C , x be Element of C ; r '&' q = P ; let i , j ; let s be State of A , x be set ; s2 . n = N . n ; set y = ( x + y ) * ( x + y ) ; \mathbb i in dom g ; l . 2 = y ; |. g . y .| <= r ; f . x in { C } ; V is non empty ; let x be Element of X ; 0 <> f . g ; f2 /* q is convergent ; f . i is_measurable_on E ; assume that \xi in { Nn1 } ; reconsider i = i as Ordinal ; r * v = 0. X ; rng f c= REAL ( ) ; G = 0 .--> ( 0 .--> 1 ) ; let A be Subset of X ; assume A1 : A2 is open & A is open ; |. f . x .| <= r ; x be Element of R ; let b be Element of L ; assume x in W1 ; P [ k , a ] ; let X be Subset of L ; let b be object of B ; let A , B be category ; set X = C ( ) ; let o be OperSymbol of S ; let R be connected non empty reflexive RelStr ; n + 1 = n ; { x } c= { Z } ; dom f = { C } ; assume [ a , y ] in X ; Re ( seq ^\ k ) is convergent ; assume a1 = b1 & a2 = b2 ; A = sssssA ; a <= b or b <= a ; n + 1 in dom f ; let F be Stop S , x be Element of S ; assume r2 > x0 ; let Y be non empty set , X be non empty set ; 2 * x in dom W ; m in dom ( - g ) ; n in dom ( - g ) ; k + 1 in dom f ; not the bound not not bound in { s } ; assume x1 <> x2 ; v1 in { 0. V } ; [ b `1 , b `2 ] in T ; func func func intpos i -> Nat equals i ; T c= right_open_halfline ( T ) ; ( l - 1 ) * ( l - 1 ) = 0 ; let n be Nat ; ( t `2 ) * ( t `2 ) `2 = r * ( t `2 ) `2 ; Annnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn set t = "/\" t ; let A , B be real-membered set ; k <= len G + 1 + 1 ; { C } misses { C } ; product s1 is non empty ; e <= f or f <= e ; cluster -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> ordinal ; assume c2 = b1 ; assume h in [. q , p .] ; 1 + 1 <= len C ; c in B . ( m1 + 1 ) ; let R be Relation of X ; p . n = H . n ; assume that v1 is sequence of X and for m being Nat st m >= k holds v . m = v ; IC s3 = 0 + 1 ; k in N or k in K ; F1 \/ F2 c= F \/ F1 Int ( G1 /\ G2 ) <> {} ; ( z - z ) `2 = 0 ; p1 <> 0. TOP-REAL 2 ; assume z in { y , z } ; MaxADSet ( a ) c= F ; ex_sup_of s , S ; f . x <= f . y ; let T be continuous continuous non empty reflexive reflexive transitive RelStr ; q1 |^ m >= 1 ; a >= X & b >= Y ; assume <* a , b , c *> <> {} ; F . c = g . c ; G is one-to-one & G is one-to-one ; A \/ { a } c= B ; 0. V = 0. V ; let I be Instruction of S , i be Instruction of S ; f1 . x = 1 ; assume z \ x = 0. X ; C = 2 |^ n ; let B be sequence of subsets of \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \leq B ; assume X1 = p .: D ; n + l in NAT ; f " . P is compact ; assume x1 in REAL ; p1 = - p2 ; M . k = {} ; \varphi . 0 in rng \varphi ; OSMMMMO is \sqcup assume z <> 0. L ; n < N . k ; 0 <= seq . 0 ; - q + p + p = v ; not v in B ; set g = f /. 1 ; for R being stable Relation of R , A being set st R is stable & R is stable holds R is stable set RR1 = Vertices R , RR1 = stable R , RR1 = stable R , RR1 = stable ; p1 c= p2 & p1 in P1 implies p1 in P1 x in [. 0 , 1 .] ; f . y in dom F ; let T be continuous continuous continuous TopAugmentation of S ; inf the carrier of S is Subset of S ; downarrow a = downarrow b ; P , C , D is_collinear ; assume x in { [ s , r ] : s in { s , t ] } ; 2 |^ i < 2 |^ m ; x + z = x + z ; x \ ( a \ x ) = x \ ( a \ x ) ; ||. \mathopen \mathclose .|| .|| <= r ; assume Y c= field Q & Y c= Z ; a \times b = b & b * a = b ; assume a in { A . i } ; k in dom q1 ; p is FinSequence of S ; i -' 1 + 1 = i - 1 ; f | A is one-to-one ; assume x in f .: { X } ; i2 - 1 = 0 ; i2 + 1 <= i2 + 1 ; g " * a in N ; K <> { [ {} , {} , {} ] } ; cluster -> -> -> -> -> -> -> -> -> -> -> -> -> -> positive for ; |. q .| ^2 > 0 ; |. p1 .| = |. p .| ; s2 - s1 > 0 ; assume x in { G } ; min ( C , m ) in C ; assume x in { G } ; assume i + 1 + 1 = len G ; assume i + 1 + 1 = len G ; 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 void for TopStruct for I be Program of S ; let f be ManySortedSet of I ; let z be Element of C , f be Function ; u in { \hbox { \boldmath $ g } } : g in A } ; 2 * n < 2 * ( 2 * n ) ; x , y , z is_collinear ; BB c= { V where V is Subset of V : V in W } ; assume I is_closed_on s , P ; U = { U } ; M /. 1 = z /. 1 ; x1 = x1 & x2 = x2 ; i + 1 < n + 1 + 1 + 1 ; x in { [ {} , <* 0 *> ] } ; f1 . ( len f1 + 1 ) <= f1 . ( len f1 + 1 ) ; l be Element of L ; x in dom ( F1 + F2 ) ; let i be Element of NAT ; r1 is Element of REAL ; assume <* o , o *> <> {} ; s . x = 1 ; card K in M ; assume that X in U and Y in U ; let D be a_partition of Omega ; set r = ]. k + 1 , k + 1 .[ ; y = W . ( 2 * x ) ; assume dom g = cod f ; let X , Y be non empty TopSpace , X be Subset of Y ; x \oplus A is even ; |. <*> A .| . a = 0 ; cluster -> sublattice for Lattice of L ; a1 in B . s1 . s1 ; let V be finite VectSp of F , F be FinSequence of V ; A * B on A * B ; f1 = NAT --> 0 ; A , B , C , D , E , F be Subset of V ; z1 = P1 . j .= P1 . j ; assume f " . P is closed ; reconsider j = i as Element of M ; a , b , c be Element of L ; assume q in A \/ ( B \/ C ) ; dom ( F * C ) = o ; set S = { REAL } , X = { REAL } , Y = { REAL } , Z = { REAL } , Z = { REAL } , Z = { REAL } z in dom ( A --> y ) ; P [ y , h . y ] ; not x0 in dom f /\ dom ( - 1 ) ; let B be non-empty ManySortedSet of I ; ( PI < 2 * PI * PI ; reconsider z1 = 0 as Nat ; LIN a , d , c ; [ y , x ] in { I } ; ( Q /. 3 ) `2 = 0 ; set j = x0 * ( m , n ) ; assume a in { x , y } ; j1 - j > 0 ; I \! \mathop { \rm \hbox { - } TruthEval } \varphi = 1 ; [ y , d ] in F ; let f be Function of X , Y ; set A2 = ( - B ) * ( A - B ) ; s1 , s2 , s1 , s2 , s2 , s2 , s2 , s2 , s2 , s2 , s2 , s2 , s2 , s2 , s2 be Element of R ; j1 -' 1 + 1 = 0 ; set m = 2 * n + 1 ; reconsider t = t as bag of n ; I . j = m . j ; i |^ s mod n = 1 & i |^ n divides n ; set g = f | D , f = g | D ; assume that X is lower and 0 <= r ; ( p1 `1 ) ^2 = 1 ; a < ( p1 + p3 ) `1 ; L \ { m } c= UBD C ; x in Ball ( x , 10 ) ; not a in { c } ; 1 <= i2 -' 1 + 1 ; 1 <= i2 -' 1 + 1 ; i + 1 <= len h ; x = W . ( len W + 1 ) ; [ x , z ] in X \times Z ; assume y in [. x0 , x .[ ; assume p = <* 1 , 2 , 3 *> ; len <* x1 *> = 1 ; set H = h . ( g . { g } ) ; card b * a = |. a .| * a ; Shift ( w , 0 ) |= v ; set h = h1 \circ ( h \circ ( h1 , h2 ) ) ; assume x in X1 /\ X2 ; ||. h .|| < d ; x in the carrier of support ( f ) ; f . y = F ( y ) ; for n holds X [ n ] ; k -' l = kl + k ; <* p , q *> /. 2 = q ; let S be Subset of the topology of Y ; P , Q , t , s , t be Point of S ; Q /\ M c= union ( F | M ) ; f = b * ( Sgm S ) ; let a , b be Element of G ; f .: X <= f . ( sup X ) ; let L be non empty reflexive transitive RelStr , X be Subset of L ; Si is x -to_power i ; let r be non negative Real ; M |= _ { x } H ; v + w = 0. V ; P [ len F ] ; assume InsCode ( i + 1 ) = 8 ; the carrier of M = { 0 , 1 } ; cluster z * seq -> summable ; let O be Subset of the carrier' of C ; |. f " .| is continuous ; x2 = g . ( j + 1 ) ; cluster -> non empty for Element of string S ; reconsider ll = ll as Nat ; v1 is vertex of the carrier of G ; f3 is SubSpace of TOP-REAL 2 ; h1 /\ ( [#] ( I[01] /\ ( ( TOP-REAL n ) | K1 ) ) ) <> {} ; let k be Nat ; q " is Element of X ; F . t is set ; assume n <> 0 & n <> 1 ; set d1 = EmptyBag n , d2 = EmptyBag n , d2 = EmptyBag n , d2 = EmptyBag n , d2 = n as Element of NAT ; let b be Element of Bags n ; assume for i holds b . i = b . i ; x is root of ( p ) * ( p * ( p * ( p * ( p * q ) ) ) ) ; r in ]. p , q .[ ; let R be FinSequence of REAL ; SI <> p2 ; IC SCM ( SCM ) <> a ; |. p - x .| >= r ; 1 * s1 = s1 * s1 + s2 * s2 * s2 ; let x be FinSequence of NAT ; let f be Function of C , D ; for a , b being Element of L holds 0. L + a + b = a + b IC s = s . IC s .= IC s ; H + G = FH - ( H - G ) ; C . x = ( C . x ) `2 ; f1 = f .= ( f + g ) . x .= ( f + g ) . x ; Sum <* p . 0 *> = p . 0 ; assume v + W = v + W ; not a1 in { a1 } ; a1 , b1 _|_ b , d ; a2 , o _|_ o , a2 ; reflexive reflexive reflexive transitive in the carrier of C ; IccC is antisymmetric in the carrier of C ; ex_sup_of rng ( H - ( H - ( H - ( H - ( H - ( H - ( H - ( H - ( H - ( H - ( H - ( H - H ) ) x = a * ( - b * ( - a * ( - b * ( - b * ( - b * ( - b * ( - b * b * ( - a * b ) * |. p1 .| ^2 >= 1 ; assume i2 -' 1 < len f ; rng s c= dom ( f1 + f2 ) /\ dom f2 ; assume support a misses support b ; let L be associative commutative loop structure ; s " + 0 < n + 1 + 1 ; p . c = ( f1 . c ) * ( f2 . c ) ; R . n <= R . n ; Directed I = { [ 0 , 1 ] } ; set f = x + ( y , r ) ; cluster Ball ( x , r ) -> bounded ; consider r being Real such that r in A ; cluster -> NAT -defined for Function ; let X be non empty directed Subset of S ; let S be non empty RelStr , x be Element of L ; cluster <* L1 . N , L1 . N *> -> complete for complete ; sqrt ( 1 - a ^2 ) = a ^2 - b ^2 ; ( q . {} ) `2 = o `2 ; n -' ( i + 1 ) > 0 ; assume that ( 1 - 2 * t ) <= t and t <= 1 ; card B = k + 1 ; x in union rng ( f1 + f2 ) ; assume x in the carrier of R ; d in X ; f . 1 = L . ( F . 1 ) ; the vertices of G = { v } ; let G be G be <= wwwwwwwwwwwww_Point ; e , v1 , v2 , v1 is_collinear ; c . ( i + 1 ) in rng c ; f2 /* q is convergent ; set z1 = - z2 , z2 = - z1 , z1 = - z2 , z2 = - z2 , z1 = - z2 , z2 = - z2 , z1 = - z2 , z2 = - z2 , z1 assume w is Element of atat-on S ; set f = p \! \! \smallfrown ( t ^ <* t *> ) ; let c be object of C ; assume ex a st P [ a ] ; let x be Element of REAL m ; let I be Subset-Family of X ; reconsider p = p as Element of NAT ; v , w , w is_collinear ; let s be State of SCM+FSA ; p is FinSequence of the carrier of K ; stop I c= P1 ; set cT1 = { f /. i where i is Nat : i < n & j < n } ; w ^ t ^ t ^ t ^ t ^ t ^ t ^ t ^ t ^ t ^ t ^ t ^ t ^ t ^ t ^ t ^ t ^ t ^ t ^ t ^ t ^ W1 /\ W2 = ( W1 + W2 ) /\ ( W1 + W2 ) ; f . j is Element of J . j ; let x , y be Element of T ; ex d st a , b // b , d ; a <> 0 & b <> 0 implies a * b = b * c ord ( x ) = 1 & x is Real ; set g2 = lim ( s ^\ k ) , g2 = lim ( s ^\ k ) ; 2 * x >= 2 * x + 2 * y ; assume ( a 'or' c ) . z <> TRUE ; f (*) g in Hom ( c , d ) ; Hom ( c , d ) <> {} ; assume 2 * ( q | m ) > m ; L1 . ( F1 . ( F1 . ( F1 . k ) ) ) = 0 ; RX \/ RX = X \/ { 0. X } ( the function sin ) . x <> 0 ; ( the function exp of ) . x > 0 ; o in { X1 where X1 is Subset of X1 : X1 in X2 } /\ X1 <> {} } ; e , v1 , v2 , v1 is_collinear ; s3 > ( 1 - r ) * ( 1 - r ) ; x in P .: ( F .: ( F .: ( F .: ( X ) ) ) ) ; J be closed under ideal ( R , n ) ) ; h . ( p1 + p2 ) = ( f . ( p1 + p2 ) ) . ( p1 + p2 ) ; Index ( p , f ) + 1 <= j ; len q1 = width ( - q2 ) ; the support of K c= A ; dom f c= union rng F ; k + 1 in Seg ( n + 1 ) ; let X be ManySortedSet of the carrier of S ; [ x , y ] in ( the InternalRel of R ) \/ ( the InternalRel of R ) ; i = D1 or i = D2 ; assume a mod n = b mod n ; h . x2 = g . x2 ; F c= 2 |^ ( n + 1 ) ; reconsider w = |. s1 .| as sequence of REAL ; ( 1 - m ) * r < p ; dom f = dom ( I * ( I * ( I * ( I * ( I * ( I * ( I * ( I * ( I * ( I * ( I * ( I * ( I * ( [#] TOP-REAL 2 = ( TOP-REAL 2 ) | K1 .= ( TOP-REAL 2 ) | K1 ; let x be Element of REAL ; then not d in A ; cluster -> non empty for TopSpace ; w be Element of M ; x be Element of \mathclose { -1 } ( n ) ; u in W1 & v in W1 & u in W2 implies u in W1 + W2 reconsider y = y as Element of L2 ; N is full SubRelStr of T ; ex_sup_of { x , y } , c "\/" d ; g . n = n |^ ( n + 1 ) .= n |^ ( n + 1 ) ; h . J = EqClass ( u , J ) ; seq be summable of X ; dist ( x , y ) < r / 2 ; reconsider m1 = m as Element of NAT ; x- x0 < r1 - x0 ; reconsider P = P as strict Subgroup of N ; set g1 = p * ( q * ( q * ( q * ( q * ( q * ( q * ( q * ( q * p ) ) ) ) ) ; let n , m , k be Nat ; assume that 0 < e and f | A is bounded ; D2 . ( I + 2 ) in { x } ; cluster -> -> -> empty for Subset of T ; let P be compact Subset of TOP-REAL 2 ; G in LSeg ( \pi , 1 ) ; 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 ; op _ 1 c= { [ {} , {} ] } ; reconsider m = mm as Element of NAT ; reconsider d = x as Element of C ; let s be 0 -started State of SCMPDS , p be Nat ; let t be 0 -started State of SCMPDS , Q be t be State of SCMPDS ; b , b , x is_collinear ; assume i = n \/ { n } & j in { n } ; let f be PartFunc of X , Y ; N1 >= sqrt ( ( sqrt ( 2 * c ) ^2 ) ; reconsider t1 = T as Point of TOP-REAL 2 ; set q = h * p ^ <* d *> ^ <* d *> ; z2 in U . ( y + z ) /\ ( ( y + z ) /\ ( y + z ) ) ; A |^ 0 = { <* A *> |^ 0 , A |^ 0 *> len W1 = len W2 + len W1 + W2 + W2 ; len ( h + c ) in dom ( h + c ) ; i + 1 in Seg ( len s + 1 ) ; z in dom ( - g ) /\ dom ( - g ) ; assume p2 = ( - 1 ) * ( - 1 ) ; len G + 1 <= len G + 1 + 1 ; f1 * f2 is_differentiable_on ]. x0 , x .[ ; let seq + seq + seq -> summable ; assume j in dom ( M1 M1 M1 ) ; let A , B , C be Subset of X ; x , y , z , u is_collinear ; b ^2 - ( a ^2 ) * ( a ^2 - c ^2 ) >= 0 ; <* xnnnnnnnnnnnnnnnny *> ^ <* x *> *> ^ <* y *> ^ <* y *> ^ <* x *> ^ <* y *> ^ <* y *> ^ <* a , b , c is_collinear ; len ( p2 - p1 ) = len p2 - len p1 + len p1 ; ex x being element st x in dom R & x = R . x ; len q = len ( K * G ) ; s1 = Initialize ( s1 ) .= s1 ; consider w being Nat such that q = z + w ; x ` ` ` is Element of L ` ; k = 0 & n <> 0 implies n = 0 then X is discrete ; for x st x in L holds x in L . x ||. f /. c - f /. c .|| <= r ; c in ]. p , q .[ & c in ]. p , q .[ ; reconsider V = the carrier of TOP-REAL n as Subset of the topology of TOP-REAL n ; N , M |= L ; then z >= \twoheaddownarrow x ; M = f & M = g implies M * ( f , g ) = g * ( f , g ) ( ( ( ( ( ( ( ( ( ( TOP-REAL 2 2 2 ) ) ) ) ) ) ) | K1 ) ) . 1 ) = ( ( ( TOP-REAL 2 ) ) | K1 ) ) . 1 ) ; dom g = dom f /\ X ; mode mode OperSymbol of G is _il of G means : DefDef: for n being odd Element of G holds it . n = 2 * n ; [ i , j ] in Indices M ; reconsider s = x " as Element of H " ; let f be Element of the carrier of Subformulae p ; F1 . ( a , b ) = F1 . ( a , b ) ; cluster AffineMap ( a , b , r ) -> compact ; let a , b , c be Real ; rng s c= dom ( f1 + f2 ) /\ dom f2 ; curry ( F1 , k ) is additive ; set k2 = card ( B \/ { x } ) ; set G = Sym ( X , Y ) ; reconsider a = [ x , s ] as Symbol of G ; let a , b be Element of M , p be Element of M ; reconsider s1 = s as Element of S ; rng p c= the carrier of L ; let d be Subset of the bound of A ; ( x | x ) = 0 implies x = 0 I in dom ( \mathop { \rm Directed } ( C , n ) ) ; let g be continuous Function of X , Y ; reconsider D = Y as Subset of TOP-REAL 2 ; reconsider i2 = len ( p - 1 ) as Element of NAT ; dom f = the carrier of S ; rng h c= union ( the carrier of J ) ; cluster 'not' All ( x , H ) -> universal ; d * N > 1 * N * ( N * N ) ; ]. a , b .[ c= [. a , b .] ; set g = f " | ( D \ { p } ) ; dom ( p | ( Seg m ) ) = Seg m ; 3 + 2 * ( 2 * x ) <= k + 2 * ( 2 * x ) ; tan is_differentiable_on Z implies ( ( ( ( ( ( ( ( ( tan ) ) (#) tan ) ) `| Z ) ) ) . x ) = ( ( ( tan ) (#) tan ) ) . x ) ^2 x in rng ( f ^ g ) ; f , g be FinSequence of D ; for p being Point of S1 holds p in the carrier of S2 & p in the carrier of S2 & p in the carrier of S1 & p in the carrier of S2 implies p in the carrier of S2 rng f " { 0 } = dom f /\ dom ( f " ) ; ( the carrier' of G ) . e = v ; width G -' 1 < width G - 1 ; assume v in rng ( S | E ) ; assume x is root or x is root ; assume 0 in rng ( ( g | A ) | A ) ; let q be Point of TOP-REAL 2 ; let p be Point of TOP-REAL 2 ; dist ( O , u ) <= |. p .| + 1 ; assume dist ( x , b ) < dist ( a , b ) ; <* S *> is with_only if for x being Element of the carrier' of C holds x in the carrier' of C i <= len ( G * ( i2 , k ) ) + 1 ; let p be Point of TOP-REAL 2 ; x1 in the carrier of I[01] | ( ( TOP-REAL 2 ) | ( ( ( TOP-REAL 2 ) | P ) ) | ( P ) ) ) ; set p1 = f /. i , p2 = f /. j ; g in { { g2 : r < g2 } ; q2 = { S } ^ <* q2 *> ; ( ( ( 1 - 2 ) (#) ( f ) ) `| Z ) . x = ( 1 - 2 * x ) * ( 1 + 2 * x ) ; - p + I + p + p c= - p + ( - p + p ) ; n < LifeSpan ( P1 , s1 ) + 1 ; CurInstr ( p1 , s1 ) = i ; A /\ Cl { x } <> {} ; rng f c= ]. r - 1 , r + 1 .[ ; let g be Function of S , V ; let f be Function of L1 , L2 ; reconsider z = z as Element of InclPoset Ids carrier of L ; let f be Function of S , T ; reconsider g = g as Morphism of c , b ; [ s , I ] in S \times A ; len ( the connectives of C ) = len the connectives of C ; let C , D be subcategory of C ; reconsider v1 = V as Subset of X | B ; attr p is valid means : DefDef: p is valid & p is valid ; assume that X c= dom f and X c= dom f and for x st x in X holds f . x = g . x ; H |^ a is Element of H |^ a ; A1 be Element of O ( ) , A2 ( ) , B2 ( ) , B2 ( ) ) , B2 ( ) , B2 ( ) , B2 ( ) , B2 ( ) , B2 ( ) , B2 ( ) ) ) , B2 ( ) p1 , p2 , p3 is_collinear & p2 , p1 , p3 is_collinear ; consider x being element such that x in v ^ K ; not x in { 0. TOP-REAL 2 } & x in { 0. TOP-REAL 2 } ; p in [#] ( ( TOP-REAL n1 ) | B ) ; 0 < ( |. E .| ) . m ; op ^ c = c ; consider c being element such that [ a , c ] in G ; a1 in dom ( F . s1 ) ; let L being distributive distributive distributive distributive distributive \mathclose empty ` for x being Element of L holds x ` ` is ` ; set i1 = the Element of the carrier of K ; let s be 0 -started State of SCM+FSA , p be Comput ( p , s , 0 ) ; assume y in ( f1 \/ f2 ) .: A ; f . ( len f + 1 ) = f . ( len f + 1 ) ; x , f . ( x , y ) \bfparallel f . ( y , x ) , f . y ; attr X c= Y means : DefDef: for x st x in X holds x in Y ; let y be upper bound of Y , x be Element of Y ; let x be Element of REAL ; set S = <* Bags n , m , k *> ; set T = [. 0 , PI / 2 .] ; 1 in dom ( mid ( f , 1 , j ) ) ; ( 4 * PI * PI ) < 2 * PI ; x2 in dom ( ( - 1 ) (#) ( f + g ) ) /\ dom ( - 1 ) (#) ( f + g ) ) ; O c= dom I & rng I c= I implies for x being element st x in I holds I . x = x ( the source of G ) . x = v ; not HT ( f , T ) in 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 ; h1 . i = f . i ; ( p `1 ) * ( - p `1 ) = - p `1 * ( - p `1 ) ; i + 1 <= len Cage ( C , n ) ; len <* P *> = len P + len <* Q *> ; set NN = the Element of the carrier of N1 ; len g-' -' ( x + 1 ) + ( x - 1 ) <= x ; a on B & b on B implies b on B reconsider rI = r * ( I * ( v , I ) ) as FinSequence of REAL ; consider d such that x = d and a [= d ; given u such that u in W and x = u + v ; len f /. ( \downharpoonright n ) = len f -' n + 1 ; set q2 = p3 - q2 , q2 = q2 - q2 , q1 = q2 - q2 , q2 = q2 - q2 , q2 = q2 - q2 , q2 = q2 ; set S = \llangle the carrier of S , the carrier of S \rrangle ; MaxADSet ( b ) /\ ( ( P /\ Q ) ) c= ( P /\ Q ) /\ ( Q /\ Q ) card ( G . ( q1 + q2 ) ) c= F . ( ( G . ( q1 + q2 ) ) ) ; f .: D meets h .: D ; reconsider D = E as non empty directed Subset of L1 ; H = ( the Sorts of A ) '&' ( the Sorts of A ) ; assume t is Element of Funcs ( X , INT ) ; rng f c= the carrier of S ( ) ; consider y being Element of X such that x = { y } ; f1 . ( a1 , a2 ) = b1 + ( b2 + a2 - a2 ) ; the carrier of G = E \/ { E } ; reconsider m = len |^ k as Element of NAT ; set Sn1 = LSeg ( n , C ) , Pn1 = LSeg ( n , d ) ; [ i , j ] in Indices M1 & [ i , j ] in Indices M1 ; assume that P c= Seg m and P is invertible ; for k st m <= k holds z . k in K ; consider a being set such that p in a and a in G ; L1 . p = p * ( p * ( 1 - p ) ) + ( p * ( 1 - p ) ) ; p2 . i = p1 . i .= p2 . i ; let P1 , P2 be a_partition of Y ; attr 0 < r & r < 1 implies r < 1 ; rng ( AffineMap ( a , X ) ) = { 0. 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 - t ) ) = len s + len t ; reconsider x1 = x1 as Element of L1 . i as Element of L1 . i ; Q in FinMeetCl ( the topology of X ) ; dom ( f1 + f2 ) c= dom ( f1 + f2 ) /\ dom ( f2 + g2 ) ; attr n divides m & n divides m ; reconsider x = x as Point of I[01] ; a in \mathop { \rm Point of [: D2 D2 , D1 :] ; y in the carrier of ( the carrier of A ) \/ { f . x } ; Hom ( a , b ) <> {} ; consider k1 such that p " < k1 and k1 < k1 ; consider c , d such that dom f = c \ d and f . c = d ; [ x , y ] in dom g \times dom g ; set S1 = \vert \vert x , y , z .| ; s3 = m1 & s3 = m2 & s3 = m1 & s3 = m2 & s3 = m2 ; x0 in dom u /\ dom ( u + v ) /\ dom ( v + u ) /\ dom ( u + v ) ; reconsider p = x as Point of TOP-REAL 2 ; \mathbb I = REAL ( 1 ) & I = 1 implies for p being Point of TOP-REAL 2 st p in B holds p . p = p . p + p . p f . p1 <= f . p2 ; ( F . ( x , y ) ) `1 <= ( F . ( x , y ) ) `1 ; ( x - y ) * ( x - y ) = ( - y ) * ( x - y ) ; for n being Element of NAT holds P [ n ] ; J , K , L , L be Subset of I ; assume 1 <= i & i < len <* a *> ; 0 |-> a = 0. K ; X . i in 2 |^ ( A * B ) ; <* 0 *> in dom ( e --> 0 ) ; then P [ a ] ; reconsider s1 = s as State of D ; ^\ ( i -' 1 ) <= len |^ j + 1 ; [#] S c= [#] ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( n ) ) D ) ) ) ) ) let V be strict strict strict Subspace of V ; assume k in dom ( mid ( f , i , j ) ) ; let P be non empty Subset of TOP-REAL 2 ; let A , B be Matrix of n , K ; - a * b * ( - b * a ) = - a * ( - b * a ) ; let A be Subset of REAL ; id ( the carrier' of C ) in <* o , o *> ; then ||. x .|| = 0 ; let N1 , N2 be strict Subgroup of G ; j >= len ( g | indx ( g , D1 , j ) ) ; b = Q . ( len Q + 1 ) ; f2 * f1 /* s is convergent ; reconsider h = f * g as Function of I[01] , TOP-REAL 2 ; assume that a <> 0 and delta ( a , b , c ) . 0 >= 0 and delta ( a , b , c ) . 0 >= 0 ; [ t , t ] in the InternalRel of A ; ( v -tree E ) | n is Element of T ; {} = the support of L1 + ( the carrier of L1 + L2 ) ; Directed I c= p1 ; Initialized ( p +* I ) = p +* I ; reconsider N2 = N as net of N1 , N2 be net of N2 ; reconsider Y = Y as Element of \langle Ids ( L ) , \subseteq \rangle ; \bigsqcap ( L \ { p } , p ) <> p ; consider j being Nat such that i2 = i1 + 1 + 1 and j in dom i1 + 1 ; [ s , 0 ] in the carrier of S ; m in ( B \wedge C ) /\ ( D \wedge E ) ; n <= len ( - ( p + q ) ) + len ( - q ) ; ( x1 - x2 ) * ( x1 - x2 ) = ( x1 - x2 ) * ( x1 - x2 ) ; InputVertices S = { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } ; let x , y be Element of FTTTTTTT1 ( n ) ; p = |[ ( ( TOP-REAL 2 ) ) * ( ( ( p `1 ) ) * ( 1 + ( p `2 ) ) ) , p `2 ]| ; g * ( - 1 ) = h * ( - 1 ) .= h * ( - 1 ) * ( - 1 ) ; let p , q be Element of PFuncs ( V , C ) ; x0 in dom ( ( f1 + f2 ) /* c ) /\ dom ( f2 + f1 ) ) ; ( R qua Function ) " = R " ; n in Seg len ( f ^ g ) ; for s being Real st s in R holds s <= s rng s c= dom ( ( - 1 ) (#) ( f1 + f2 ) ) ; synonym Sgm X -> Subset of X means : Def: for x being Element of X holds x in it iff x in X ; - 1 * ( - 1 ) * ( - 1 ) = - 1 * ( - 1 ) * ( - 1 ) ; set S = Segm ( A , P , Q ) ; ex w st e = sqrt ( f . w ) & f . w in F ; curry ( P1 , k ) . x = ( P1 + P2 ) . x ; let T be non empty TopSpace ; len ( f1 + f2 ) = len f1 + len f2 + ( - f2 ) .= len f1 + len f2 + ( - f2 ) .= len f1 + len f2 + ( - f2 ) .= len f1 + len f2 + ( - f2 ) .= len f2 + len f2 sqrt ( i * p ) < ( i * p ) * p ; let x , y be Element of Sub Sub ( U0 ) ; b1 , c1 // b1 , c1 ; consider p being element such that c1 . j = { p } ; assume that f " ( { 0 , 1 } ) = {} and for x st x in X holds f . x = 1 ; assume IC Comput ( F , s , k ) = n ; Directed J ( card I + card J ) ; Stop SCM+FSA ( ) , card I ( ) -> Program of SCM+FSA ( ) ; set m1 = LifeSpan ( p1 , s1 ) , m2 = p1 ; IC Comput ( P1 , s1 , i ) in dom I ; dom t = the carrier of ( the carrier of K ) \/ { 0 , 1 } .= { 1 } ; ( E-max L~ f ) .. f = 1 ; let a , b be Element of PFuncs ( V , C ) ; Cl union ( union F ) c= union ( union F ) ; the carrier of X1 union X2 \/ Y2 \/ { X2 } misses the carrier of X1 union X2 ; assume not LIN a , f . a , f . b ; consider i be Element of M such that i = d * i + 1 ; then Y c= { x } or Y c= { y } ; M |= _ { x , y } ( ( { y , x ) ) . y ) ; consider m be element such that m in Intersect ( { F } ) ; reconsider A1 = support u as Subset of X ; card A \/ B = k + ( 2 * ( 2 * ( 2 * ( 2 * x ) ) ) ; assume that a <> PI and a <> PI and a <> PI and a <> PI and a <> PI and a <> PI and a <> PI and a <> PI and a <> PI and a <> PI & a <> PI & - PI < - PI < - PI and - PI cluster s \! \mathop { \rm \hbox { - } count } ( V ) -> ( S , V ) -valued for ( string of S ) ; Ln /. ( n + 1 ) = ( n + 1 ) * ( n + 1 ) ; let P be compact Subset of TOP-REAL 2 ; assume r2 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ; let A be non empty Subset of TOP-REAL 2 ; assume [ k , m ] in Indices ( the carrier of K ) ; 0 <= ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 1 1 1 ) ) ) ) ) ) ) ) ) ) ) ) ) * ( ( ( ( ( ( 1 - 2 ) ) ) * ( ( ( - 1 ) ( F . N ) . x = ( F . N ) . x ; attr X c= Y & X c= Y implies X c= Y ( y * z ) * ( y * z ) <> 0. K ; 1 + card { \kern1pt X1 + X2 \kern1pt } <= card X1 + ( X2 + x3 ) ) + 1 ; set g = z .. z .. z ; then k = 1 ; let C be Element of Funcs ( X , INT ) ; reconsider B = A as non empty Subset of TOP-REAL 2 ; let a , b , c be Function of Y , BOOLEAN ; L1 . i = ( i .--> g ) . i .= ( i .--> g . i ) ; Plane ( x1 , y1 , y2 ) c= P ; n <= indx ( D2 , D1 , j ) + 1 ; ( g2 . O ) `2 = - ( ( g2 . I ) `2 ) `2 ; j + p .. f + ( len f -' p .. f ) = len f + len f + p .. f + p .. f ; set W = W -|^ ( m + 1 ) , E = W |^ ( m + 1 ) ; Sa . ( a , e ) = a + b . ( a , e ) .= a + b ; 1 in Seg width ( M * ( p , q ) ) ; dom ( i * f ) = dom ( i * f ) ; ( field ( a * x ) ) = W ; set Q = LE g , f , h ; cluster -> -> MSsorted for Relation of U1 ; attr F = { A } ; reconsider z9 = reproj ( 1 , z ) as Element of product G ; rng f c= rng ( f1 + f2 ) \/ rng ( f2 + g2 ) ; consider x such that x in f .: A and x in C ; f = <*> the carrier of C ; E |= _ { x , y , z } ( H , x ) ) ; reconsider n1 = n as Morphism of n , m ; assume that P is commutative and P is commutative and P is commutative ; card ( B \/ { x } ) = k + 1 + 1 ; card ( x \ ( B \ ( B \ ( B \ ( B \ A ) ) ) ) = 0 ; g + R in { s : gx0 < s + g } ; set q1 = ( q , s ) -\hbox \hbox { - 1 , 1 } ; for x being element st x in X holds ( - 1 ) * ( x - 1 ) in X h /. ( i + 1 ) = h . ( i + 1 ) ; set w = max ( B , A ) , A = max ( B , A ) , B = max ( B , A ) ; t in Seg width ( I ^ <* n *> ) ; reconsider X = dom f as Element of Fin C ( ) ; IncAddr ( i , k ) = i + ( k + 1 ) ; S is \hbox { - } bound ( L~ f ) : f is continuous & f . ( p , f . p ) <= f . p & f . p in P ; attr R is condensed means : DefDef: for x being Element of R st x in X holds x in X & x in X ; attr 0 <= a & a <= 1 implies a <= 1 ; u in ( ( c /\ d ) /\ ( b /\ d ) ) /\ ( ( c /\ d ) /\ ( b /\ e ) ) ; u in ( ( c /\ d ) /\ ( b /\ ( d /\ e ) ) ) /\ ( f /\ ( d /\ e ) ) ; len C + ( - 1 ) >= 9 + ( - 1 ) ; x , y , z is_collinear ; a |^ ( n + 1 ) + a |^ ( n + 1 ) = a |^ ( n + 1 ) + a |^ ( n + 1 ) ; <* x , y , c *> /. n in Line ( x , y ) ; set y = <* y , c *> , c = <* y , c *> , d = <* c , d *> , f = <* d , d *> ; F1 /. 1 in rng F1 ; p . m joins r , s . m ; ( p `1 ) * ( ( p `2 ) * ( p `2 ) ) `2 = ( p `2 ) * ( p `2 ) `2 ; W is with_min ; 0 + ( p + ( p + ( p + q ) ) ) * ( p + q ) ) * ( p + q ) * ( p + q ) ) * ( p + q ) * ( p + q ) ) * ( p + q ) * ( p + q x in dom g & x in dom g implies x in dom ( g " ) f1 /* ( seq ^\ k ) is convergent ; reconsider u2 = u as VECTOR of V ; p \! \! \smallfrown ( <* ( ( ( ( ( ( ( ( ( the carrier of X ) \ { {} } ) ) \ { 0 } ) ) ) ) ) = p ; len <* x *> < i + 1 + 1 + 1 ; assume that I is non empty and I c= { x } ; set i2 = \overline { \kern1pt I \kern1pt } + 2 * ( card I + 2 * ( card I + 2 * ( card I + 2 * ( card I + 2 * ( card I + 2 * ( card I + 2 * ( card I + 2 * ( card I + 2 x in { x , y } & h . x = y implies h . y = x consider y being Element of F such that y in B and y <= x ; len S = len ( the Sorts of A ) ; reconsider m = M , n = I as Element of X ; A . ( j + 1 ) = B . ( j + 1 ) \/ B . ( j + 1 ) ; set NN = \alpha=0 : for n holds PN . n = ( N . n ) * ( N . n ) ; rng F c= the carrier of gr { { a } , { b } } ; Comput ( Q , Comput ( Q , t , n ) ) . ( n + 1 ) ) = Q . ( n + 1 ) ; f . k , f . ( k + 1 ) are_relative_prime ; h " ( P /\ ( ( P /\ Q ) ) ) = f " ( P /\ ( Q /\ ( Q /\ ( Q /\ ( Q /\ ( Q /\ ( Q /\ ( Q /\ ( Q /\ ( Q /\ ( Q /\ ( Q /\ ( Q /\ ( Q /\ ( Q /\ ( g in dom ( ( - 1 ) (#) ( ( - 1 ) (#) ( g - f ) ) ) ; g` /\ ` = ( - 1 ) ` + ( - 1 ) ` ; consider n be element such that n in NAT and Z = G . n ; set d = \bf min ( ( ( |. x1 - x2 .| , |. x2 - x3 .| , |. x1 - x3 .| , |. x2 - x3 .| , |. x1 - x3 .| , |. x2 - x3 .| , |. x1 - x3 .| , |. x2 - x3 .| , |. x1 - x3 b `1 + ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) ) < 1 ; reconsider f1 = f as VECTOR of X ; attr i <> 0 implies i * ( i - j ) = i * ( j - i ) ; j1 in Seg ( len ( g . i ) ) ; dom ( i - j ) = Seg ( len ( i - j ) ) .= Seg n ; cluster sec | [. - 1 , 1 .] -> continuous ; Ball ( u , e ) = Ball ( u , e ) ; reconsider x1 = x1 as Function of S , T ; reconsider R1 = x , R2 = y as Relation of L ; consider a , b being Element of A such that x = [ a , b ] and b in A ; ( <* 1 *> ^ <* n *> ) ^ <* n *> in { 1 } ; SS1 +* S2 = S1 +* S2 +* S2 +* S2 .= S1 +* S2 +* S2 +* S2 +* S2 ; ( the function exp of Z ) * ( ( the function exp of Z ) ) is_differentiable_on Z ; let C be Function of C , D ; set C = 1GateCircStr ( <* z , x *> , f ) , f = 1GateCircStr ( <* z , x *> , f ) , g = 1GateCircStr ( <* x , y *> , f ) ; E . ( v2 , v1 ) = E . ( v2 , v2 ) .= E . ( v2 , v1 ) ; ( ( ( the function arctan ) ) `| Z ) . x = ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ex_sup_of A , B , C & for x being Element of A holds x in C * ( x , y ) ; F ( dom f , g ) is Morphism of F ( ) , g ( ) ; reconsider p9 = { q where q is Point of TOP-REAL 2 : p `1 >= 0 & q `2 >= 0 & q <> 0. TOP-REAL 2 } as Subset of TOP-REAL 2 ; g . ( W . ( W . ( W . ( W . n ) ) ) ) in [#] Y & g . ( W . n ) in Y ; let C be compact non vertical non horizontal Subset of TOP-REAL 2 ; LSeg ( f ^ g , j ) = LSeg ( f ^ g , j ) ; rng s c= dom ( f + g ) /\ dom ( f + g ) ; assume x in { ( idseq ( 2 ) * ( i + 1 ) ) where i is Nat : i < len ( ( h | i ) * ( i + 1 ) ) ) * ( i + 1 ) ) ; reconsider n1 = n , n2 = m as Element of NAT ; for y being ExtReal st y in rng ( seq ^\ k ) holds g . y <= g . y for k st P [ k ] holds P [ k + 1 ] m = m1 + ( m2 + ( m1 + m2 ) ) .= m1 + ( m2 + ( m1 + m2 ) ) .= m1 + ( m2 + ( m1 + m2 ) ) .= m1 + ( m2 + ( m1 + m2 ) ) ; assume that for n holds H . n = G . n and H . n = H . n ; set B = f .: ( the carrier of X ) ; ex d being Element of L st d in D & x << d ; assume R ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ assume assume t in ]. r , s .[ or t in ]. r , s .[ ; z + v in W & x + v in W ; x2 in y2 iff ex y st y in y2 & y in y2 & y in y2 & y in y2 & y in y2 ; attr x1 <> x0 & x1 <> 0 & x1 <> 0 & x1 <> 0 & x1 <> 0 & x1 <> 0 & x1 <> 0 & x1 <> 0 & x1 <> 0 & x1 <> 0 & x1 <> 0 & x1 <> 0 & x1 <> 0 & x1 <> 0 & x2 <> 0 & x1 <> 0 & x1 <> 0 & x1 <> 0 & x1 assume that p1 - p2 - p1 and p1 - p2 in the carrier of TOP-REAL 2 ; set q = ( 'not' f ^ <* 'not' p *> ) ^ <* p *> ; let f be PartFunc of REAL , REAL ; ( n -' 2 ) mod n = n - 2 * ( n - 2 * ( n - 2 ) ) ; dom ( T * ( dom t ) ) = dom ( T * ( dom t ) ) ; consider x being element such that x in { w where w is element : w in A } ; assume ( F * G ) . v = v * ( F * ( x , y ) ) ; assume the carrier of D c= the carrier of D ( ) ; reconsider A1 = [. a , b .] as Subset of 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 \! \smallfrown ( <* W *> ^ <* W *> ) ; n1 + 1 -' len f + len g + 1 <= len f + len g + len g + len g + 1 ; partdiff ( q , O , u ) = [ u , v ] ; set B1 = ( card G ) + ( k + 1 ) ; Sum ( L * p ) = 0. V .= ( L * p ) * p .= 0. V ; consider i be element such that i in dom p and t = p . i ; defpred Q [ Nat ] means for n being Nat st n <= $1 holds Q . n = ( n + 1 ) * ( n + 1 ) ; set s3 = Comput ( P1 , s1 , i ) , P1 = P1 , P2 = P2 ; let l be k k k and for k being Nat st k in dom l holds A . k = l . k ; reconsider U = union { union { G where G is Subset of T : G is open } as Subset of T ; consider r such that r > 0 and Ball ( p , r ) c= Q ; ( h | n + 1 ) /. i = p `1 + ( p `1 ) * ( i + 1 ) ) ; reconsider B = the carrier of X as Subset of X ; p1 = <* - ( - c ) , - ( c - c ) , - ( c - c ) , - ( c - c ) *> ; synonym f -> -> -> -> -> -> NAT for Function of NAT , REAL ; consider b being element such that b in dom F and a = F . b ; x0 < ( for x st x in X holds ( x + x0 ) * ( x - x0 ) ) * ( x - x0 ) + ( x - x0 ) * ( x - x0 ) * ( x - x0 ) ) ; attr X c= { x1 : x1 in X & x2 in X & x3 in X & x3 in X & x3 in X & x3 in X & x3 in X & x4 in X & x4 in X & x4 in X & x4 in X ; then w in Cl ( ( x + r ) * ( x + r ) ) ; angle ( x , y , z ) = angle ( x , y , z ) ; attr 1 <= len s & len s = len s implies ex n st n <= len s & n <= len s & n <= len s & n <= len s & n <= len s & n <= len s & n <= len s & n <= len s & n <= len s & n <= len t & n f1 c= f . ( k + 1 ) ; the carrier of { \bf 1 } , { 1 } , { 1 } } , { 1 } , { 1 } \rbrace , { 1 } , { 1 } , { 1 } } , { 1 } , { 1 } } , { 1 } } , { 1 } \rbrace = { 1 } , { 1 } , attr p '&' q in TAUT ( Al , p ) means : : : p in TAUT ( Al ) & p in TAUT ( Al ) ; - ( t - ( t - ( t - t ) ) ) < t - ( t - ( t - t ) ) ; U . ( 1 + 1 ) = U . ( 1 + 1 ) .= U . ( 1 + 1 ) + 1 .= U . ( 1 + 1 ) ; f .: ( the carrier of X ) = the carrier of X ; the carrier of [: O , Seg n :] = [: the carrier of O , the carrier of O :] ; for n being Element of NAT holds G . n c= G . n then V in M ^ { x } ; ex f being Element of F st f is w.r.t. F & for p being Element of F st p in A holds f . p = f . p ; [ h . 0 , h . 3 ] in the InternalRel of G ; s +* ( intloc 0 , 1 ) = ( s +* ( intloc 0 ) .--> ( 1 + 1 ) ) .= s +* ( 1 + 0 ) ; [ x1 , y1 ] in [: the carrier of TOP-REAL 2 , the carrier of TOP-REAL 2 :] ; reconsider t = t as Element of REAL ; C \/ P \/ ( [#] ( G \ { 0. G } ) ) c= ( the carrier of G ) \ ( { 0. G } \ { 0. G } ) ; f " . V in ( the topology of X ) /\ ( ( the topology of X ) . V ) ; x in [#] ( ( [#] ( \alpha ) ) /\ ( A ` ) ) ; g . x <= h . x & h . x <= h . x ; InputVertices S = { x , y , z } \/ { y , z } ; for n being Nat st P [ n ] holds P [ n + 1 ] set R = ( the carrier of K , a , b , c , d be Element of K ; assume M1 is being_line & M1 = M2 implies M1 * M2 = M2 * M1 + M1 * M2 + M1 * M2 reconsider a = ( f . ( i + 1 ) ) as Element of K ; len ( B ^ <* A *> ) = len ( ( A ^ <* B *> ) + len ( B ^ <* A *> ) ; len ( the qua finite seq ) = n & len ( the |-> of seq ) = n ; dom ( - f ) = dom ( - f ) /\ dom ( - f ) ; ( the InternalRel of Y ) . n = ( the InternalRel of Y ) . n ; dom ( p1 ^ p2 ) = dom ( p1 ^ p2 ) \/ dom p2 .= dom ( p1 ^ p2 ) \/ dom p2 \/ { p2 } .= dom ( p1 ^ p2 ) \/ { p2 } ; M . ( [ 1 , 1 ] , y ] ) = 1 * ( 1 , 1 ) .= 1 * ( 1 , 1 ) * ( 1 , 1 ) .= 1 ; assume that W is trivial and W is trivial and W is open and W is open and W is open and W is open and W is open and W is open and W is open and W is open and W is open and W is open and W is open and W is open and W is open and W is open and W is C /. ( i + 1 ) = G * ( i , j ) `1 .= G * ( i , j ) `1 ; C |- 'not' 'not' p => ( 'not' p => q ) ; for b st b in rng g holds ( - fb ) * ( - fb ) <= - ( - b ) * ( - b ) - ( ( ( ( q `1 ) ) ) ) * ( ( q `1 ) * ( - q `1 ) ) ) * ( - ( q `1 ) * ( - q `1 ) ) ) = - ( q `1 ) * ( - q `1 ) ; ( LSeg ( c , m ) \/ { d } ) \/ { c } c= R ; consider p being element such that p in LSeg ( x , p ) and p in LSeg ( x , p ) ; the carrier of X c= the carrier of X & ( the carrier of X ) \/ { 0. X } ; let s be => ( q => s ) ; Im ( ( Partial_Sums F ) . m ) = ( Partial_Sums F ) . m ; let f . x1 , x2 be Element of D ; consider g being Function such that g = F . t and for t being element st t in F holds g . t = t ; p in LSeg ( |[ - 1 , 1 ]| , |[ - 1 , 1 ]| ) ; set R1 = R |^ ( b + 1 ) , R2 = R |^ ( b + 1 ) ; IncAddr ( I , k ) = ( I , k ) --> ( I , k ) ; seq . m <= ( ( the Sorts of A ) . m ) * ( ( the Sorts of A ) . m ) ; a + b = ( a + b ) " * ( a + b ) ; id X /\ ( X /\ Y ) = X /\ Y /\ ( Y /\ Y ) .= X /\ Y /\ ( Y /\ Y ) for x being element st x in dom h holds h . x = f . x ; reconsider H = H1 \/ ( ( the carrier of N1 ) \/ ( the carrier of N1 ) \/ ( the carrier of N1 ) as non empty Subset of N1 ; u in ( ( d /\ ( e /\ ( d /\ b ) ) ) /\ ( d /\ ( e /\ j ) ) ) ; consider y being element such that y in Y and y in Y and x = [ y , x ] ; consider A being finite Subset of R such that card A = card ( A \/ { x } ) and A is finite ; p2 in rng ( f ^ <* p *> ) \ rng ( f ^ <* p *> ) ; len s1 > len s1 & s1 > 0 implies s1 * s1 = s2 * s1 + s2 * s2 ( E-max L~ Cage ( C , n ) ) `2 = ( E-max L~ Cage ( C , n ) ) `2 ; Ball ( e , r ) c= LeftComp Cage ( C , n ) ; f . a1 * ( f . a1 ) = f . a1 * ( f . a2 * ( f . a2 ) ) ; ( seq ^\ k ) . n in ]. - \infty , x0 + r .[ ; g1 . ( s . ( s . ( s . ( n + k ) ) ) ) = g . ( s . ( n + k ) ) ; the internal of S is Relation of the carrier of S , the carrier of S ; deffunc F ( Ordinal , Ordinal ) = $2 $2 $2 $2 $2 $2 $2 $2 ; F . ( s1 . a ) = F . ( s1 . a ) .= F . ( s1 . a ) ; x `1 = A . ( o , a ) .= ( the Sorts of A ) . ( o , a ) .= ( the Sorts of A ) . ( o , a ) ; card ( f " { P } ) c= ( f " { P } ) " { P } ; FinMeetCl ( the topology of T ) c= the topology of T ; synonym o is constructor for for for o is OperSymbol of S means : Def\infty : o is constructor & o <> {} & o <> {} & o <> {} & o <> {} & o <> {} & o <> {} & o <> {} & o <> {} & o <> {} & o <> {} & o <> {} & o <> {} & o <> {} & o <> assume that X + Y = X and Y + Z = Y + Z and X + Z in Y + Z and Y + Z in Z ; the carrier of ( s * the carrier of S ) ) <= 1 + ( s * the carrier of S ) * ( the carrier of S ) ; LIN a , a1 , a2 or b , c // a1 , a2 , b1 ; e . 1 = 0 & e . 2 = 0 & e . 3 = 1 ; E in { { E : E : E is non empty } ; set J = ( l , u ) -TruthEval , J = ( l , u ) -TruthEval ; set A1 = state ( s1 , State ( ) , State ( ) , cp ) , A2 = State ( s1 , State ( ) , cp ) , dp = [ s1 , bp *> , and2 ] , dp = [ s1 , bp *> , and2 ] , cin = [ s1 , bp *> , and2 ] , dp = [ s1 , bp *> , and2 ] , set c9 = [ <* c , d *> , <* d *> ] , <* d , c *> = [ d , d ] , <* d , c *> ] ; x * z * x * ( z * x ) in x * ( z * x ) ; for x being element st x in dom f holds ( g * f ) . x = g . x ; right ( f , g ) c= ( L~ f \/ L~ g ) \/ ( L~ g ) \/ ( L~ g ) \/ ( L~ f \/ L~ g ) ; U is Morphism of ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( C ) ) | ( C ) ) ; set f1 = f ^ <* g *> , f2 = g ^ <* f *> ; attr S1 is convergent means : DefDef: for n st n <= m holds ( for n holds ( ( - S2 ) * ( n - 1 ) ) * ( n - 1 ) ) * ( n - 1 ) ; f . ( 0 qua Nat ) = ( 0 qua Nat ) * ( 0 qua Nat ) .= ( 0 qua Nat ) * ( 0 qua Nat ) .= ( 0 qua Nat ) * ( 0 qua Nat ) .= ( 0 qua Nat ) * ( 0 qua Nat ) .= ( 0 qua Nat ) * ( 0 qua Nat ) .= ( cluster the carrier of W1 -> with_reflexive for RelStr ; consider d being element such that R reduces b , d and for i being Nat st i in dom R holds R . i = d ; b in dom Start-At ( ( card I + 2 , SCMPDS ) + 2 ) ; ( z + a ) + x = z + ( a + x ) .= z + ( a + x ) + x .= z + ( a + x ) + x ; len ( l . ( a , b ) ) = len ( l . ( a , b ) ) + len ( l . ( a , b ) ) ; t1 is ( X \/ { t1 } ) -valued for X being finite Subset of ( X \/ { t1 } ) st X c= ( X \/ { t1 } ) holds t1 in ( X \/ { t1 } ) \/ { t1 } t = <* F . t *> ^ ( C ^ ( p ^ q ) ) ; set p1 = ( 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 ) ) ) ) ) ) ; k2 -' 1 = k2 - 1 ; consider u being Element of L such that u = u and u in D ; len ( ( the InternalRel of K ) * ( a , b ) ) = len ( a * b ) + ( b * ( a , b ) ) ; F1 . x in dom ( ( G * ( o , x ) ) ) ; set H = the carrier of H , I = the carrier of H , H = the carrier of I , I = { H } , I = { H } , I = { H } , H = { H } , I = { H } , H = { H } , I = { H } , H = { H } , I = set H = the carrier of H , I = the carrier of H , I = the carrier of H , H = the carrier of H , I = { I } , I = { I } , H = { I } , I = { I } , { I } , H = { I } , I = { I } , { H } ( Comput ( P , s , 6 ) ) . ( m + 1 ) = s . ( m + 1 ) ; IC Comput ( P3 , s3 , t ) = ( l + 1 ) + 1 .= l + 1 ; dom ( ( ( ( ( ( ( the carrier of K ) ) ) * ( f ) ) ) * ( ( - 1 ) * ( f ) ) ) ) = dom ( ( - 1 ) * ( f ) ) ; cluster <* l *> ^ \varphi -> ( 1 + 1 ) -element for ( S ) -valued ( string of S ) ; set b1 = [ <* b1 , b2 *> , <* b1 , b2 *> ] , b2 = [ b1 , b1 , b2 ] , b1 = [ b1 , b2 ] , b3 = [ b1 , b1 , b2 ] , b3 = [ b1 , b2 ] , b3 = [ b1 , b1 , b3 ] , b3 = [ b1 , b2 ] , s3 = [ b1 , b3 Line ( M , i ) = L * ( i , j ) ; n in dom ( ( the Sorts of A ) * ( the Sorts of A ) ) ; cluster f1 + f2 -> continuous for PartFunc of REAL , REAL ; consider y being Point of X such that a = y and ||. x - y .|| < r ; set x1 = t . intpos ( 8 + 3 ) , y1 = t . intpos ( 8 + 3 ) , y2 = t . intpos ( 8 + 3 ) , y1 = t . intpos ( 8 + 3 ) ; set p1 = LifeSpan ( p1 , s1 ) , p2 = p1 ; consider a being Point of D2 such that a in W1 and b in W1 and a in W2 and b in W1 ; not A , B , C , D , E , F , J , J , F , J , J , F , J , F , J , F , J , E , F , J , F , J , F , J , F , J , F , J , F , J , F , J , F , J , let A , B , C be Element of B ; |. p2 .| ^2 - ( |. p2 .| - sn ) ^2 >= 0 ; l -' 1 + 1 + ( l - 1 ) = l - 1 + 1 + 1 ; x = v + ( a * b ) + ( b * c ) ; the topology of L = the topology of L ; consider y being element such that y in dom ( H . i ) and x = H . i ; { f } \ { n } = ( ( \mathop { \rm Free } ( A , v ) ) ) \/ { H } ; let Y be Subset of X ; 2 * n in { N : 2 * n + 1 * n + 1 * n * n + 1 * n + 1 * n * n + 1 } ; let s be FinSequence of REAL ; for x st x in Z holds ( ( - 1 ) (#) ( ( - 1 ) (#) ( x + a ) ) ) `| Z ) . x = - 1 / ( x + a ) rng ( ( h * ( f * ( f * g ) ) ) ) c= the carrier of ( ( ( ( ( h * g ) * f ) ) ) ; j + 1 + len f <= len f + len g + len g + len g + len g ; reconsider R1 = R * ( I , J ) as Function of REAL , REAL n , m , n be Nat ; C . x = ( s . a ) * ( s . b ) .= ( s . b ) * ( s . b ) .= ( s . b ) * ( s . b ) .= ( s . b ) * ( s . b ) ; ( - z ) * ( z , n ) = - ( z * ( z , n ) ) .= - ( z * ( z , n ) ) .= - ( - z * ( z , n ) ) .= - ( - z * ( z , n ) ) .= - ( - z * ( z , n ) ) .= - ( - z * ( t is_at ( C , s ) ; support ( f + g ) c= dom ( f + g ) \/ dom g \/ dom g \/ dom g ; ex N st N = ( - 1 ) * ( N - 1 ) & N * ( - 1 ) > 0 ; for y , p , q being Element of TOP-REAL 2 st p = p & q in P & p in P holds p . y = q . p { [ x1 , x2 ] : [ x1 , x2 ] in [: { x1 , x2 , x3 , x4 } , { x1 , x2 , x3 , x4 } :] } ; h = ( i |-> h ) . ( i , j ) .= ( i |-> j ) . ( i , j ) .= ( i |-> j ) . ( i , j ) .= ( i , j ) ; ex x1 be Element of G st x1 = x & x1 * x1 in A & x2 in A & x1 in A ; set X = ( ( \mathop { \rm EqClass ( 1 , succ ( succ ( succ ( succ ( succ ( succ ( succ ( succ ( succ ( succ O O ) ) ) ) ) ) ) ) ) . ( O , A ) ) ; b . n in { g1 : x0 < g1 & x0 < g1 & g1 < x0 } ; f /* s1 is convergent & f /. x0 = f /. x0 ; the lattice of Y = the lattice of Y & the lattice of Y = the carrier of Z implies the carrier of Y = the carrier of Z ( 'not' a . x ) '&' ( b . x ) = ( a . x ) '&' ( b . x ) ; q2 = ( ( ( ( ( ( ( ( ( the the carrier of TOP-REAL 2 ) ) ) | ( A ) ) ) ) ) ) * ( ( ( ( the carrier of ( ( ( ( A ) ) | B ) ) ) ) ) * ( ( ( ( A ) ) | B ) ) * ( ( ( A - B ) | B ) ) ) * ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( - 1 ) ) ) ) * ( - 1 ) * ( - 1 ) * ( - 1 ) ) ; set K = upper upper upper ( A , B ) , L = { upper ( A , B , C ) : A in F & B in F & C in F & C in F & C in F & A in F & B in F } ; assume e in { ( ( ( ( w - w ) / 2 ) ) * ( ( - w ) / 2 ) ) * ( ( - w ) / 2 ) ) * ( ( - w ) / 2 ) ) ; reconsider d = a , e = b as Element of dom F ; LSeg ( f , q ) /\ LSeg ( f , j ) = { f /. ( j + 1 ) } ; assume X in { T . ( N . ( N . ( N . ( N . ( N . ( N . ( N . ( N . . N . N . N . . N . N ) ) ) ) ) : N . ( N . ( N . N . N . N ) ) } ; assume that Hom ( d , c ) <> {} and Hom ( d , c ) <> {} and Hom ( d , c ) <> {} ; dom S29 = dom S /\ dom ( the carrier of K ) .= dom ( the carrier of K ) /\ dom ( the carrier of K ) .= dom ( the carrier of K ) /\ dom ( the carrier of K ) /\ dom ( the carrier of K ) /\ dom ( the carrier of K ) .= dom ( the carrier of K ) /\ dom ( the carrier of K ) /\ dom x in H implies ex a st x = a * ( x , a ) ( a * ( a , b ) ) * ( a , b ) = a * ( a , b ) .= a * ( a , b ) .= a * ( a , b ) .= a * ( b * ( a , b ) ) .= a * ( b * ( a , b ) ) .= a * ( b * ( a , b ) ) .= a * ( b D2 . j in { r : r <= ( 1 - r ) * ( 1 - r ) & r <= 1 } ; ex p being Point of TOP-REAL 2 st p = x & p `1 >= 0 & p `1 >= 0 & p `1 <= 0 & p `1 <= 0 & p `1 <= 0 & p `1 <= 0 & p `1 <= 0 & p `1 <= 0 & p `1 = 0 & p `1 = 0 & p `1 = 0 & p `1 = 0 & p `1 = 0 & p `1 = for c holds f . c = g . c iff f . c = g . c dom ( ( f1 + f2 ) (#) ( f2 + g2 ) ) /\ X c= dom ( f1 + f2 ) /\ X ; 1 = ( p * ( p * ( p * ( p * ( p * q ) ) ) ) * ( p * ( p * q ) ) .= p * ( p * q ) + ( p * q ) * ( p * q ) ) .= p * ( p * q ) + ( p * q ) * ( p * q ) ; len g = len f + len <* x *> .= len f + len <* y *> + len <* z *> .= len f + len <* y *> + len <* z *> .= len f + len <* z *> + len <* y *> + len <* z *> .= len f + len <* y *> + len <* z *> + len <* z *> + len dom ( F - G ) = dom ( F - G ) /\ dom ( F - G ) .= dom ( F - G ) /\ dom ( - G ) ; dom ( f . t ) = dom ( f . t ) ; assume a in ( ( id the carrier of T ) ) .: D ; assume that g is one-to-one and g is one-to-one and g is one-to-one ; ( x \ y ) \ ( x \ y ) = 0. X ; consider f such that f * ( f * ( b * f ) ) = f * ( b * f ) and f * ( b * f ) = f * ( b * f ) and f * ( b * f ) ) = f * ( b * f ) and f * ( b * f ) = f * ( b * f ) ; ( the function cos ) | [. - 1 , 1 .] is continuous ; Index ( p , co ) + 1 <= len ( p - co ) + len ( - p .. LS ) + len ( - p .. LS ) ; t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 , t1 , t2 be Element of S ( ) ; cluster -> inf ( ( the carrier of K ) .: ( the carrier of K ) ) ; then P [ f . ( i + 1 ) ] ; Q [ ( D . ( x , y ) ) , F . ( x , y ) ] ; consider x being element such that x in dom ( F . s ) and y = F . x ; l . i < r . i & l . i < l . i ; the Sorts of A = ( the Sorts of A ) +* ( the Sorts of A ) .= ( the Sorts of A ) +* ( ( the Sorts of A ) +* ( the Sorts of A ) ) ; consider s being Function such that s is one-to-one and rng s = { 0 , 1 } and rng s = { 1 } and rng s = { 1 } and rng s = { 1 } and rng s = { 1 } ; dist ( ( b1 , c1 ) , ( ( - ( c1 - c2 ) ) ) ) <= ( - ( c1 - c2 ) ) * ( ( - c2 ) * ( ( - c2 ) * ( - c2 ) ) ) ; ( Gauge ( C , n ) ) * ( i , j ) `1 = ( ( Gauge ( C , n ) * ( i , j ) ) `1 ; q <= ( 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 ) ) ) ) ) ) ) LSeg ( f | i2 , i2 ) /\ LSeg ( i2 , j2 ) = {} ; given a being ExtReal such that a <= I and a <= I and a <= I ; consider a , b being complex number such that z = a * y and z = b * z + b * z ; set X = { b where b is Element of NAT : b < n & n < m } ; ( x * y ) * z = ( x * y ) * z ; set x9 = [ <* x , y *> , f1 ] , y9 = [ <* y , z *> , f2 ] , f3 = [ <* z , x *> , f3 ] , a4 = [ <* x , y *> , f3 ] , a5 = [ <* y , z *> , and2 ] , a5 = [ <* z , x *> , and2 ] , a5 = [ <* z , x *> lnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn ( ( ( ( q `1 / |. q .| - sn ) ) ) / ( 1 + sn ) ) ^2 ) ^2 = 1 ; ( ( - p ) * ( - p ) ) * ( - p ) ) * ( - p ) < 1 ; ( the carrier of S ) \/ { ( the carrier of S ) \/ { {} } = { {} } ; ( s1 - s1 ) . k = ( s1 - s1 ) . k + s1 . k .= s1 . k - s1 . k ; rng ( h + c ) c= dom ( ( h + c ) (#) ( f + c ) ) ; the carrier of X = the carrier of X & the carrier of Y = the carrier of X implies for x being Point of X holds x in the carrier of Y ex p1 st p1 = p1 & |. p1 - p2 .| = r & |. p1 - p2 .| = r ; set h = { \raise .4ex \hbox { - } , A , B , C , D , E , F , J , M , N , F , J , M , N , F , N , F , J , F , J , M , F , N , F , J , M be Element of S ; R |^ ( 0 * n ) = R |^ ( 0 * n ) .= R |^ ( 0 * n ) .= R |^ ( 0 * n ) .= R |^ ( 0 * n ) ; ( Partial_Sums ( F ) ) . n = ( ( Partial_Sums ( F ) ) . n ) * ( ( Partial_Sums ( F ) ) . n ) ; f2 = C . ( EEEEEEEV ( ) , EV ( ) , EV ( ) ) ) ; SI . b = ( the Sorts of A ) . b .= ( the Sorts of A ) . b .= ( the Sorts of A ) . b .= ( the Sorts of A ) . b ; p2 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ; dom ( f . t ) = Seg n & dom ( f . t ) = n ; assume o = ( the connectives of S ) . 11 & ( the connectives of S ) . 12 in ( the connectives of S ) . 12 ; set \varphi = ( l , ( l , 1 ) ) , \varphi = l , l = ( l , 1 ) , l = ( l , 1 ) , t = ( l , 1 ) , l = ( l , 1 ) .[ ; synonym p is non-zero means : DefDef: for for p being Polynomial of n , L holds p . ( p , q ) = p . ( p , q ) ; ( Y - 1 ) * ( ( Y - 1 ) * ( Y - 1 ) ) = ( Y - 1 ) * ( Y - 1 ) & Y - 1 * ( Y - 1 ) * ( Y - 1 ) ) * ( Y - 1 ) = Y * ( Y - 1 ) ; defpred X [ Nat ] means for n being Nat st n <= $1 holds P [ n ] implies P [ n + 1 ] ; consider k be Nat such that for n being Nat st n <= k holds s . n < x0 + r ; Det ( I |^ ( m -' n ) ) = - ( - ( m |^ n ) |^ n ) .= - ( - ( m |^ n ) |^ n ) ; sqrt ( b - sqrt ( b ^2 - a ^2 ) ) < ( b ^2 - a ^2 ) * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * a ) - b ^2 ) ) ) ; C . d = C . d .= C . d ; attr X1 is open means : only : X1 is open & X2 is open & X1 is open & X2 is open & X1 is open & X2 is open & X1 is open & X2 is open implies X1 is open & X2 is open & X2 is open & X1 is open & X2 is open & ( X2 is open & ( X2 is open implies implies implies implies implies implies implies implies implies X1 is open ) ; defpred F ( Element of E , Element of E ) = the Element of E ( ) * ( $1 , $2 ) ; t ^ <* n *> in { t ^ <* n *> ^ <* n *> ^ <* n *> ; ( x \ y ) \ ( x \ y ) = ( x \ y ) \ ( x \ y ) .= 0. X ; let X be non empty set ; synonym A is open for B is open for B is open ; len M1 = len M1 & M1 = M2 implies M1 @ M2 @ = M1 @ M1 @ M2 @ M1 @ M1 @ M2 @ M1 @ M1 @ M1 @ M1 @ M1 @ M2 @ M1 @ M1 @ M1 @ M2 @ M1 @ M1 @ M1 @ M1 @ M1 @ M1 @ M1 @ M1 @ M1 @ M1 @ M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 v = { x where x is Element of K : 0 < x & x < 0 } ; ( Sgm m ) . ( d - e ) = ( m gcd ( m - e ) ) . ( d - e ) ; inf divset ( divset ( D2 , k + 1 ) ) = D2 . ( k + 1 ) ; g . r1 = - ( 2 * r1 ) & g . r2 = - 1 ; |. a .| * |. f .| = ( |. a .| * ( |. f .| ) ) * ( |. a .| * ( |. f .| ) ) .= ( |. a .| * |. f .| ) * ( |. a .| ) ; f . x = ( h . x ) `1 & g . x in { ( h . x ) `1 } ; ex w st w in dom B & <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* 1 *> ^ <* [ 1 , <* d *> , <* d *> ] in ( { [ d , e ] } ) \/ { d , e } ; IC Comput ( i , s1 , n ) = n + 1 ; IC Comput ( P , s , 1 ) = succ IC Comput ( P , s , 1 ) .= succ IC s .= succ IC s ; ( IExec ( B1 , Q , t ) ) . a = t . a .= t . a ; LSeg ( f , i ) misses LSeg ( f , j ) ; assume for x , y being Element of L st x in C holds x + y in C ; integral ( integral ( f , C ) , ( f `| C ) `| C ) = ( f `| C ) . x ; let F be FinSequence of REAL ; ||. R /. ( h + c ) - R /. ( h + c ) .|| < e * ( 1 + c ) + ( 1 - c ) * ( 1 - c ) ; assume a in { q where q is Element of M : dist ( z , q ) <= r } ; set p1 = |[ 2 , 1 ]| , p2 = |[ 2 , 1 ]| , p1 = |[ 2 , 1 ]| , p1 = |[ 2 , 1 ]| , q1 = |[ 2 , 1 ]| ; consider x , y being Element of X such that [ x , y ] in F and x in F and y in G ; for y being Element of REAL st y in Y holds y in Y + ( x + y ) func |. p .| -> o of A equals |. p .| ; consider t being Element of S such that x , y , z is_collinear and t , z is_collinear and t , t , z is_collinear ; dom ( x1 - x2 ) = Seg len x1 & dom ( x1 - x2 ) = Seg len x1 ; consider y being Real such that ( x - y ) = ( y - x ) * y and 0 <= y and 0 <= y and 0 <= y and 0 <= y and 0 <= y and 0 <= y and 0 <= y and 0 <= y and 0 <= y and 0 <= y and 0 <= y and 0 <= y and 0 <= y and 0 <= y and 0 <= y and 0 <= y and 0 <= y and 0 <= y and 0 <= y and 0 <= y ||. f /* s1 .|| = ||. f /* s1 .|| ; ( the InternalRel of A ) ~ \/ ( the InternalRel of A ) ~ = {} ; assume that i in dom p and i + 1 in dom p and i + 1 in dom p and i + 1 in dom p ; reconsider h = f | [: X , Y :] as Function of X , Y :] , ( X \/ Y ) , ( X \/ Y ) | X ; u in the carrier of W1 + W2 & u in the carrier of W1 + W2 & u in the carrier of W1 + W2 & u in the carrier of W1 + W2 & v in the carrier of W1 + W2 ; defpred P [ Element of L ] means f . $1 <= f . $1 & f . $1 <= f . $1 ; T . ( u , v ) = s * x + s * y .= s * ( u , v ) + s * y .= s * ( u , v ) + s * y ; - ( - ( - ( - ( ( - ( x ) ) ) ) ) ) = - ( - ( x - 1 ) ) ) .= - ( - 1 ) * ( - 1 ) * ( - 1 ) ) .= - 1 * ( - 1 ) * ( - 1 ) .= - 1 * ( - 1 ) * ( x - 1 ) ) .= - 1 * ( - 1 * ( given a being Point of G1 , x being Point of G1 , y being Point of G1 , a being Point of G1 st x = a & y in G1 & x in G1 & y in G1 & x in G1 & y in G1 & x in G1 & y in G1 & x in G1 & y in G1 holds x = y ; reconsider f1 = [ the carrier of G , the carrier of G ] , f2 = [ the carrier of G ] , f3 = [ the carrier of G , the carrier of G ] ; let k be Nat , n be Nat ; for x being element holds x in A implies x in ( A ` ) ` & x in A ` consider u , v being Element of R such that l /. i = u * v + u * v ; 1- ( ( p `1 ) / |. p .| - sn ) ) * ( - sn ) * ( - sn ) ) * ( - sn ) * ( - sn ) * ( - 1 ) ) * ( - 1 ) * ( - 1 ) ) * ( - 1 ) * ( - 1 ) * ( - 1 ) ) * ( - 1 ) * ( - 1 ) ) * ( - 1 L1 . k = L1 . ( F . k ) & L1 . k in L1 . ( F . k ) ; set i2 = i2 i2 i2 i2 i2 i2 i2 i2 = i2 i2 + 1 ; attr B is universal means : DefDef: for x , y , z being Element of B holds ( x + y ) * ( x + y ) = ( x + y ) * ( z + y ) ; { a } "/\" D = { a where a is Element of N : a in D } ; ( \square , p2 ) * ( ( ( ( ( ( ( ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( - 1 ) ( - f ) . ( - f . ( - f . x ) ) = ( - f . x ) * ( - f . x ) ; ( G * ( i2 , k ) ) `1 = ( G * ( i2 , k ) ) `1 ; ( Proj ( i , n ) ) . i = <* ( proj ( i , n ) . i ) *> ; f1 + f2 * ( ( f1 + f2 ) * ( f1 + f2 ) ) ) is_differentiable_on Z ; attr the function cos ( ( ( ( ( the function tan of tan ) ) ) `| Z ) ) . x = ( ( the function tan ) . x ) ^2 ; ex t being SortSymbol of S st t = s & ( ( the Sorts of A ) . s = ( the Sorts of A ) . s & ( ( the Sorts of A ) . s = ( the Sorts of A ) . s ) . t ; defpred C [ Nat ] means for P being Nat st P . ( $1 + 1 ) is consistent & P [ $1 ] holds P [ $1 + 1 ] ; consider y being element such that y in dom p and p . i = p . i and p . i = p . i ; reconsider L = product ( { x1 } --> ( x2 , y2 ) ) as Subset of \prod ( A , B ) ; for c being Element of C holds ( id C ) . c = id C LIN f . n , p . n .= ( f | n ) . n ; ( f * g ) . x = f . x & ( f * g ) . x = g . x ; p in { ( ( 1 - i ) * ( ( i - j ) * ( i - j ) ) ) * ( i - j ) ) * ( i - j ) ; f - p = ( f - g ) /* ( c + d ) .= f - g ; consider r being Real such that r in rng ( f | divset ( D , j ) ) and r < m + r ; f1 . ( ( ( ( ( ( ( ( ( ( ( r r r ) ) * f1 ) ) * f1 ) ) * f1 ) ) * f1 ) . x ) in ( ( ( ( ( ( ( ( ( - r ) * f1 ) * f1 ) ) * f1 ) ) * ( ( ( - r ) * f1 ) * f1 ) ) * ( ( - r ) * f1 ) ) ) * ( ( - r ) * eval ( a | n , x ) = a * ( ( a | n ) * ( x , x ) ) .= a * ( a , x ) .= a * ( a , x ) * ( x , x ) .= a * ( a * x , x ) .= a * ( a * x , x ) .= a * ( x , x ) ; z = Comput ( { t , ( ( ( ( ( ( ( ( ( ( ( ( ( { t { t ) ) * ( ( f1 f1 ) * f1 ) ) ) ) * ( ( f1 , f2 ) * ( f1 , f2 ) ) ) ) ) ) ) ) ) * ( ( ( ( ( f1 * f1 ) * ( f1 * f2 ) ) * ( ( f1 * f2 ) ) ) ) ) ) ) .= ( ( set H = { [ the Sorts of A , the Sorts of A ] : for s being SortSymbol of S holds ( the Sorts of A ) . s c= { [ s , the carrier of S ] } ; consider S19 being Element of D such that S = S ^ <* n1 *> and <* n1 *> ^ <* n2 *> = <* n1 *> ^ <* n2 *> ; assume that x1 in dom f and x2 in dom f and x1 in dom f and x2 in dom f and f . x2 = f . x1 ; - 1 - 1 <= ( - 1 ) * ( - 1 ) ; 0. V is Linear_Combination of A & Sum ( L ) = Sum ( L ) + Sum ( L ) ; let k1 , k2 be Element of NAT , k1 , k2 be Element of NAT ; consider j be element such that j in dom a and x = g . j and a . j = g . j ; H . ( x1 , x2 ) c= H . ( x1 , x2 ) or H . ( x1 , x2 ) c= H . ( x1 , x2 ) ; consider a being Real such that p = a * ( - p ) and a * ( - p ) = - p * ( - p ) ; assume that a <= c and b <= d and a <= c and b <= d and d <= c and c <= d and d <= d and d <= d and d <= d and d <= d and d <= d and d <= d and d <= d and d <= d and d <= d and d <= d and d <= d and d <= d and d <= d and d <= d and d <= d and d <= d and d <= d and d <= d and cell ( Gauge ( C , m ) , i ) , ( Gauge ( C , m ) , j ) ) is not bounded ; A2 in { ( S . i ) where i is Element of NAT : i in dom ( S . i ) } ; ( T * b1 ) . y = L . ( b1 * y ) .= L . ( b1 * y ) .= L . ( b1 * y ) ; g . ( s , I ) = s . ( s , I ) & g . ( s , I ) = s . ( s , I ) ; ( log ( 2 * k ) ) ^2 + ( 2 * k ) ^2 + ( 2 * k ) ^2 ) >= ( 2 * k ) ^2 + ( 2 * k ) ^2 ; then p => q in S & p => q in S ; dom ( the addF of K ) misses dom ( the addF of K ) ; synonym f is -> -> -> -> -> -> -> -> -> real for Nat ; assume for a being Element of D holds f . ( a , b ) = f . ( a , b ) ; i = len ( p1 + p2 ) .= len p1 + len p2 + len p3 .= len p3 + len p3 + ( - p2 ) .= len p3 + len p3 + ( - p2 ) .= len p3 + len p3 + ( - p2 ) .= len p3 + len p3 + ( - p2 ) + ( - p2 ) .= len p3 + len p3 + ( - p2 ) .= len p3 + ( - p2 ) + ( - p2 ) + ( - p2 ) .= len p3 + ( - p2 ) + ( l /. ( k + 1 ) ) `1 = ( g /. ( k + 1 ) ) `1 + ( g /. ( k + 1 ) ) `1 ; CurInstr ( P1 , Comput ( P1 , s1 , i + 1 ) ) = halt SCM+FSA ; assume for n being Nat holds ( for n being Nat holds ( ( seq . n ) - seq . n ) ) . n = ( seq . n ) - ( seq . n ) ; ( sin ( r ) ) . r2 = cos ( r ) .= cos ( r ) * ( cos ( r ) ) .= cos ( r ) * ( cos ( r ) ) .= cos ( r ) * ( cos ( r ) ) .= cos ( r * ( r * r ) ) .= cos ( r * ( r * r ) ) .= cos ( r * r ) * ( cos ( r * r ) ) .= cos ( r * r ) * ( cos ( r * r ) ) .= set q = [ t1 , t2 , t1 , t2 ] , t1 = [ t1 , t2 ] , t2 = [ t1 , t2 ] , t1 = [ t1 , t2 ] , t2 = [ t1 , t2 ] , t2 = [ t1 , t2 ] , t1 = [ t1 , t2 ] , t2 = [ t1 , t2 ] , t1 = [ t1 , t2 ] , t2 = [ t1 , t2 ] , t2 = [ t1 , t2 ] , t1 = [ t1 , t2 ] , t2 = [ consider G being sequence of S such that for n being Element of NAT holds G . n in { G . n where n is Element of NAT : n <= n & n <= m } ; consider G such that F = G and ex H st H in { { H } & H in { H } and H in { H } ; the root of tree of ( x , s ) . s in ( the Sorts of A ) . s ; Z c= dom ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Z Z Z ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) for k be Element of NAT holds ( ( Im ( f , g ) ) . k ) . k = ( Im ( ( Im ( f , g ) ) . k ) * ( Im ( f , g ) ) . k ) assume that - 1 < ( 1 - ( q `1 / |. q .| - sn ) ) and 1 < - sn and 0 <= ( 1 - sn ) and 1 <= ( 1 - sn ) and - sn * ( 1 - sn ) and 0 <= 0 and 1 <= ( 1 - sn ) and 0 <= 1 and 1 <= 1 and 1 <= ( 1 - sn ) * ( 1 - sn ) and 1 <= 1 and 0 <= 1 and 1 assume that f is continuous and for a , b st a < b & b < d & d < b holds f . b = f . a + f . b and f . b = f . b - f . b and f . b = f . b - f . b and f . b = f . b - f . b and f . b = f . b - f . b ; consider r being Element of NAT such that s1 = ( ( ( ( ( ( ( ( 1 - r ) * ( s1 ) ) ) ) ) ) * ( ( ( - s1 ) * ( s1 - s1 ) ) ) ) * ( 1 - s1 ) ) ) and s1 * ( 1 - s1 * ( s1 - s1 ) ) ) = 1 * ( 1 - s1 * ( s1 - s1 ) ) ; LE f /. i + f /. j , f /. j , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) ) , f /. ( i + 1 ) , f /. ( i + 1 ) , f /. ( i + 1 ) ) , f /. ( i + 1 ) , f /. ( i + 1 ) ) , f /. ( i + 1 ) , assume that x in the carrier of K and y in the carrier of K and x in the carrier of K and y in the carrier of K ; assume f ^ <* ( the InternalRel of F ) . ( i + 1 ) *> in ( the carrier of F ) . ( i + 1 ) ; rng ( ( ( the carrier of M ) | ( the carrier of M ) ) \/ ( the carrier of M ) ) c= the carrier of M ; assume z in { ( the carrier of G ) \ { t where t is Element of G : t in F } ; consider l being Nat such that for m being Nat st n <= m holds |. ( s . m ) - s . m .| < g ; consider t being VECTOR of product G such that for s being VECTOR of product G st s = t holds ||. ( s - t ) * ( s - t ) .|| < 1 ; attr the carrier of the carrier of K is such that the carrier of K = { v } and for p being Element of K holds p . p = v . p ; consider a being Element of the Point of X such that a on the carrier of X and a on A ; ( - x ) * ( - x ) = ( - x ) * ( - x ) ; let D be set , p be FinSequence of D ; defpred R [ element , element ] means ex x , y being element st x = [ x , y ] & $2 = [ x , y ] ; L~ ( p1 , p2 ) = { p1 : p1 in LSeg ( p1 , p2 ) & p1 in { p2 } & p1 in { p1 } & p1 in { p2 } & p1 in { p2 } & p1 in { p2 } & p1 in { p2 } ; i -' len ( h + c ) + len ( h + c ) + ( - ( i + c ) ) + ( - ( i + c ) ) ) + ( - ( i + c ) ) ) < i - ( - ( i + c ) + ( - ( i + c ) ) + ( - ( i + c ) ) ; for n be Element of NAT st n in dom F holds F . n = ( n + 1 ) * ( n + 1 ) for r , s , t being Element of REAL st t in [. 1 , t .] & t < 1 holds t <= s & t <= 1 holds t <= s assume v in { G where G is Subset of T : G in B & G in B } ; g be non-empty -> non-empty Function of A , REAL ; min ( g . ( x , y ) ) = ( g . ( x , y ) ) . ( g . ( x , y ) ) ; consider q1 being sequence of ExtREAL such that for n being Nat holds q1 . n = ( n + 1 ) * ( ( n + 1 ) * ( n + 1 ) ) ; consider f being Function such that dom f = NAT and for n being Nat holds f . n = F ( n ) ; reconsider Bnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn consider j be Element of NAT such that x = the Element of j and j < n and j < n ; consider x such that z = x and card ( x * ( x * ( x * ( x * ( x * ( x * ( x * p ) ) ) ) ) and x in A and x in A and x in A and x in A ; ( C * ( k + 1 ) ) * ( C * ( k + 1 ) ) = C * ( ( C * ( k + 1 ) ) ) * ( C * ( k + 1 ) ) ; dom ( X --> ( X --> 0 ) ) = X & dom ( X --> 0 ) = X /\ dom ( X --> 0 ) ; ( S -\hbox { - } bound ( C ) ) . ( 1 + 1 ) <= ( S - 1 ) / ( 1 + ( 1 - ( 2 * ( 2 * ( - 1 ) ) ) ) ; attr x is Point of S means : DefDef: ex y st x = y & y in S & x in S ; consider X being element such that X in dom ( f | X ) and X in Y and for n being Nat st n <= m holds f . n = ( f | X ) . n ; assume that Im k is continuous and for x being Element of REAL st x in dom ( x - y ) holds x + y in dom ( x - y ) ; ( 1 / 2 * ( ( ( AffineMap ( n , 0 ) ) ) ) ) * ( ( ( AffineMap ( n , 0 ) , 1 ) ) ) * ( ( ( AffineMap ( n , 0 , 1 ) ) , ( ( AffineMap ( n , 0 , 0 ) , 1 ) ) * ( ( AffineMap ( n , 0 , 1 ) , 1 ) ) ) ) ) * ( ( ( AffineMap ( n , 0 , 0 ) , 1 ) ) ) ) * ( ( ( ( 1 / 2 ) ) ) ) * ( ( ( ( 0 , 0 ) ) defpred P [ Element of omega ] means ( the Element of $1 = ( the \alpha=0 of $1 ) * ( $1 + 1 ) ) * ( a , b ) ; IC Comput ( P , s , 2 ) = succ ( IC Comput ( P , s , 2 ) ) .= ( a , I ) ; f . x = f . ( - g . x ) .= f . ( - g . x ) * ( - g . x ) .= f . ( - g . x ) * ( g . x ) .= f . ( - g . x ) * ( g . x ) .= f . ( - g . x ) * ( g . x ) .= f . ( - g . x ) * ( g . x ) .= f . ( - g . x ) * ( g . x ) ; ( M * F ) . n = ( M * F ) . n .= ( M * F ) . n ; the support of L1 + ( the carrier of L1 + L2 ) c= ( the carrier of L1 + L2 ) \/ ( the carrier of L1 + L2 ) ; attr a , b , c , d , x , y , z , x , y , z be Element of A ; ( the partial of product F ) . n <= ( the carrier of F ) . n ; attr 1 - ( 1 - ( ( - ( ( ( - ( ( ( ( ( ( ( ( ( ( ( ( ( ( - ( p - p - ( p - p ) - - ( ( - ( p - p ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) * ( ( - ( ( ( - ( ( ( ( - ( ( ( ( - ( ( ( - ( ( ( ( - ( ( ( - ( ( ( ( - ( ( ( - ( ( ( ( - ( ( ( s in { p ^ <* n *> where p is Element of NAT : p in T & p in T } ; [ x1 , x2 ] in ( x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x6 , x5 , x5 , x5 , x6 , x5 *> ) . 2 ; attr m is Nat means : Def: for n being Nat holds ( F . n ) . n = ( F . n ) * ( F . n ) ; len ( ( ( ( ( ( ( ( ( ( ( ( ( G G , ( ( ( ( ( G ( ( ( ( ( ( ( G G ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( consider u , v being VECTOR of V such that x = u + v and u in W and v in W and u + v in W ; given F being FinSequence of NAT such that F = x and for k being Nat st k in dom F holds F . k = ( x - k ) * F . k ; 0 = .| * ( 1- BPoint ) + ( - BPoint Point Point Point Point Point of n1 ) ; consider n be Nat such that for m being Nat st n <= m holds |. ( f # x ) . m - x .| < e ; cluster non empty for TopSpace ; \bigsqcap ( B , A ) = {} .= ( the carrier of S ) . ( the carrier of S ) .= ( the carrier of S ) . ( the carrier of S ) .= ( the carrier of S ) . ( the carrier of S ) .= ( the carrier of S ) . ( the carrier of S ) .= ( the carrier of S ) . ( the carrier of S ) .= ( the carrier of S ) . ( the carrier of S ) .= ( the carrier of S ) . ( the sqrt ( r ^2 + ( r ^2 + 1 ) ^2 ) + r ^2 ) + r ^2 + r ^2 ) ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 ^2 ) ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r ^2 + r for x being element st x in A holds ( f `| A ) . x = ( f `| A ) . x 2 * ( 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 ) ) * reconsider p = P /. ( 1 , j ) , q = ( - 1 ) * ( 1 , j ) as Element of K ; consider x1 , x2 being element such that x1 in ]. x1 - x2 , x2 + x3 .[ and x1 in ]. x1 - x2 , x3 .[ and x2 in ]. x1 - x2 , x3 .[ and x1 in ]. x1 - x2 , x3 .[ ; for n being Nat st 1 <= n & n <= len ( g * f ) holds ( g * f ) . n = ( g * f ) . n consider y , z being Element of A such that y in the carrier of A and z in the carrier of A and z in the carrier of A and z in the carrier of A and z in the carrier of A and z in the carrier of A and z in the carrier of A and z in the carrier of A and z in the carrier of A and z in the carrier of A and z in the carrier of A ; given H1 , H2 being strict Subgroup of G such that x = H1 & y = H2 & x = H2 and y in H ; let S be non empty reflexive RelStr , T be non empty RelStr , x be Element of S ; [ a + i , b + c ] in ( the carrier of V ) /\ ( the carrier of V ) ; reconsider m1 = max ( ( ( ( ( p . n ) - p ) ) * ( n + 1 ) ) * ( n + 1 ) ) * ( n + 1 ) ) as Element of NAT ; I <= width ( the InternalRel of K ) * ( i , j ) + ( the carrier of K ) * ( i , j ) ; f2 /* q = ( f2 /* q ) ^\ k .= ( f2 /* q ) ^\ k .= ( f2 /* q ) ^\ k ; attr A1 : A1 : ( A \/ B ) c= ( A \/ B ) \/ ( ( A \/ B ) \/ ( B \/ C ) ; func A -|^ -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> empty Subset of equals equals equals equals ( the carrier of it ) |^ ( n ) ; dom ( ( the Line of K , i ) ) = dom ( F , m ) ; cluster |[ x `1 , x `2 ]| -> |[ x `1 , x `2 ]| -> |[ x `1 , x `2 ]| ; E |= All ( x , y , z ) => ( x , y ) ; F .: ( ( id X ) ) . x = F . ( x , y ) .= F . ( x , y ) .= F . ( x , y ) ; R . ( h . m ) = F . ( h . m ) + R . ( h . m ) ; cell ( G , i2 , j2 ) + ( i2 , j2 ) in cell ( GoB f , i2 , j2 ) ; IC Comput ( P1 , s1 , i + 1 ) = IC Comput ( P1 , s1 , i + 1 ) .= IC Comput ( P1 , s1 , i + 1 ) ; sqrt ( ( ( ( ( ( q `1 / |. q .| - sn ) ) ) ) ^2 ) ) ) + ( ( ( ( q `1 / q `1 ) ) ) ^2 ) ) ^2 ) > 0 ; consider x1 being element such that x1 in dom g and g . x1 = g . x2 and g . x2 = g . x2 ; dom ( ( ( ( ( ( ( ( r - 1 ) (#) f ) ) `| Z ) ) ) `| Z ) = dom ( ( ( r (#) f ) `| Z ) ) `| Z ) .= dom ( ( r (#) f ) `| Z ) ; d . ( y , z ) = ( y , z ) `2 ; attr i is Nat means : Def: for C being Subset of TOP-REAL 2 st C . i in C holds C . i in C . i ; assume that x0 in dom ( f + g ) and for x st x in dom ( f + g ) holds f . x = f . x + g . x ; p in Cl A implies for x being Point of T st x in A holds p . x in A for x being Element of REAL n st x in REAL holds ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 1 1 1 / n ) ) ) ) ) ) ) ) ) ) `| REAL ) ) . x = ( ( 1 - x ) * ( ( 1 - x ) * ( ( 1 - x ) * ( 1 - x ) ) ) ) ) . x ) func mode mode <% -> -> -> -> <% f_e -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> -> empty NAT equals by by by by by the carrier of it is defined by the carrier of it is defined by the term ( Def . 1 ) & for n being Nat holds it . n = n ; [ a1 , a2 ] in ( the carrier of A ) \/ ( the carrier of A ) ; ex a , b being object of C st a in the carrier of C & b in the carrier of C & a in the carrier of C & b in the carrier of C & b in the carrier of C ; ||. ( v . n ) - ( v . n ) .|| < ( |. v . n ) - ( v . n ) .| ; then for Z being set st Z in { Y where Y is Subset of Z : Y in Z } holds Y in { Y where Y is Subset of Z : Y in Z } ; ex_sup_of s , ( the carrier of S ) . s , ( the carrier of S ) . s ] in { [ s , t ] } ; consider i , j being Element of NAT such that i < j and i < j and j < n and i < j ; let D be non empty set , p be FinSequence of D ; consider e being Element of the carrier of X such that e , e // e , d and e in the carrier of X and e in the carrier of X and e in the carrier of X and e in the carrier of X and e in the carrier of X and e in the carrier of X and e in the carrier of X and e in the carrier of X and e in the carrier of X and e in the carrier of X and e in the carrier of X and e in the carrier of X and e in the carrier of X and e in the carrier of X and e in the carrier of X set U = I \! \hbox { - } \! \mathop { \rm \hbox { - } TruthEval } ( I , u ) , I = I \! \mathop { \rm \hbox { - } TruthEval } ( I , u ) , I = I \! \! \mathop { - } \! \mathop { - } \! \! \mathop { \rm \hbox { - } TruthEval } ( I , u ) ) ; |. q1 .| ^2 = ( |. q1 .| ) ^2 + ( |. q2 .| ) ^2 + ( |. q2 .| ) ^2 + ( |. q2 .| ) ^2 + ( |. q2 .| ) ^2 + ( |. q2 .| ) ^2 + ( |. q2 .| ) ^2 + ( |. q2 .| ) ^2 + ( |. q2 .| ) ^2 + ( |. q2 .| ) ^2 + ( |. q2 .| ) ^2 ) ; let T be non empty TopSpace , x be Point of T , y be Point of T ; dom ( the Sorts of A ) = dom ( the Sorts of A ) & dom ( the Sorts of A ) = dom ( the Sorts of A ) ; dom ( h | X ) = dom ( h | X ) /\ X .= X /\ ( dom ( h | X ) /\ X ) .= X /\ ( X /\ Y ) .= X /\ Y /\ Y .= X /\ Y /\ Y /\ Y .= X /\ Y /\ Y /\ Y .= X /\ Y /\ X /\ Y /\ X .= X /\ Y /\ Y ; for N1 , N2 being Element of N1 , N2 being Element of N2 st N1 in dom ( ( ( ( ( h + c ) * ( h + c ) ) * ( ( h + c ) * ( f + c ) ) ) ) holds N1 = N2 ( mod u ) mod ( m , n ) = ( m mod ( m , n ) ) mod ( m mod n ) .= ( m mod n ) mod n ; - ( ( q `1 ) ) * ( - ( q `1 ) * ( - q `1 ) ) ) * ( - ( q `1 ) * ( - q `1 ) ) ) * ( - q `1 / ( q `1 / q `2 ) ) ) * ( - q `1 / q `1 ) ) * ( - q `1 / q `1 ) * ( - q `1 / q `1 ) * ( - q `1 / q `1 ) ) * ( - q `1 ) * ( - q `1 / q `1 ) + ( - q `1 / q `1 ) * ( - q `1 / q `1 ) * ( - q `1 ) * ( - attr r1 = ( r1 + r2 ) * ( r1 + r2 ) & r1 in dom ( r1 + r2 ) * ( r1 + r2 ) ; v1 . m is Function of X , REAL ; attr a <> b & a <> b & a <> 0. ( K , n ) & a * ( - b * ( - b * ( - a * a ) ) ) = - a * ( - b * a ) ; consider i , j being Nat such that i = [ i , j ] and i < j and j < n ; |. p .| ^2 + ( |. p .| - p .| ) ^2 + ( |. p .| ) ^2 + ( |. p .| ) ^2 + ( |. p .| ) ^2 + ( |. p .| ) ^2 + ( |. p .| ) ^2 + ( |. p .| ) ^2 ) ^2 ; consider p1 , p2 being Element of X such that y = p1 and p1 in { p1 where p1 is Point of X : p1 in P & p1 in P & p1 in P & p1 in P & p1 in P & p1 in P & p1 in P & p1 in P & p1 in P & p1 in P & p1 in P and p1 in P and p1 in P and p1 in P and p1 in P and p1 in P and p1 in P and p1 in P and p1 in P and p1 in P and p1 in P and p1 in P and p2 in P and p1 in P and p1 in P and p1 |[ 1 / 2 * ( |[ - 1 , 1 ]| ) , |[ 1 , 1 ]| ]| in |[ 1 , 1 ]| ; ( TOP-REAL 2 ) | ( ( ( TOP-REAL 2 ) | ( ( ( ( TOP-REAL 2 ) | ( ( ( TOP-REAL 2 ) | ( K ) ) ) ) ) ) is closed & ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( K ) ) ) = ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( K ) ) ) ) ; s |= H1 implies H |= ( H '&' ( H '&' ( ( H '&' ( H '&' ( H '&' ( H '&' ( H '&' ( H '&' ( H '&' ( H '&' ( H '&' ( H '&' ( H '&' ( H '&' ( H '&' ( H ) ) ) ) ) ) ) ) ) ) len s + ( - 1 ) * ( - 1 ) = len s + ( - 1 ) * ( - 1 ) .= len s + 1 + 1 * ( - 1 ) * ( - 1 ) .= 1 ; consider z being Element of L1 such that z >= x and z >= y and z >= x ; LSeg ( |[ - ( ( TOP-REAL 2 ) * ( ( TOP-REAL 2 ) * ( ( - 2 ) * ( ( - 2 ) * ( ( - 2 ) * ( ( - 2 ) * ( - 2 ) ) ) ) , ( - 2 * ( ( - 2 ) * ( ( - 2 ) * ( ( - 2 ) * ( ( - 2 ) * ( ( - 2 ) * ( - 2 ) ) ) ) ) ) ) ) /\ { {} } ; lim ( ( f `| N ) /* ( h ^\ N ) ) = ( ( f `| N ) /* ( h ^\ N ) ) . n ; P [ i , f . ( i + 1 ) ] ; for r be Real st 0 < r holds ( s * ( s - r ) ) * ( 1 - r ) < r let X be set , x be Element of X , y be Element of X ; Z c= dom ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( f f f ) ) ) ) ) ) ) ) ) ) ) ) ) ) `| Z ) ) ; ex j be Nat st j in dom ( l ^ <* x *> ) & ( l ^ <* y *> ) . j = ( l ^ <* y *> ) . j ; for u , v , w being VECTOR of V st u < v & u + u in M holds u + v in M A , B , C , D is_collinear ; - Sum <* v , u , v *> = - ( - u ) .= - ( u - v ) .= - ( u - v ) .= - ( u - v ) .= - ( u - v ) .= - ( u - v ) .= - ( u - v ) .= - ( u - v ) .= - ( u - v ) .= - ( u - v ) .= - ( u - v ) ; ( Exec ( a , b ) ) . ( a , b ) = ( a , b ) . ( a , b ) .= ( a , b ) . ( a , b ) .= ( a , b ) . ( a , b ) ; consider h being Function such that f . a = h and for x being element st x in I holds h . x = ( the InternalRel of J ) . x ; let D be non empty set , W1 , W2 be non empty Subset of the carrier of W1 union ( the carrier of W2 ) | ( the carrier of W1 + W2 ) ; card X = 2 implies for x being Element of X holds x in X & x in X ( E-max L~ Cage ( C , n ) ) in rng Cage ( C , n ) ) ; let T be tree , p be FinSequence of dom T , q be Element of dom p , r be Element of dom p ; [ i2 + 1 , 1 ] in Indices G & [ i2 , 1 ] in Indices G & [ i2 , 1 ] in Indices G ; let k be Nat , n be Nat ; dom F " = the carrier of X & F " . x = ( the carrier of X ) " . x ; consider C being finite Subset of V such that C c= A and C c= C and C c= A and C c= C and C c= A and C c= A and C c= A and C c= A and C c= A and C c= B and C c= B and C c= B and C c= B and C c= B and C c= B and C c= B and C c= B and C c= B and C c= B and C c= B and C c= B and C c= B is open and C c= B and C c= B and C c= B and C c= B and C c= B c= B and C c= B and C c= B and C c= B implies for V being prime Lin of the carrier of T holds the carrier of V = { the carrier of T : ex p is Point of T st p in A & p in A & p in A & p in A & p in A & p in A & p in A & p in A & p in A & p in A & p in A & p in A & p in A & p in A & p in A & p in A & p in A & p in A & p in A & p in B & p in B & p in B & p in B & p in B & p in B & p in set X = { F ( v1 ) where v is Element of V : not contradiction } , Y = { { { { v where v is Element of V : not contradiction } ; angle ( p1 , p2 , p3 ) = |[ - ( ( - p1 ) * ( - p1 ) * ( - p1 ) ) , ( - p1 ) * ( - p1 ) * ( - p1 ) * ( - p1 ) * ( - p1 ) * ( - p1 ) * ( - p1 ) ) ]| ; - sqrt ( ( ( ( ( ( q `1 / |. q .| - sn ) ) ) ) ^2 ) = ( - ( ( q `1 / |. q .| - sn ) ) ^2 ) .= - ( - ( q `1 / |. q .| ) ) ^2 ) ; ex f being Function of I[01] , TOP-REAL 2 st f is continuous one-to-one & f . 0 = p1 & f . 1 = p2 & f . 0 = p1 & f . 1 = p2 & f . 0 = p2 & f . 1 = p1 & f . 0 = p2 & f . 1 = p2 & f . 0 = p1 & f . 0 = p2 & f . 1 = p2 & f . 0 = p1 & f . 1 = p2 & f . 1 = p2 & f . 1 = p2 & f . 0 = p2 & f . 1 = p3 & f . 0 = p3 & f . 1 = p3 & f . attr f is differentiable means : only : for x st x in N holds f . x = f . x + f . x ; ex r st x = [ r , s ] & r < r & s < 1 & r < 1 & r < 1 & s < 1 & r < 1 & s < 1 & r < 1 & s < 1 & t < 1 & t < 1 & t < 1 & t < 1 & t < 1 & t < 1 & t < 1 & t < 1 & t < 1 & t < 1 & t < 1 & t < 1 & t < 1 & t < 1 & t < 1 & t < 1 & t < 1 & t < 1 & t < 1 & t < 1 & t < 1 & attr f is FinSequence of D means : DefDef: for p being FinSequence of D st p in D holds p . p = f . p ; attr i in dom G means : : : r * ( i , j ) = r * ( i , j ) ; consider c1 , c2 being bag of n , b being bag of n , L being b being bag of n , p being Element of n , p being Polynomial of n , L being Element of n , p being Polynomial of n , L such that ( p + b ) = p + ( b + p ) and p = <* p . ( p + T ) *> and p . ( p + T ) = p . ( p + T ) and p . ( p + T ) = p . ( p + T ) and p . ( p + T ) = p . ( p . T ) and p . T ) and p . T . T u in { |[ r1 , r2 ]| : r1 < r2 & r2 < 1 & s1 < 1 & s1 < 1 & s1 < 1 } ; card X \/ Y . k = the carrier of X + ( Y + k ) .= ( Y + ( k + 1 ) ) . k + ( Y + k ) . k .= ( Y + ( k + 1 ) ) . k + ( Y + k ) . k ; attr len M1 = len M1 & M1 = M2 implies M1 @ M2 = M1 @ M1 @ M2 @ M1 @ M1 @ M2 @ M1 @ M1 @ M1 @ M1 @ M1 @ M2 @ M1 @ M1 @ M1 @ M1 @ M1 @ M1 @ M1 @ M1 @ M1 @ M1 @ M1 @ M1 @ M1 @ M1 @ M1 @ M1 @ M1 @ M1 @ M1 @ M1 @ M1 @ M1 @ M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 consider g being Real such that 0 < g and for y being Point of S st y in { g where g is Real : ||. g - f /. y - f /. y .|| < g } c= N ; assume x < ( - a ) * ( - b ) or x in - a * ( - b ) ; ( H1 '&' H2 ) . i = ( H1 '&' H2 ) . i & ( H1 '&' H2 ) . i = ( H1 '&' H2 ) . i & ( H1 '&' H2 ) . i = ( H1 '&' H2 ) . i ) ; for i , j st [ i , j ] in Indices M & [ i + j ] in Indices M holds ( - ( i - j ) ) * ( i , j ) = - ( i - j ) * ( i - j ) let f be FinSequence of NAT , x be Element of NAT ; assume F = { [ a , b ] : c in A & b in B & c in C } ; b * ( - ( q * ( - q * ( - q * ( - q * q ) ) ) ) = - ( - ( q * q - q * q ) ) * ( - ( q * q - q * q ) ) * ( - ( q * q - q * q ) ) * ( - ( q * q - q * q ) ) * ( - q * q - q * q ) ) * ( - q * q - q * q ) ) ; Int ( D \ { p where p is Point of T : p in D & p in D & p in D } c= { p where p is Point of T : p in D } attr s is summable means : DefDef: for n holds ( for n being Nat holds ( n >= 1 ) * ( n + 1 ) ) * ( n + 1 ) = n * ( n + 1 ) * ( n + 1 ) ; dom ( ( TOP-REAL 2 ) | D ) = ( TOP-REAL 2 ) | D .= D ; [ X \to Z , X ] is full & Z is full implies for x being Element of Z holds x in X & x in Z ( G * ( i2 , 1 ) ) `2 = ( G * ( i2 , 1 ) `2 ) `2 & ( G * ( i2 , 1 ) `2 ) `2 = ( G * ( i2 , 1 ) `2 ) `2 ; synonym m1 c= m2 for for for for for for m being Nat st m in dom m1 holds ( for n being Nat st n <= m holds ( m . n ) * ( n + 1 ) ) * ( n + 1 ) = ( m * ( n + 1 ) ) * ( n + 1 ) ; consider a being Element of B such that x = F . ( a , b ) and for i being Element of B st i in { [ a , b ] } holds a in { [ a , b ] } ; We mode multiplicative loop over R -> non empty set means : the carrier of it = the carrier of it & the carrier of it = the carrier of it implies the carrier of it = the carrier of it & the carrier of it = the carrier of it & the carrier of it = the carrier of it & the carrier of it = the carrier of it & the carrier of it = the carrier of it & the carrier of it = the carrier of it & the carrier of it = the carrier of it = the carrier of it & the carrier of it = the carrier of it = the carrier of it it it & the carrier of it = the carrier of it it = the carrier of it = the carrier of it & the carrier of it = the carrier of it = the carrier of Comput ( a , b , 1 ) + ( b + 1 ) = [ a , b + 1 ] .= [ a , b + 1 ] + [ b , 1 ] ; cluster -> -> empty for Relation of REAL , REAL ; - ( ( - 1 ) * ( - 1 ) ) * ( - 1 ) * ( - 1 ) ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) ) = - 1 * ( - 1 * ( - 1 ) * ( - 1 ) ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) ) ; eval ( a | ( n , L ) ) = a * ( n , L ) .= a * ( n , L ) .= a * ( n , L ) .= a * ( n , L ) .= a * ( n , L ) * ( n , L ) .= a * ( n , L ) * ( n , L ) .= a * ( n , L ) ; assume the carrier of ( TOP-REAL 2 ) | D = the carrier of ( TOP-REAL 2 ) | D ; assume 1 <= k + 1 & k + 1 <= len ( <* q *> ^ <* p *> ^ <* p *> ) ; 2 * ( a + b ) + ( 2 * ( a + b ) ) + 2 * ( a + b ) + 2 * ( a + b ) + 2 * ( a + b ) + 2 * ( a + b ) ) + 2 * ( a + b ) + 2 * ( a + b + 2 * ( a + b ) ) + 2 * ( a + b + 2 * ( a + b ) ) + 2 * ( a + b + 2 * ( a + b + 2 * ( a + b ) ) + 2 * ( a + b + 2 * ( a + b ) ) + 2 * ( a + b + 2 * ( a + b + 2 * ( a + b ) ) + 2 * ( a + b ) ) M |= _ { x , y , z } ( x , y ) ) implies for x , y , z being Element of M holds ( x . ( y , z ) ) . x = ( x . ( y , z ) ) . y assume that f is differentiable and for x st x in N holds f . x = f . x ; let G be finite Group , x be Element of G ; { c where where where c is Real : 0 <= c & c <= 1 & c <= 1 & c <= 1 & c <= 1 & not ( ex p , q , r st p in { |[ p , q ]| & p in { |[ p , q ]| : p in { p , q ]| } ) & not p in { p , q } & p in { p , q } & p in { p , q } ; the carrier of ( TOP-REAL 2 ) | ( ( ( ( TOP-REAL 2 ) | ( ( ( ( n n ) n ) ) ) ) ) ) = ( the carrier of ( ( TOP-REAL 2 ) | ( TOP-REAL 2 ) ) ) ) /\ ( ( TOP-REAL 2 ) | ( ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) ) ) ) ) .= { ( ( TOP-REAL 2 ) ) | ( ( TOP-REAL 2 ) ) | ( ( TOP-REAL 2 ) ) ) ) ; let G be Group , F be Element of G , G be Element of G ; UsedIntLoc ( IExec ( I , P , s ) ) . a = ( Initialize s ) . a .= s . a ; for f1 being FinSequence of F , f2 being FinSequence of the carrier of F st f1 = ( p ^ f2 ) ^ ( f1 ^ f2 ) holds f1 ^ f2 = f2 ^ ( f2 ^ f1 ) ( p `1 ) * ( ( p `1 ) * ( p `1 ) ) + ( p `1 ) * ( p `1 ) ) = ( p `1 ) * ( p `1 ) + ( p `1 ) * ( p `1 ) ; let x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x6 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 be Element of A ; for x st x in dom ( ( - ( x - a ) ) * ( x - a ) ) holds ( - ( x - a ) * ( x - a ) ) = ( - ( x - a ) * ( x - a ) ) let T be non empty TopSpace , x be Point of T ; ( a 'or' b ) . x = ( a 'or' b ) . x .= ( a 'or' b ) . x .= ( a 'or' b ) . x .= ( a 'or' b ) . x ; for e being set st e in A & e in { A } holds ex x being Element of X st x in A & x in A & x in A & x in A & y in A & x in A & y in A & x in A & x in A for i be set st i in the carrier of S holds H . i = H . i & H . i = H . i & H . i = H . i & H . i = H . i & H . i = H . i & H . i = H . i & H . i = H . i & H . i = H . i & H . i = H . i & H . i = H . i & H . i = H . i = H . i = H . i & H . i = H . i = H . i = H . i & H . i = H . i = H . i = H . i = H . i & H . i = H . i & H for v , w being Element of NAT st x <> v & w <> v holds ( J . v ) . w = J . v card D = card ( D + 1 ) + 1 .= ( D + 1 ) + 1 .= ( D + 1 ) + 1 ; IC Comput ( i , s , 0 ) = ( i + 1 ) + ( i + 1 ) .= ( i + 1 ) + ( i + 1 ) .= i + ( i + 1 ) ; len f /. ( i + 1 ) = len f + len g + len g .= len f + len g + len g ; for a , b being Element of NAT st 1 <= a & a <= b & b <= 1 holds a <= b + ( 1 - a ) let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 ; lim ( ( ( ( ( - ( ( ( - ( ( ( ( - ( ( ( ( - ( ( ( ( - ( ( ( - ( ( - ( ( ( ( ( - ( ( - ( ( ( ( - ( ( ( - ( ( ( ( - ( ( - ( ( ( - ( ( ( ( - ( ( ( - ( ( ( - ( ( ( ( - ( ( ( ( ( - ( ( ( - ( ( ( ( - ( ( ( ( - ( ( ( ( - ( ( - ( ( ( ( ( ( - ( ( ( ( ( ( ( ( ( ( ( ( - ( ( ( ( ( ( ( ( ( ( ( - ( ( - ( ( ( ( ( ( ( ( ( z2 = g /. ( i + 1 ) .= g /. ( i + 1 ) .= g /. ( i + 1 ) ; [ f . 0 , f . 1 ] in the InternalRel of G or [ f . 1 , f . 2 ] in the InternalRel of G ; let G be Subset-Family of B ; CurInstr ( P1 , Comput ( P1 , s1 , i + 1 ) ) = ( P1 , Comput ( P1 , s1 , i + 1 ) ) . ( a + 1 ) .= ( P1 + P2 ) . ( a + 1 ) .= ( P1 + P2 . ( a + 1 ) ) . ( a + 1 ) .= ( P1 + P2 ) . ( a + 1 ) ; assume a on M & b on M & a on M & b on M & a on M & b on M & a on M & b on M & b on M & a on M & b on M & b on M & a on M & b on M & b on M & a on M & b on M & b on M & a on M & b on M & b on M & a on M & b on M & b on M & a on M & b on M & b on M & b on M & a on M & b on M & b on M & b on M & a on M & b on M & b on M & a on M on M & b on M & a on M & b on M & b on M & a on M & b on M & a on M & b on M & a on M & b on M & b on attr T is with_B_T means : Def: for x , y being Point of T st x in F holds x in F & y in F & x in F & y in F & x in F & y in F & x in F & y in F & x in F & y in F & x in F & y in F & x in F & y in F ; for g1 , g2 being Real st g1 in ]. x0 - r , x0 + r .[ & for n st n <= n holds |. ( g1 - g2 ) . n - r .| < g1 holds g1 . n < g1 ( \neg ( ( ( ( ( ( 1 / 2 ) * ( ( ( f f ) ) ) * ( ( f ) ) * ( ( f ) ) * ( ( - 1 ) * ( ( f ) ) * ( ( - 1 ) * ( ( f ) ) * ( ( - 1 ) * ( ( f ) ) * ( ( - 1 ) * ( ( f ) ) * ( ( - 1 ) * ( ( - 1 ) * ( ( ( ( 1 ) ) ) ) ) ) ) ) ) ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( - 1 ) ) ) ) ) ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) ) ) ) * ( ( - 1 ) ) ) * ( ( - 1 ) ) ) * ( ( F . i = F . i .= F . i .= ( F . i ) * ( F . i ) .= ( F . i ) * ( F . i ) .= ( F . i ) * ( F . i ) .= ( F . i ) * ( F . i ) .= ( F . i ) * ( F . i ) .= ( F . i ) * ( F . i ) ; ex y being set st y = f . n & for x being set st x in A holds f . x = F ( x ) ; func f * F -> FinSequence of V means : Def: for x being Element of it holds x * x = x * ( x * F ) + ( x * F ) . x ; not x1 in { x , y , z } \/ { y , z } \/ { z , x , y , z } \/ { z , x , y } \/ { z , x , z } \/ { z , x , y } \/ { z , x , z } \/ { z , y , z } \/ { z , x , z } \/ { z , y , z } } \/ { z , x , y } \/ { z , x , z } \/ { z , y , z } \/ { z , x , z } \/ { z , y , z } \/ { z , x , z , z } \/ { z , x , z , z } \/ { z , y , z } \/ { z , x , z } \/ { z , x , z } \/ { z , x , z for n being Nat for x being set st x in h . n holds h . n = o . n & h . n = o . n + h . n ex SPN1 being Element of [: the carrier of Al , the carrier of Al :] st PN1 = ( the Sorts of A ) . ( ( the Sorts of A ) . s ) & ( the Sorts of A ) . s = ( the Sorts of A ) . s ; consider P being FinSequence of the carrier of G such that P = ( the carrier of G ) \/ { p } and P is line and P is line and P is line and P is line ; let T be TopSpace , A be Basis of T , B be Basis of T ; attr f is partial u0 u0 u0 u0 u0 u0 u0 u0 u0 u0 means : Defmeans : : : for i be Nat holds ( f . i ) * ( f . i ) = r * ( f . i ) ; defpred P [ Nat ] means for n being Nat st n <= $1 holds ( for p being Element of NAT st p in F holds p . p = ( n + $1 ) * p . p ; ex j st 1 < j & j < width GoB f & ( GoB f ) * ( i , j ) `1 = ( GoB f ) * ( i , j ) `1 ; defpred U ( set , set ) = { $2 where $2 is Element of $2 : ex F being sequence of T st F . $2 = F . $2 & for n being Nat st n <= m holds F . n in F . n ; for p1 being Point of TOP-REAL 2 st p1 in P holds LE p1 , p2 , P & LE p1 , p2 , P & p1 , p1 , P & LE p1 , p2 , P & p1 , p1 , p2 , P & P = P & P = P & P = Q & P = Q & P = Q & P = Q & P = Q & P = Q implies P = Q & P = Q & P = Q = Q & P = Q & P = Q & P = Q = Q & P = Q & P = Q & P = Q = Q & P = Q = Q & P = Q & P = Q = Q & P = Q = Q implies P = Q & P = Q = Q & P = Q & P = Q & P = Q & P = Q & P = Q & P = Q = Q = Q & P = Q = Q , Q & P , p1 f . ( H . ( H . ( x , y ) ) ) in the Sorts of A & for x st x in the carrier of A holds f . x in the Sorts of A & f . ( H . x ) in the Sorts of A ; ex p1 being Point of TOP-REAL 2 st x = p1 & ( |. p1 .| - sn ) / ( 1 + sn ) >= 0 & ( 1 - sn ) / ( 1 + sn ) >= 0 & ( 1 - sn ) / ( 1 + sn ) >= 0 & ( 1 - sn ) / ( 1 + sn ) >= 0 & ( 1 - sn ) / ( 1 + 1 ) >= 0 & 1 + sn * ( 1 + sn ) >= 0 & 1 + sn * ( 1 + sn ) * ( 1 + sn ) * ( 1 + sn ) * ( 1 + sn ) * ( 1 + sn ) * ( 1 + sn ) * ( 1 + sn ) * ( 1 + sn ) * ( 1 + sn ) * ( 1 + sn ) * ( 1 + sn ) * ( 1 + sn ) * ( 1 + sn ) * ( 1 + sn ) * ( 1 + sn ) * ( 1 assume for d being Element of NAT holds ( ( s - d ) * ( ( - d ) * ( ( - d ) * ( - d ) ) ) * ( ( - d ) * ( - d ) ) ) * ( ( - d ) * ( - d ) ) ) ) * ( - d * ( - d * ( - d ) ) ) ) * ( - d * ( - d * ( - d * ( - d * ( - d * ( - d * ( - d * ( - d * ( - d * ( - d * ( - d * ( - d * ( - d * ( - d * ( - d * ( - d * ( - d * ( - d * ( - d * ( - d * ( d * ( - d * ( - d * ( - d * ( - d * ( - d * ( - d * ( - d * ( - d * assume that s <> t and for x being Point of TOP-REAL n st x in { p } holds x in { p } and x in { p } and p <> 0. TOP-REAL n and p <> 0. TOP-REAL n in dom p and p <> 0. TOP-REAL n and p <> 0. TOP-REAL n and p <> 0. TOP-REAL n and p <> 0. TOP-REAL n and p <> 0. TOP-REAL n and p <> 0. TOP-REAL n and p <> 0. TOP-REAL n and p <> 0. TOP-REAL n and p <> 0. TOP-REAL n and p <> 0. TOP-REAL n and p <> 0. TOP-REAL n and p <> 0. TOP-REAL n and p <> 0. TOP-REAL n and p <> 0. TOP-REAL n and p <> 0. TOP-REAL n and p <> 0. TOP-REAL n and p <> 0. TOP-REAL n and p <> 0. TOP-REAL n and p <> 0. TOP-REAL n and p <> 0. TOP-REAL n and p <> 0. TOP-REAL n and p <> 0. TOP-REAL n and p <> 0. TOP-REAL n and p <> 0. TOP-REAL n and p <> 0. given r such that 0 < r and for n being Nat st n <= m holds |. ( f /* s1 ) . n - f . n .| < r ; ( p | ( x , y ) ) | ( x , y ) = p | ( x , y ) ; assume that x + h + x in dom ( ( - h ) (#) ( ( - h ) (#) ( ( - h ) (#) ( ( - h ) (#) ( - h ) ) ) ) and x in dom ( - h (#) ( - h ) ) ) and x in dom ( - h (#) ( - h ) ) ; attr i in dom A & i in dom ( A * B ) & i in dom ( A * B ) ; for i be non zero Element of NAT st i in Seg n holds h . i = ( ( ( ( ( a |^ n ) ) * ( i |^ n ) ) ) * ( i |^ n ) ) ) * ( i |^ n ) ) ( ( ( ( ( b - c ) ) (#) ( ( - c ) (#) ( ( - a ) ) (#) ( ( - b ) (#) ( ( - b ) (#) ( - b ) ) ) ) ) ) * ( ( - b ) (#) ( ( - b ) (#) ( - b ) ) ) ) * ( ( - b ) (#) ( - b ) (#) ( - b ) ) ) ) ) * ( ( - b * ( - b * ( - b * ( - b * ( - b * ( - b * ( - a * ( - a * ( - ( b * ( - a * ( - a * ( - a * ( - b * ( - a * ( - a * ( - a * ( - a * ( - a * ( - a * ( b * ( - b * ( - b * ( - b * ( - b * ( b * ( - b * ( - b * assume that for x st x in Z holds ( ( ( ( for x st x in Z ) holds ( ( ( tan ) `| Z ) ) `| Z ) . x = ( ( tan . x ) / ( cos . x ) ^2 ) and for x st x in Z holds ( ( tan . x ) ^2 ) ^2 + ( cos . x ) ^2 ) ^2 ) ^2 and ( ( ( tan . x ) ^2 ) ^2 ) ^2 ) holds ( ( ( ( tan . x ) ^2 ) ^2 = ( ( cos . x ) ^2 ) ^2 and ( for x st x in Z holds ( ( x + cos . x ) ^2 ) ^2 ) ^2 ) ^2 ) ^2 ) ^2 ) ^2 ) ^2 ) ^2 + ( ( cos . x ) ^2 ) ^2 ) ^2 ) ^2 ) ^2 = ( ( ( 1 + ( cos . x ) ^2 ) ^2 ) ^2 + ( cos . x ) ^2 + ( cos . x ) ^2 consider Rnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn ex k be Element of NAT st for x be Element of NAT st x in X holds ( ( f , x ) . k ) * ( ( f , x ) * ( f , x ) ) . k = ( f , x ) * ( f . x ) ; x in { x1 , x2 , x3 , x4 } \/ { x1 , x2 , x3 , x4 } \/ { x3 , x4 } \/ { x3 , x4 } \/ { x4 , x5 , x5 , x6 } \/ { x1 , x2 , x3 , x4 } \/ { x2 , x3 , x4 } \/ { x4 , x5 } \/ { x1 , x2 , x4 } \/ { x1 , x2 , x4 } \/ { x1 , x2 , x3 , x4 } \/ { x3 , x5 } \/ { x1 , x5 } \/ { x1 , x2 , x4 } \/ { x3 , x5 } \/ { x3 , x5 } \/ { x4 , x5 } \/ { x1 , x5 , x5 } \/ { x1 , x5 , x5 } \/ { x1 , x5 , x5 } \/ { x1 , x5 } \/ { x1 , x5 } \/ { x1 , x5 } \/ { x2 , x5 , x5 } \/ { x1 , x5 , x5 } \/ { x1 , x5 , ( G * ( i , j ) ) `2 = ( G * ( i , j ) `2 ) `2 .= ( G * ( i , j ) `2 ) `2 .= ( G * ( i , j ) `2 ) `2 ; f1 * ( the Arity of S ) . o = p . o .= ( the carrier' of S ) . o ; func <* T *> -> tree of T means : DefDef: for p , q being FinSequence of T st p in P & p in T holds p . q in T . p ; F /. k = F . ( p + 1 ) .= F . ( p + 1 ) + F . ( p + 1 ) .= F . ( p + 1 ) + F . ( p + 1 ) ; let A , B , C be Matrix of n , K ; 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 . ( k + 1 ) ) * ( seq . ( k + 1 assume that x in ( the carrier of C ) and y in the carrier of C and x in the carrier of C and y in the carrier of C and x in the carrier of C and y in the carrier of C and y in the carrier of C and x in the carrier of C and y in the carrier of C and y in the carrier of C and y in the carrier of C and y in the carrier of C and y in the carrier of C and y in the carrier of C and y in the carrier of C and y in the carrier of C and y in the carrier of C and y in the carrier of C and y in the carrier of C and y in the carrier of C and y in the carrier of C and y in the carrier of C and y in the carrier of C and y in the carrier of C and y in the carrier of C and y in the carrier of C and y in the carrier of C and y in the carrier of defpred P [ Element of NAT ] means for k being Element of NAT st k < $1 holds ( f /. k ) * ( f /. ( k + 1 ) ) = ( f /. ( k + 1 ) ) * ( f /. ( k + 1 ) ) ; assume that 1 <= k and k + 1 <= len G and k + 1 <= width G and k + 1 <= width G and k + 1 <= width G and G * ( 1 , j ) `1 = G * ( 1 , j ) `1 and G * ( 1 , k ) `1 and G * ( 1 , k ) `2 = G * ( 1 , k ) `1 and G * ( 1 , k ) `2 = G * ( 1 , k ) `1 and G * ( 1 , k ) `1 = G * ( 1 , k ) `1 and G * ( 1 , k ) `1 and G * ( 1 , k ) `1 and G * ( 1 , k ) `1 and G * ( 1 , k ) `1 and G * ( 1 , k ) `1 and G * ( 1 , j + 1 , j + 1 ) `1 and G * ( 1 , j + 1 , G * ( 1 , j + 1 , j + 1 , j attr s < 1 & s < 1 & s < 1 implies ex p , q st p = p * q & p in X & p in X & p in X & p in X & q in X & p in X & p in X & p in X & p in X & p in X & p in X & p in X & p in X & p in X & p in X & p in X & p in X & p in X & p in X & p in X & p in X & p in X & p in X & p in X & p in X & p in X & p in X & p in X & p in X & p in X & p in X & p in X & p in X & p in X & p in X implies p in X & p in X & p in X implies p in X & p in X & p in X & p in X & p in X implies p in let M be non empty Subset of TopSpaceMetr ( M ) ; defpred P [ Element of omega ] means for x being Element of omega st x in Z holds ( ( a * x ) * ( a * x ) ) * ( a * x ) = a * ( a * x ) + ( a * x ) * ( a * x ) ; defpred P1 [ Nat ] means for n being Nat st n < $1 holds ( f /. $1 ) `1 - ( f /. x0 ) `1 / ( n + 1 ) `1 - ( f /. x0 ) `1 ) / ( n + 1 ) ; ( f ^ g ) . ( i + 1 ) = ( f ^ g ) . ( i + 1 ) .= ( f ^ g ) . ( i + 1 ) .= ( f ^ g ) . i ; ( - 1 ) * ( ( - 1 ) * ( - 1 ) ) = ( - 1 ) * ( - 1 ) * ( - 1 ) .= - 1 * ( - 1 ) * ( - 1 ) * ( - 1 ) .= - 1 * ( - 1 ) * ( - 1 ) .= - 1 * ( - 1 ) * ( - 1 ) * ( - 1 ) .= - 1 * ( - 1 * ( - 1 ) * ( - 1 ) ; defpred P [ Nat ] means for x being Element of NAT st x in $1 holds x in the carrier of G & x in the carrier of G & x in the carrier of G & x in the carrier of G & x in the carrier of G ; assume that f /. 1 in Ball ( u , r ) and for i being Nat st i in dom f holds f /. i = f /. i ; defpred P [ Element of NAT ] means ( for x being Element of REAL n holds ( ( ( ( x - 1 ) * ( x - 1 ) ) * ( x - 1 ) ) ) . x = ( ( x - 1 ) * ( x - 1 ) ) * ( x - 1 ) ) * ( x - 1 ) ) ; for x being Element of product F st x in ( the carrier of F ) . i holds x in ( the carrier of F ) . i ( x |^ ( n + 1 ) ) * ( x |^ ( n + 1 ) ) = ( x |^ ( n + 1 ) ) * ( x |^ ( n + 1 ) ) .= ( x |^ ( n + 1 ) ) * ( x |^ ( n + 1 ) ) .= ( x |^ ( n + 1 ) ) * ( x |^ ( n + 1 ) ) .= x |^ ( n + 1 ) * ( x |^ ( n + 1 ) ) ; DataPart Comput ( P +* I , s , k ) = DataPart Comput ( P +* I , s , k + 1 ) ; given r such that 0 < r and for g st g in dom ( f + g ) holds g . g in dom ( f + g ) /\ dom ( f + g ) and g . g in dom ( f + g ) /\ dom ( f + g ) and g . g in dom ( f + g ) /\ dom ( f + g ) and g . g in dom ( f + g ) /\ dom ( f + g ) ; assume that X c= dom ( f1 + f2 ) and for x st x in X holds f1 . x = ( f1 + f2 ) . x and f2 . x = ( f1 + f2 ) . x and f2 . x = ( f1 + f2 ) . x and f2 . x = ( f1 + f2 ) . x and for x st x in X holds f1 . x = ( f1 + f2 ) . x ; let L be continuous continuous continuous continuous continuous non empty Subset of L ; consider i being Element of NAT such that i in dom A and i < p \ast ( m \ast ( p \ast ( p ) ) ) ; ( f1 /* s1 ) . n = ( f1 /* s1 ) . n .= ( f1 /* s1 ) . n ; ex p1 being Element of NAT st F . ( p , p1 ) = g . ( p , p1 ) & F . ( p , p1 ) = g . ( p , p1 ) ; ( mid ( f , i , j ) ) /. ( i + 1 ) = ( f /. i ) * ( f /. ( i + 1 ) ) .= ( f /. i ) * ( f /. ( i + 1 ) ) .= ( f /. i ) * ( f /. ( i + 1 ) ) .= ( f /. ( i + 1 ) ) * ( f /. ( i + 1 ) ) .= ( f /. i ) * ( f /. ( i + 1 ) ) ; ( p ^ <* p *> ) . ( k + 1 ) = ( p ^ <* p *> ) . ( k + 1 ) .= p . ( k + 1 ) ; len ( mid ( D2 , D1 , j ) ) + 1 = len D2 + len D1 + len D2 + 1 .= len D2 + len D2 + 1 ; x * y = ( - 1 ) * ( x * y ) .= ( - 1 ) * ( x * y ) .= - 1 * ( x * y ) .= - 1 * ( x * y ) .= - 1 * ( x * y ) .= 1 * ( x * y ) .= 1 * ( x * y ) + 1 * y * y .= 1 * ( x * y ) + 1 * y * y .= 1 * ( x * y ) + 1 * y .= 1 * ( x * y ) + 1 * ( x * y ) + 1 * ( x * y ) + 1 * y * x * y * y * y + 1 * y * y * y * y * y * y * y + 1 * y * y * y + 1 * y * x * y + 1 * ( x * y ) .= 1 * ( x * y * x * x + 1 * x * x * x * y * x * x * y * y * x * x * x + 1 * x * x * x * y + 1 * y * x * x + 1 * y v . ( reproj ( i , y ) ) * ( ( reproj ( i , y ) ) * ( ( reproj ( i , y ) ) ) ) . ( ( reproj ( i , y ) ) * ( ( reproj ( i , y ) ) ) . ( ( reproj ( i , y ) ) ) . ( ( reproj ( i , y ) ) * ( ( reproj ( i , y ) ) ) . ( ( reproj ( i , y ) ) ) . ( ( ( reproj ( i , y ) ) ) . ( ( reproj ( i , y ) ) ) . ( ( reproj ( i , y ) ) . ( ( i , y ) ) . ( ( reproj ( i , y ) ) . ( ( i , y ) ) ) ) ) = ( ( i , y ) ) . ( ( ( ( ( x , y ) ) . ( ( i , y ) ) . ( ( reproj ( i , y ) ) . ( ( ( i , y ) ) . ( ( i , y ) ) . ( ( ( ( ( ( x , y ) ) ) . ( ( i , y ) ) i * ( i - j ) = <* 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 0 , 0 , 1 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 1 .[ ) ; Sum ( L * F ) = Sum ( L * F ) + ( L * F ) .= Sum ( L * F ) + ( L * F ) .= Sum ( L * F ) + ( L * F ) .= Sum ( L * F ) + ( ( L * F ) * F ) .= Sum ( L * F ) + ( L * F ) .= Sum ( L * F ) + ( L * F ) * F ) .= Sum ( L * F ) + ( ( L * F ) .= Sum ( L * F ) + ( ( L * F ) .= Sum ( L * F ) + ( ( L * F ) .= Sum ( L * F ) * F + ( L * F ) .= Sum ( L * F ) .= Sum ( L * F ) .= ( L * F ) + ( ( ( F * F ) .= Sum ( L * F ) + ( ( L * F ) .= Sum ( L * F ) .= Sum ( L * F ) .= Sum ( L * F ) + ( ( L * F ) .= Sum ( L * F ) + ( ( L * F ) .= ex r being Real st for Y being Real st Y in X holds Y in Y & for p being Real st p in Y holds p in Y holds p >= r ( the Go-board of GoB f ) * ( i , j ) = ( the GoB f ) * ( i , j ) `1 & ( GoB f ) * ( i , j ) `2 = ( GoB f ) * ( i , j ) `2 ; ( the carrier of TOP-REAL 2 ) * ( ( ( ( ( ( ( ( 1 1 - 1 ) ) (#) f ) ) ) `| Z ) ) . x = ( ( ( - 1 ) (#) f ) `| Z ) . x ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( x + 1 ) ) ) ) .= ( - 1 * ( x + 1 ) ) * ( x + 1 ) ) .= - 1 ; that that that that that that that delta ( a , b , c ) - delta ( a , b , c ) and delta ( a , b , d ) - delta ( a , c , d ) and delta ( a , b , d ) - delta ( a , c , d ) and delta ( a , b , d ) - delta ( a , c , d ) and a - delta ( a , b , d ) - delta ( a , c , d ) = - delta ( a , b , d ) and a - delta ( a , b , d ) = - delta ( a , d , d ) and a - delta ( a , d , d ) = - delta ( a , d , d ) and a - delta ( a , d , d , d ) and a - delta ( a , b , d , d ) and a - delta ( a , b , d ) = - delta ( a , c , d , d ) and a - delta ( a , b , d ) = - delta ( a , b , d , d , d ) and a - delta ( a , b , attr : : : : : : \bigsqcap ( L , X ) for x being Element of L holds ( x in X implies x in X & x in X & x in X ) ; ( the carrier of B ) . ( i , j ) = ( the carrier of B ) . ( i , j ) .= ( the carrier of B ) . ( i , j ) .= ( the carrier of B ) . ( i , j ) ;