thesis ; thesis ; thesis ; thesis ; contradiction ; thesis ; thesis ; contradiction ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; contradiction ; thesis ; thesis ; contradiction ; 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 ; contradiction ; thesis ; thesis ; thesis ; contradiction ; thesis ; thesis ; thesis ; thesis ; thesis ; contradiction ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; contradiction ; thesis ; thesis ; thesis ; thesis ; contradiction ; thesis ; contradiction ; 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 ; contradiction ; 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 ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; contradiction ; thesis ; thesis ; contradiction ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; thesis ; contradiction ; 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 ; assume not thesis ; assume not thesis ; thesis ; assume not thesis ; x <> b D c= S let Y ; S `2 is Cauchy ; q in P ; V ; y in N ; x in T ; m < n ; m <= n ; n > 1 ; let r ; t in I ; n <= 4 ; M is finite ; let X ; Y c= Z ; A // M ; let U ; a in D ; q in Y ; let x ; 1 <= l ; 1 <= w ; let G ; y in N ; f = {} ; let x ; x in Z ; let x ; F is one-to-one ; e <> b ; 1 <= n ; f is special ; S misses C ; t <= 1 ; y divides m ; P divides M ; let Z ; let x ; y c= x ; let X ; let C ; x _|_ p ; o is monotone ; let X ; A = B ; 1 < i ; let x ; let u ; k <> 0 ; let p ; 0 < r ; let n ; let y ; f is onto ; x < 1 ; G c= F ; a >= X ; T is continuous ; d <= a ; p <= r ; t < s ; p <= t ; t < s ; let r ; D <= E ; assume e > 0 ; assume 0 < g ; p in X ; x in X ; Y `2 in Y ; assume 0 < g ; not c in Y ; not v in L ; 2 in z `2 ; assume f = g ; N c= b ` ; assume i < k ; assume u = v ; I = J ; B `2 = b `2 ; assume e in F ; assume p > 0 ; assume x in D ; let i be element ; assume F is onto ; assume n <> 0 ; let x be element ; set k = z ; assume o = x ; assume b < a ; assume x in A ; a `2 <= b `2 ; assume b in X ; assume k <> 1 ; f = Product l ; assume H <> F ; assume x in I ; assume p is prime ; assume A in D ; assume 1 in b ; y is generated of squares ; assume m > 0 ; assume A c= B ; X is lower assume A <> {} ; assume X <> {} ; assume F <> {} ; assume G is open ; assume f is dilatation ; assume y in W ; y \not <= x ; A `2 in B `2 ; assume i = 1 ; let x be element ; x `2 = x `2 ; let X be BCK-algebra ; assume S is non empty ; a in REAL ; let p be set ; let A be set ; let G be _Graph , v be Vertex of G , v be Vertex of G , x be set ; let G be _Graph , v be Vertex of G , v be Vertex of G , x be set ; a be UNKNOWN of V ; let x be element ; let x be element ; let C be FormalContext , a , b be object of C ; let x be element ; let x be element ; let x be element ; n in NAT ; n in NAT ; n in NAT ; thesis ; y be Real ; X c= f . a let y be element ; let x be element ; i be Nat ; let x be element ; n in NAT ; let a be element ; m in NAT ; let u be element ; i in NAT ; let g be Function ; Z c= NAT ; l <= [: NAT , NAT :] ; let y be element ; r2 in X ; let x be element ; k1 be Integer ; let X be set ; let a be element ; let x be element ; let x be element ; let q be element ; let x be element ; assume f is being_homeomorphism ; let z be element ; a , b // K ; let n be Nat ; let k be Nat ; B `2 c= B `2 ; set s = s1 ; n >= 0 + 1 ; k c= k + 1 ; R1 c= R ; k + 1 >= k ; k c= k + 1 ; let j be Nat ; o , a // Y ; R c= Cl G ; Cl B = B ; let j be Nat ; 1 <= j + 1 ; arccot is_differentiable_on Z ; exp_R is_differentiable_in x ; j < i0 ; let j be Nat ; n <= n + 1 ; k = i + m ; assume C meets S ; n <= n + 1 ; let n be Nat ; h1 = {} ; 0 + 1 = 1 ; o <> b2 ; f2 is one-to-one ; support p = {} ; assume x in Z ; i <= i + 1 ; r1 <= 1 ; let n be Nat ; a "/\" b <= a ; let n be Nat ; 0 <= r1 ; let e be Real , x be Real ; not r in G . l c1 = 0 ; a + a = a ; <* 0 *> in e ; t in { t } ; assume F is not discrete ; m1 divides m ; B * A <> {} ; a + b <> {} ; p * p > p ; let y be ExtReal ; let a be Int-Location , I be Program of SCM+FSA ; let l be Nat ; let i be Nat ; let r ; 1 <= i2 ; a "\/" c = c ; let r be Real ; let i be Nat ; let m be Nat ; x = p2 ; let i be Nat ; y < r + 1 ; rng c c= E Cl R is boundary ; let i be Nat ; R2 ; cluster uparrow x -> simplex-like ; X <> { x } ; x in { x } ; q , b // M ; A . i c= Y ; P [ k ] ; 2 to_power x in W ; X [ 0 ] ; P [ 0 ] ; A = A |^ i ; o >= s ; G . y <> 0 ; let X be RealNormSpace , x be Point of X , r be Real ; a in X ; H . 1 = 1 ; f . y = p ; let V be RealUnitarySpace , W be Subspace of V , v be VECTOR of V ; assume x in - M ; k < s . a ; not t in { p } ; let Y be functional set , x be Element of Y ; M , L are_isomorphic ; a <= g . i ; f . x = b ; f . x = c ; assume L is lower-bounded & L is lower-bounded ; rng f = Y ; G c= L & G c= L ; assume x in Cl Q ; m in dom P ; i <= len Q ; len F = 3 ; Free p = {} ; z in rng p ; lim b = 0 ; len W = 3 ; k in dom p ; k <= len p ; i <= len p ; 1 in dom f ; b `2 = a `2 + 1 ; x `2 = a * y `2 ; rng D c= A ; assume x in K1 ; 1 <= i0 - 1 ; 1 <= i0 - 1 ; pU c= cos .: { p } 1 <= ii & jj <= k ; 1 <= ii & jj <= k ; UMP C in L ; 1 in dom f ; let seq ; set C = a * B ; x in rng f ; assume f is Lipschitzian on X ; I = dom A ; u in dom p ; assume a < x + 1 ; seq is bounded ; assume I c= P1 ; n in dom I ; let Q ; B c= dom f ; b + p _|_ a ; x in dom g ; F-14 is continuous ; dom g = X ; len q = m ; assume A2 is closed ; cluster R \ S -> real-valued ; upper_bound D in S ; x << sup D ; b1 >= Y1 & b2 >= Y2 ; assume w = 0. V ; assume x in A . i ; g in exists ClU of X ; y in dom t ; i in dom g ; assume P [ k ] ; chi C c= f ; x9 is increasing ; let e2 be element ; - b divides b ; F c= \tau ( F ) ; G is non-decreasing ; G is non-decreasing ; assume v in H . m ; assume b in [#] B ; let S be non void ManySortedSign , X be non-empty ManySortedSet of S ; assume P [ n ] ; assume that union S is linearly-independent and finite ; V is Subspace of V ; assume P [ k ] ; rng f c= NAT ; assume inf X in X ; y in rng f ; let s , I be set , x be set ; b `2 c= b `2 ; assume not x in cQ ; A /\ B = { a } ; assume len f > 0 ; assume x in dom f ; b , a // o , c ; B in B-24 ; cluster Product p -> non empty ; z , x // x , p ; assume x in rng N ; cosec is_differentiable_in x & cosec is_differentiable_in x ; assume y in rng S ; let x , y be element ; i2 < i1 & i2 < i2 ; a * h in a * H ; p , q in Y ; cluster \sqrt I -> left ideal ; q1 in A1 & q2 in A2 ; i + 1 <= 2 + 1 ; A1 c= A2 & A2 c= A1 ; |. \hbox { \boldmath $ p } .| < n ; assume A c= dom f ; Re ( f ) is_integrable_on M ; let k , m be element ; a , a // b , b ; j + 1 < k + 1 ; m + 1 <= n1 ; g is_differentiable_in x0 - r ; g is continuous ; assume that O is symmetric and O is transitive ; let x , y be element ; let jj be Nat ; [ y , x ] in R ; let x , y be element ; assume y in conv A ; x in Int V ; let v be VECTOR of V ; P3 is_closed_on s , P ; d , c // a , b ; let t , u ; let X be set ; assume k in dom s ; let r be non negative Real ; assume x in F | M ; let Y be Subset of S ; let X be non empty TopSpace , x be Point of X , y be Point of X ; [ a , b ] in R ; x + w < y + w ; { a , b } >= c ; let B be Subset of A , x be Element of B ; let S be non empty ManySortedSign ; let x be variable , f be FinSequence of REAL ; let b be Element of X , c be Element of X ; R [ x , y ] ; x ` ` = x ` ; b \ x = 0. X ; <* d *> in D |^ 1 ; P [ k + 1 ] ; m in dom ( ( n + 1 ) -tuples_on NAT ) ; h2 . a = y ; P [ n + 1 ] ; cluster G * F -> pre\kern1pt ; let R be non empty multiplicative RelStr , a , b be Element of R ; let G be _Graph ; let j be Element of I ; a , p // x , p ; assume f | X is lower ; x in rng ( go ^' pion1 ) ; let x be Element of B ; let t be Element of D ; assume x in Q .vertices() ; set q = s ^\ k ; let t be VECTOR of X ; let x be Element of A ; assume y in rng p `2 ; let M be mamaid ; let N be non empty multiplicative loop of M ; let R be RelStr with finite finite Anatural number ; let n , k be Nat ; let P , Q be RelStr ; P = Q /\ [#] S ; F . r in { 0 } ; let x be Element of X ; let x be Element of X ; let u be VECTOR of V ; reconsider d = x as Int-Location ; assume I is not destroy a ; let n , k be Nat ; let x be Point of T ; f c= f +* g ; assume m < v8 ; x <= c2 . x ; x in F ` ; cluster S --> T -> o -valued ; assume t1 <= t2 & t2 <= t1 ; let i , j be even Nat ; assume F1 <> F2 & F2 <> F1 ; c in Intersect ( union R ) ; dom p1 = c & dom p2 = c ; a = 0 or a = 1 ; assume A1 <> A2 & A2 <> A1 ; set i1 = i + 1 ; assume a1 = b1 & a2 = b2 ; dom g1 = A & rng g1 c= A ; i < len M + 1 ; assume not - rng G c= rng G ; N c= dom ( f1 + f2 ) ; x in dom ( sec | X ) ; assume [ x , y ] in R ; set d = sqrt ( x , y ) ; 1 <= len g1 & 1 <= len g2 ; len s2 > 1 & len s2 > 1 ; z in dom ( f1 | X ) ; 1 in dom D2 & D2 is increasing ; ( p `2 ) ^2 = 0 ; j2 <= width G & j <= width G ; len cos > 1 + 1 ; set n1 = n + 1 ; |. q9 .| = 1 ; let s be SortSymbol of S ; i gcd ( i , i ) = i ; X1 c= dom f & X1 c= dom f ; h . x in h . a ; let G be lower_bound Group ; cluster m * n -> invertible ; let k9 be Nat ; i -' 1 > m - 1 ; R is transitive ; set F = <* u , w *> ; p-2 c= P3 & P-2 c= P3 ; I is_closed_on t , Q & I is_halting_on t , Q ; assume [ S , x ] is directed-sups-preserving ; i <= len ( f2 | i ) ; p is FinSequence of X ; 1 + 1 in dom g ; Sum R2 = n * r ; cluster f . x -> complex-valued ; x in dom ( f1 + f2 ) ; assume [ X , p ] in C ; BX c= X0 & BX c= X ; n2 <= ( n2 + 1 ) - 1 ; A /\ ( P ` ) c= A ` ; cluster x -valued for Function ; let Q be Subset-Family of S , x be Element of T ; assume n in dom g2 & n in dom g2 ; let a be Element of R ; t `2 in dom ( e2 | n ) ; N . 1 in rng N ; - z in A \/ B ; let S be SigmaField of X , X be set , F be Subset-Family of X ; i . y in rng i ; REAL c= dom f & f | X is bounded ; f . x in rng f ; NAT <= sqrt ( r ^2 - 2 ) ; s2 in r-5 ( s2 ) ; let z , z be complex number ; n <= N . m ; LIN q , p , s ; f . x = \twoheaddownarrow x /\ B ; set L = [ S \to T ] ; let x be non positive Real ; let m be Element of M ; f in union rng ( F | X ) ; let K be add-associative right_zeroed right_complementable associative distributive non empty doubleLoopStr , f , g be FinSequence of K , a , b be Element of K ; let i be Element of NAT ; rng ( F * g ) c= Y ; dom f c= dom x & f . x = f . x ; n1 < n1 + 1 - 1 ; n1 < n1 + 1 - 1 ; cluster \bf T ( X ) -> limit_ordinal ; [ y2 , 2 ] `2 = z ; let m be Element of NAT ; let S be Subset of R ; y in rng ( S * ) ; b = sup dom ( f | X ) ; x in Seg ( len q ) ; reconsider X = [: D , D :] as set ; [ a , c ] in E1 ; assume n in dom h2 & n in dom h2 ; w + 1 = a1 + b1 & w + 1 = b1 ; j + 1 <= j + 1 ; k2 + 1 <= k1 + 1 ; let i be Element of NAT ; Support u = Support p ; assume X is complete Sub] of m ; assume that f = g and p = q ; n1 <= n1 + 1 - 1 ; let x be Element of REAL ; assume x in rng s2 & y in rng s1 ; x0 < x0 + 1 / 2 ; len ( Carrier ( L ) ) = W ; P c= Seg ( len A ) ; dom q = Seg n & dom q = Seg n ; j <= width M ^ B ; let r8 be real-valued sequence ; let k be Element of NAT ; Integral ( M , P ) < +infty ; let n be Element of NAT ; assume z in ( o \tt 0 ) ; let i be set ; n -' 1 = n - 1 ; len ( n |-> 0 ) = n ; (# Z , c #) c= F assume x in X or x = X ; x is midpoint of b , c , d ; let A , B be non empty set , f , g be FinSequence of A , i , j be Nat ; set d = dim ( p ) ; let p be FinSequence of L ; Seg i = dom q & dom q = Seg i ; let s be Element of E -tuples_on NAT ; B1 be Basis of x , y ; Carrier ( 3 /\ L2 ) = {} ; L1 /\ L2 = {} ; assume { x } = \mathopen { \downarrow y } ; assume b , c , c is_collinear ; LIN q , c , c ; x in rng ( f | reflexive ) ; set n8 = n + j ; let D1 be non empty set , f be FinSequence of D1 , g be FinSequence of D2 ; let K be add-associative right_zeroed right_complementable associative distributive non empty doubleLoopStr , V be Subset of K ; assume f `2 = f & h `2 = h ; R1 - R2 is total & R2 is total ; k in NAT & 1 <= k & k <= len f ; let a be Element of G ; assume x0 in [. a , b .] ; K1 ` is open & ( ( TOP-REAL 2 ) | K1 is open ) ; assume that a , b ] is_a_maximal in C ; a , b be Element of S ; reconsider d = x as Vertex of G ; x in ( s + f ) .: A ; set a = Integral ( M , f ) ; cluster strict n_Str ; not u in { \hbox { \boldmath $ g } } ; the support of f c= B & f is one-to-one ; reconsider z = x as VECTOR of V ; cluster the InitS of L -> strict ; r (#) H is partially function of X , Y ; s . intloc 0 = 1 ; assume that x in C and y in C ; let U0 be strict non-empty MSAlgebra over S , x be Element of U0 ; [ x , Bottom T ] is compact ; i + 1 + k in dom p ; F . i is stable Subset of M ; r-35 in { ( y ) `2 , ( y ) `2 } ; let x , y be Element of X ; A , I be with_0. X ; [ y , z ] in O ; Reloc ( i , k ) = 1 ; rng Sgm A = A ; q |- All ( y , q ) ; for n holds X [ n ] ; x in { a } & x in d ; for n holds P [ n ] ; set p = |[ x , y ]| ; LIN o , a , b ; p . 2 = Z / Y ; ( D1 | ( D2 ) ) `2 = {} ; n + 1 + 1 <= len g ; a in [: Al , Al :] ; u in Support ( m *' p ) ; let x , y be Element of G ; let I be Ideal of L ; set g = f1 + f2 , h = f2 + g2 , i = f1 + f2 , i = f2 + g2 , j = f2 + g2 , i = f2 + g2 a <= max ( a , b ) ; i-1 < len G + 1 ; g . 1 = f . i1 ; x `2 , y `2 in A2 ; ( f /* s ) . k < r ; set v = VAL g ; i - k + 1 <= S - k ; cluster non empty multiplicative for Group ; x in support ( t ) ; assume a in [: the carrier of G , { x } :] ; i `2 <= len ( y9 | i ) ; assume p divides b1 + b2 & p divides b2 ; M1 <= upper_bound M1 & M2 <= sup M1 implies M1 + M2 <= M2 assume x in ( W-min X ) .: X ; j in dom ( z | n ) ; let x be Element of D ( ) ; IC Comput ( 5 , s , m ) = l1 .= ( 0 + 1 ) ; a = {} or a = { x } ; set u9 = Vertices G , G2 = G , G1 = G , G2 = G , G2 = G , G2 = G ; seq " is non-zero & seq " is non-zero ; for k holds X [ k ] ; for n holds X [ n ] ; F . m in { F . m } ; h-4 c= hF & hF c= h ; ]. a , b .[ c= Z ; X1 , X2 are_separated & X2 , X1 are_separated ; a in Cl ( union F ) ; set x1 = [ 0 , 0 ] ; k + 1 -' 1 = k - 1 ; cluster real-valued for Relation ; ex v st C = v + W ; let IT be non empty 1-sorted , x be Element of IT . x ; assume V is Abelian add-associative right_zeroed right_complementable distributive ; X1 \/ Y in natural ( L ) ; reconsider x = x as Element of S ; max ( a , b ) = a ; upper_bound B is upper Subset of B ; let L be non empty reflexive antisymmetric RelStr , X be Subset of L , x be Element of L ; R is reflexive & R is transitive ; E , g |= All ( y , H ) ; dom G ' _ { y } = a ; sqrt ( 1 - 4 ) >= - r ; G . x0 in rng G ; let x be Element of F , y be Element of F ; D [ Initialize Comput ( P1 , s1 , 0 ) ] ; z in dom id B & z in dom ( id B ) ; y in the carrier of N & y in the carrier of N ; g in the carrier of H & h in the carrier of H ; rng ( f | X ) c= NAT & rng ( f | X ) c= { 0 } ; j `2 + 1 in dom s1 ; let A , B be strict Subgroup of G ; let C be non empty Subset of REAL ; f . z1 in dom h & h . z2 in dom h ; P . k1 in rng P ; M = ( A +* {} ) +* ( A .--> {} ) ; let p be FinSequence of REAL ; f . n1 in rng f & f . n1 in rng f ; M . ( F . 0 ) in REAL ; ind [. a , b .] = b ; assume that the distance of V , Q is symmetric and v in Q ; let a be Element of ^ V ; let s be Element of P ( ) ; let PP be non empty reflexive RelStr ; n be Nat ; the support of g c= B & the support of g c= B ; I = halt SCM R & I = p ; consider b being element such that b in B ; set B3 = BCS ( K , K ) ; l <= Sum ( F . j ) ; assume x in \mathopen { [ s , t ] } ; ( x `2 ) ^2 in { t `2 where t is Element of t : t `2 <= t `2 } ; x in product ( ( JumpParts T ) . s ) ; let h be Morphism of c , a ; Y c= [: the_rank_of Y , the_rank_of Y :] ; A2 \/ ( A1 \/ A2 ) c= Carrier ( L1 ) ; assume LIN o , a , b ; b , c // d1 , d2 ; x1 , x2 in Y & x2 in Y ; dom <* y *> = Seg 1 & dom <* y *> = Seg 1 ; reconsider i = x as Element of NAT ; set l = |. ar .| ; [ x , x `2 ] in X ~ ; for n be Nat holds 0 <= x . n |[ a , b ]| = [. a , b .] ; cluster empty -> non empty for Subset of T ; x = h . ( f . z1 ) ; q1 , q2 in P & q1 , q2 in P ; dom M1 = Seg n & dom M1 = Seg n ; x = [ x1 , x2 ] ; R , Q be ManySortedSet of A ; set d = sqrt ( 1 + ( n + 1 ) ) ; rng g2 c= dom W & rng g2 c= dom W ; P ( [#] Sigma \ B ) <> 0 ; a in field R & a = b ; let M be non empty Subset of V , V be Subset of V , v be VECTOR of V ; let I be Program of SCM+FSA , a be Int-Location ; assume x in rng ( R * S ) ; let b be Element of the lattice of T ; dist ( e , z ) > re ; u1 + v1 in W2 & v1 in W1 + W2 ; assume support L misses rng G ; let L be lower-bounded antisymmetric transitive RelStr ; assume [ x , y ] in [: { x } , { y } :] ; dom ( A * e ) = NAT ; a , b be Vertex of G ; let x be Element of ( \mathop { M } ) . x ; 0 <= Arg a * PI ; o , a9 // o , y ; { v } c= the support of l & { v } c= the support of l ; let x be variable of A ; assume x in dom ( uncurry f ) ; rng F c= ( product f ) |^ X assume D2 . k in rng D ; f " . p1 = 0 ; set x = the Element of X , y = the Element of Y ; dom ( Ser G ) = NAT & dom ( G . 0 ) = NAT ; n be Element of NAT ; assume LIN c , a , e1 ; cluster finite for FinSequence of NAT ; reconsider d = c as Element of L1 ( ) ; ( v2 |-- I ) . X <= 1 ; assume x in the support of f & y in the carrier of f ; conv ( @ A ) c= conv ( A ) ; reconsider B = b as Element of the carrier of T ; J , v |= P ! l ( P , l ) ; cluster J . i -> non empty for TopSpace ; ex_sup_of Y1 \/ Y2 , T & Y is directed ; W1 is_field ( W1 + W2 ) & W2 is_field ( W2 + W1 ) ; assume x in the carrier of R & y in the carrier of R ; dom ( n |-> 0 ) = Seg n & dom ( n --> 0 ) = Seg n ; s2 misses s2 & s1 misses s2 implies s1 \/ s2 misses s2 & s1 \/ s2 misses s2 assume ( a 'imp' b ) . z = TRUE ; assume that X is open and f = X --> d ; assume [ a , y ] in Indices ( f ) ; assume that that that Directed I c= J and card I c= K and card I = card J ; Im ( ( lim seq ) (#) ( Im ( seq ) ) ) = 0 ; ( ( - 1 ) (#) sin ) . x <> 0 ; sin is_differentiable_on Z & cos is_differentiable_on Z implies cos is_differentiable_on Z & for x st x in Z holds cos . x = cos . x / cos . x - cos . x t2 . n = t2 . n .= t2 . n ; dom ( cos (#) F ) c= dom F ; W1 . x = W2 . x .= W2 . x ; y in W .vertices() \/ W .vertices() ; kX <= len vX & k <= len vX & k in dom v & k in dom v & k in dom v & v . ( len v ) = v . ( len v ) ; x * a \equiv y * ( a mod m ) ; proj2 .: S c= proj2 .: P & proj2 .: P c= proj2 .: P ; h . p3 = g2 . I .= g2 . I ; G * ( 1 , 1 ) `1 = U * ( 1 , 1 ) `1 .= G * ( 1 , 1 ) `1 ; f . r1 in rng f & f . r2 in rng f ; i + 1 - 1 <= len \hbox { - 1 } ; rng F = rng ( F | n ) .= dom F /\ dom ( F | n ) .= dom F ; mode unital of G is well unital ; [ x , y ] in A ~ { a } ; x1 . o in L2 . ( o . o ) ; the support of m c= B & m is one-to-one ; not [ y , x ] in id X ; 1 + p .. f <= i + len f ; seq ^\ k1 is lower & lim seq = ( lim seq ) - ( lim seq ) ; len ( F . m ) = len ( F . m ) ; let l be Linear_Combination of B \/ { v } ; let r1 , r2 be complex number ; Comput ( P , s , n ) . x = s . x ; k <= k + 1 & k + 1 <= len p ; reconsider c = {} as Element of L ; let Y be with_carrier of T ; cluster strict for Function of L , L ; f . j1 in K . j1 & f . j1 in K ; cluster J => y -> total for Function ; K c= 2 -tuples_on the carrier of T ; F . ( b1 , b2 ) = F . ( b2 , b3 ) ; x1 = x or x1 = y or x1 = z ; attr a <> {} means : Def1 : sqrt a = 1 ; assume that cf a c= b and b in a ; s1 . n in rng s1 & s1 . n in rng s1 ; { o , b2 } on C2 & { o , b1 } on C2 ; LIN o , b , b9 & LIN o , b , c9 ; reconsider m = x as Element of ^ V ; let f be special non empty FinSequence of D ; let F2 be non empty non empty TopSpace ; assume that h is being_homeomorphism and y = h . x ; [ f . 1 , w ] in F . ( f . 2 ) ; reconsider p9 = x as Point of m , P ; A , B be Element of R ; cluster non empty strict for \mathopen { 0. X } ; rng c ' misses rng e & rng e c= rng e & e in rng e ; z is Element of gr { x } ; not b in dom ( a .--> p1 ) ; assume that k >= 2 and P [ k ] ; Z c= dom ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( the component of Q c= UBD A & A c= ( UBD A ) ` ; reconsider E = { i } as finite Subset of I ; g2 in dom ( 1 / 2 ) & g2 in dom ( 1 / 2 ) ; attr f = u * v ; for n holds P1 [ n ] ; { x . O : x in L } <> {} ; let x be Element of V . s ; a , b be Nat ; assume that S = S2 and p = S2 and p = S2 and p = S2 and p = S2 ; gcd ( n1 , n2 ) = 1 & gcd ( n2 , n1 ) = 1 ; set o1 = 2 * PI / 2 , o2 = 2 * PI / 2 , o2 = 2 * PI / 2 , o2 = 2 * PI / 2 ; seq . n < |. r1 - r2 .| ; assume that seq is increasing and r < 0 ; f . ( y1 , x1 ) <= a ; ex c being Nat st P [ c ] ; set g = { n / 1 where n is Element of NAT : n in n } ; k = a or k = b or k = c ; [: a , b :] , [: b , c :] are_equipotent ; assume that Y = { 1 } and s = <* 1 *> ; I1 . x = f . x .= 0 .= 0 ; W4 .first() = W . ( 1 + 1 ) .= W . ( 1 + 1 ) ; cluster -> trivial for Vertex of G , finite _Graph ; reconsider u = u as Element of Bags X ; A in B ^ implies A , B are_equipotent & A , B are_isomorphic x in { [ 2 * n + 3 , k ] } ; 1 >= sqrt ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ; f1 is_Initialize s , f2 & f2 is_Initialize s , f1 ; ( f . q ) `2 <= ( q `2 ) ^2 ; h is Element of the carrier of Cage ( C , n ) ; ( b `2 ) ^2 <= ( p `2 ) ^2 / ( p `2 ) ^2 ; let f , g be membership of X , Y ; S /. ( k , l ) <> 0. K ; x in dom max ( - ( f , g ) ) ; p2 in N1 . ( p1 , p2 ) & p2 . ( p1 , p2 ) in N1 ; len ( H ) < len ( H ) ; F [ A , F ( ) ] ; consider Z such that y in Z and Z in X ; attr 1 in C means : Def1 : A c= C & A c= C ; assume that r1 <> 0 or r2 <> 0 and r1 < r2 ; rng ( q1 - q2 ) c= rng ( q1 - q2 ) ; A1 , L , L is_collinear implies A1 , A2 are_separated & A2 , L is_collinear y in rng f & y in { x } ; f /. ( i + 1 ) in L~ f ; b in element ( p , S ) & c in { p , q } ; then S is negative & P [ S ] ; Cl ( [#] T ) = [#] T & A is open ; f12 | ( A2 \/ B2 ) = f2 | ( A2 \/ B2 ) ; 0. M in the carrier of W & the carrier of W c= the carrier of V ; v , v be Element of M ; reconsider K = union rng K as non empty set ; X \ V c= Y \ V & Y \ V c= Y \ Z ; let X be Subset of S , T be Subset of T ; consider H1 such that H = 'not' H1 and H1 is strict ; 1_ 1 c= One * t * exp ( r , t ) ; 0 * a = 0. R .= a * 0. R .= a * 0. R ; A |^ 2 = A |^ ( 2 + 1 ) ; set vY = v4 /. n , vY = v4 /. n , vY = v5 ; r = 0. \langle \cal E , \Vert * \Vert *> ; ( f . p3 ) `1 >= 0 ; len W = len ( W | ( len W ) ) .= len W ; f /* ( s * G ) is divergent_to+infty ; consider l being Nat such that m = F ( l ) ; t in ( W1 /\ W2 ) & not t in ( W1 /\ W2 ) & t in ( W1 /\ W2 ) ; reconsider Y1 = X1 as SubSpace of X ; consider w such that w in F and not x in w ; let a , b , c be Real ; reconsider i = i - 1 as non zero Element of NAT ; c . x >= id ( L . x ) ; ( natural T ) \/ omega ( T ) is Basis of T ; for x being element st x in X holds x in Y cluster [ x1 , x2 ] -> pair ; downarrow a /\ { t } is Subset of T ; let X be with NAT , f be non empty NAT set ; rng f = IExec ( S , X ) ; let p be Element of B , x be SortSymbol of S ; max ( N1 , 2 ) >= N1 & max ( N1 , 2 ) >= 0 ; 0. X <= b / ( m * ( n + 1 ) ) ; assume that i in I and R1 . i = R ; i = j1 & p1 = q2 & p2 = q2 & q1 = q2 & q1 = q2 ; assume gR in the InternalRel of g & gR in the InternalRel of g ; let A1 , A2 be Point of S , A2 be Subset of T ; x in h " ( P /\ [#] ( T | P ) ) /\ [#] ( T | P ) ; 1 in Seg 2 & 1 in Seg 3 & 2 in Seg 3 implies 1 in Seg 3 reconsider X1 = X as non empty Subset of Tu ; x in ( the Arrows of B ) . i ; cluster E . n -> ( the vertices of G ) -valued ; n1 <= i2 + len g2 & i2 + len g2 <= len g2 ; ( i + 1 ) + 1 = i + ( 1 + 1 ) ; assume v in the carrier' of G2 & v in the carrier' of G2 ; y = Re ( y ) + Im ( y ) ; ( 8 / ( - 1 ) ) gcd p = 1 ; x2 is_differentiable_on ]. a , b .[ & y2 is_differentiable_on ]. a , b .[ implies ex x st x in dom ( f `| Z ) & ( f `| Z ) . x = ( f `| Z ) . x rng ( M2 * M1 ) c= rng ( M2 * M1 ) ; for p being Real st p in Z holds p >= a \bf X \bf Y ( ) = proj1 * ( f | X ) ; ( seq ^\ m ) . k <> 0 ; s . ( G . ( k + 1 ) ) > x0 ; ( p \! \mathop { x } ) . 2 = d ; A \oplus ( B \ominus C ) = ( A \oplus B ) ^^ C ; h \equiv gg . ( mod P ) , g . ( mod P ) ; reconsider i1 = i-1 as Element of NAT ; let v1 , v2 be VECTOR of V , v be VECTOR of V , w be VECTOR of V , y be VECTOR of V , x be VECTOR of V , a be VECTOR of V , b be VECTOR of V , a be VECTOR for V being Subspace of V holds V is Subspace of V reconsider ii = i - 1 as Element of NAT ; dom f c= [: C , D :] & dom f = [: D , D :] ; x in ( the Sorts of B ) . n ; len reconsider l = len ( f1 ^ f2 ) - len ( f2 ^ g2 ) as Element of NAT ; p1 c= the topology of T & the topology of T c= the topology of T ; ]. r , s .[ c= [. r , s .] ; be Basis of T2 , B be Basis of T2 ; G * ( B * A ) = ( id o1 ) * ( B * A ) ; assume that p , u are_collinear and u , v V V and u , v V V V are collinear ; [ z , z ] in union rng ( F | X ) ; 'not' ( b . x ) 'or' b . x = TRUE ; deffunc F ( set ) = $1 .. S - $1 ; LIN a1 , a4 , b1 & LIN a1 , a4 , b3 , b3 ; f " { f . x } = { x } ; dom w2 = dom ( r (#) ( w | n ) ) .= dom ( r (#) ( w | n ) ) ; assume that 1 <= i and i <= n and j <= n and i <= n ; ( ( g2 ) . O ) `2 <= 1 ; p in LSeg ( E . i , F . i ) ; Ii * ( i , j ) = 0. K ; |. f . ( s . m ) - g .| < g1 ; q9 . x in rng ( q | n ) ; Carrier ( L1 ) misses Carrier ( L2 ) ; consider c being element such that [ a , c ] in G ; assume N\mathfrak o = o & Nc1 = o & Nc1 = o ; q . ( j + 1 ) = q /. ( j + 1 ) ; rng F c= ( F |^ C ) " { C } ; P . ( B2 \/ D2 ) <= 0 + 0 ; f . j in [. f . j , f . j .] ; attr 0 <= x & x <= 1 implies x / ( 1 + 1 ) <= x / ( 1 + 1 ) ; p `2 <> 0. TOP-REAL 2 & p `2 <> 0. TOP-REAL 2 ; cluster _ aaaaaaaaaaaaaaaaaaaaS ( S , T ) -> non empty ; let x be Element of S ~ ; <^ F , F ^> is one-to-one ; |. i .| <= - ( 2 |^ n ) / ( 2 |^ n ) ; the carrier of I[01] = dom P & P is closed ; ! * ( n + 1 ) ! > 0 * n ; S c= ( A1 /\ A2 ) /\ ( A2 /\ A1 ) ; a3 , a4 // b2 , a4 & a4 , a4 // a4 , a4 ; then dom A <> {} & dom A <> {} & dom A = {} ; 1 + ( 2 * k + 4 ) = 2 * k + 5 ; x Joins X , Y , G , i , j , k , G ; set v2 = ( v /. i ) * ( v /. ( i + 1 ) ) ; x = r . n .= ( r . n ) * ( r . n ) ; f . s in the carrier of S2 & f . s in the carrier of S2 ; dom g = the carrier of I[01] & rng g = the carrier of I[01] ; p in Lower_Arc ( P ) /\ Lower_Arc ( P ) ; dom d2 = [: A2 , B2 :] & dom d2 = [: A2 , B2 :] ; 0 < sqrt ( p ^2 - ( p `2 - 1 ) ) ^2 + ( p `2 - 1 ) ^2 ; e . ( m + 1 ) <= e . ( m + 1 ) ; B \ominus X \/ B \ominus Y c= B \ominus X - +infty < Integral ( M , Im ( g | B ) ) ; cluster O := F -> U -defined for OperSymbol of X ; let U1 , U2 be non-empty MSAlgebra over S , f be Function of U1 , U2 ; Proj ( i , n ) * g is_differentiable_on X ; x , y , z be Point of X , x be Point of X , y be Point of X , z be Point of X , x be Point of X , y be Point of X , z be Point of X , x be reconsider p9 = p . x as Subset of V ; x in the carrier of Lin ( A ) & y in A ; let I , J be parahalting Program of SCM+FSA , a , b be Int-Location ; assume - a is lower & b is lower ; Int Cl A c= Cl Int A & Int Cl A c= Cl Int A ; assume for A being Subset of X holds Cl A = A & A is closed ; assume q in Ball ( |[ x , r ]| , r ) ; ( p2 `2 ) ^2 <= ( p2 `2 ) ^2 / ( p2 `2 ) ^2 ; Cl Q ` = [#] ( T | A ) ; set S = the carrier of T , T = the carrier of T ; set I8 = IC f , I8 = width f , R8 = width f , R8 = width f , R8 = width f , R8 = width f , I8 = width f , I8 = width f , I8 = width len p -' n = len q - n .= len q - n ; A is permutation of Funcs ( A , x ) ; reconsider n6 = ni - 1 as Element of NAT ; 1 <= j + 1 & j + 1 <= len ( s | n ) ; q9 , q9 be Element of M , q be Element of M ; a1 in the carrier of S1 & a2 in the carrier of S1 & a3 in the carrier of S1 ; c1 /. ( n + 1 ) = c1 . ( n + 1 ) ; let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 ; y = ( ( f * S ) * ( x , y ) ) . x ; consider x being element such that x in \mathop { \rm many Cl } A ; assume r in ( dist ( o ) ) .: P ; set i2 = ( TOP-REAL 2 ) * ( ( N + 1 ) * ( N + 1 ) ) ; h2 . ( j + 1 ) in rng h2 ; Line ( M1 , k ) = M . i ; reconsider m = sqrt ( x ^2 - 2 ) as Element of REAL n ; U1 , U2 be Subspace of U0 , U2 be Subspace of U0 , f be Function of U1 , U2 ; set P = Line ( a , d ) ; len p1 < len p2 + 1 & len p2 = len p1 + 1 ; T1 , T2 be Scott topological - of L , T ; then x <= y & ( { x } c= { y } ) & x in { x } ) ; set M = n -tuples_on the carrier of K , C = n -tuples_on the carrier of K ; reconsider i = x1 , j = x2 as Nat ; rng ( the_arity_of o ) c= dom ( the Arity of S ) & rng ( the Arity of S ) c= dom ( the Arity of S ) ; z1 " = z1 " * ( z " ) .= z " * ( z " ) .= z " * ( z " ) ; x0 - r in L /\ dom f & f | X in L /\ dom f ; then w is strict for S being non empty Subset of S , T , f be Function of S , T ; set x9 = x9 ^ <* Z *> , y9 = y9 ^ <* Z *> , x9 = y9 ^ <* Z *> , y9 = Z ^ <* Z *> , y9 = Z ^ <* Z *> , x9 = Z ^ <* Z *> , y9 = Z ^ <* Z *> len w1 in Seg len w1 & len w2 = len w2 & len w1 = len w2 ; ( uncurry f ) . ( x , y ) = g . y ; let a be Element of PFuncs ( V , { k } ) ; x . n = sqrt ( |. a . n .| - |. b .| ) ; ( p `1 ) ^2 <= ( G * ( 1 , 1 ) ) `1 ; rng ( g | X ) c= L~ ( g | X ) ; reconsider k = i-1 * j + 1 as Nat ; for n be Nat holds F . n is with_finite ; reconsider x9 = x as VECTOR of M , x9 = y as VECTOR of M ; dom ( f | X ) = X /\ dom f .= X /\ dom f .= X ; p , a // p , c & b , a // c , c ; reconsider x1 = x as Element of REAL m -tuples_on REAL ; assume i in dom ( a * p ^ q ) ; m . ( \hbox { \boldmath $ g $ } , p ) = p . ( \hbox { \boldmath $ g $ } , p ) ; a / ( s . m ) - 1 / ( s . n ) <= 1 / ( s . m ) ; S . ( n + k ) c= S . ( n + k ) ; assume that B1 \/ B2 = B2 and B1 \/ B2 = B2 and B2 \/ B2 = B1 \/ B2 ; X . i = { x1 , x2 } & X . i = { x1 , x2 } ; r2 in dom ( h1 + h2 ) & r2 in dom ( h1 + h2 ) ; u1 - 0. R = a & bO = b - a ; F8 is_closed_on t , Q & P8 is_halting_on t , Q implies Q is closed set T = IExec ( X , x0 , x1 ) ; Int Cl ( Int R ) c= Int Cl R & Int Cl ( Int R ) c= Cl Int R ; consider y being Element of L such that c . y = x ; rng ( ( F . x ) | ( F . x ) ) = { F . x } ; G " { c } c= B \/ S \/ S \/ S \/ S \/ S ; fo is_convergent & X is Relation of [: X , Y :] , Y ; set RP = the Point of P , R = the Point of Q , S = the Point of Q ; assume that n + 1 >= 1 and n + 1 <= len M ; let k2 be Element of NAT ; reconsider pmax = u as Element of ( TOP-REAL n ) | K1 ; g . x in dom f & x in dom g implies x in dom g & g . x in dom g assume that 1 <= n and n + 1 <= len f1 and f1 /. n = f1 /. ( n + 1 ) ; reconsider T = b * N as Element of G / ( N * N ) ; len ( PX2 ) <= len ( PX2 ) - 1 ; x " in the carrier of A1 & y " in the carrier of A1 & x " in the carrier of A2 ; [ i , j ] in Indices ( A @ ) & [ i , j ] in Indices ( A @ ) ; for m be Nat holds Re ( F . m ) is simple ; f . x = a . i .= a1 . ( i + 1 ) ; let f be PartFunc of REAL , REAL-NS i , REAL ; rng f = the carrier of \bf SCM ( A ) & rng f c= the carrier of \bf SCM ( A ) ; assume s1 = sqrt 2 / ( 2 * r ) ; attr a > 1 & b > 0 & a > 1 & b > 0 ; let A , B be Subset of on ( I ) ; reconsider X0 = X , Y1 = Y as RealNormSpace ; let f be PartFunc of REAL , REAL , x be Element of REAL ; r * ( v1 |-- I ) . X < r * 1 ; assume that V is Subspace of X and X is Subspace of V ; t , t9 be Relation of T , the carrier of S , X be set ; Q [ e1 , f ] \/ { v } = f . ( v , f . v ) ; g \circlearrowleft ( L~ z ) = z & g /. ( len z ) = z ; |. |[ x , v ]| - [ x , v ] .| = vu1 ; - f . w = - ( L * w ) ; z - y <= x - y iff z <= x + y & z <= y - x & z <= y - x & z <= x - y ; sqrt ( 7 / ( e / ( e / ( e / ( e / ( e / ( e / ( e / ( e / ( e / ( e / ( e / ( e / ( e / ( e / ( e / ( e / ( assume that X is BCK-algebra and 0 in X and 0 in X and 0 in X ; F . 1 = v1 & F . 2 = v2 & F . 3 = v2 ; ( f | X ) . x2 = f . x2 ; ( ( - 1 ) (#) ( tan * tan ) ) `| Z in dom ( sec * tan ) ; i2 = ( f /. len f ) `2 .= ( f /. len f ) `2 ; X1 = X2 \/ X1 & X2 = X1 \/ X2 & X1 \/ X2 = X2 ; [. a , b .] = 1_ G & [. b , c .] = { b } ; let V , W be non empty VectSpStr over F_Complex , f , g be FinSequence of V , h be FinSequence of V , a be Real ; dom g2 = the carrier of I[01] & rng g2 = the carrier of I[01] ; dom f2 = the carrier of I[01] & rng f2 = the carrier of I[01] ; ( proj2 | X ) .: X = proj2 .: X .= proj2 .: X ; f . ( x , y ) = h1 . ( x , y ) ; x0 - r < a1 . n - r / 2 & x0 - r < a1 . n ; |. ( f /* s ) . k - ( f /* s ) . k .| < r ; len Line ( A , i ) = width A & width A = width B ; Sa3 / 2 = ( S * g ) / 2 .= ( S * g ) / 2 ; reconsider f = v + u as Function of X , the carrier of Y ; intloc 0 in dom Initialized ( p ) & Initialized ( p ) in dom Initialized ( p ) ; i1 , i2 , x3 , x4 is_collinear & ( i1 , i2 , x4 is_collinear & i2 , j2 , x4 is_collinear & F is closed & F is closed & F is closed & F is closed & F is closed & G is closed & F is closed & G is closed & F is closed & G is closed arccos r + arccos r = sqrt ( 2 * PI + 0 ) ; for x st x in Z holds f2 is_differentiable_in x & f2 is_differentiable_in x & f1 . x > 0 ; reconsider q2 = sqrt ( q `2 / |. q .| - sn ) as Element of REAL ; ( 0 qua Nat ) + 1 <= i + ( j + 1 ) ; assume f in the carrier of [ X \to \Omega Y , Omega Y ] ; F . a = H / ( ( x , y ) / ( x , z ) ) ; true T -not ( ex u st u in T & not ( ex C st C in T & u in C ) & C in T ) & ( not C in T ) ; dist ( ( a * seq ) . n , h ) < r ; 1 in the carrier of [. 0 , 1 .] & 1 in dom f & f . 0 = f . 1 ; ( p2 `1 ) ^2 - ( p2 `2 ) ^2 > - ( p2 `2 ) ^2 ; |. r1 - r2 .| = |. a1 - a2 .| * |. q1 - q2 .| ; reconsider S-14 = 8 as Element of Seg 8 ; ( A \/ B ) |^ b c= A |^ b \/ B |^ c ; DW .first() = DW .( ) + 1 ; i1 = [: the carrier of K , { n } :] & i2 = [: the carrier of K , { n } :] ; f . a [= f . ( f . a "\/" f . b ) ; attr f = v & g = u u u , v - u ; I . n = Integral ( M , F . n ) ; [: { 1 } , S :] . s = 1 ; a = VERUM ( A ) or a = VERUM ( A ) ; reconsider k2 = s . b2 as Element of NAT ; ( Comput ( P , s , 4 ) ) . GBP = 0 ; L~ M1 meets L~ M2 & L~ M1 misses L~ M2 implies M1 \/ M2 meets L~ M2 set h = the continuous Function of X , R , x be Point of S , y be Point of T ; set A = { L . ( n + 1 ) where L is FinSequence : L . ( n + 1 ) = L . ( n + 1 ) } ; for H st H is negative holds P [ H ] ; set b21 = [: S , { i } :] , b13 = [: S , { i } :] , b13 = [: S , { i } :] ; Hom ( a , b ) c= Hom ( a , b ) ; sqrt ( 1 / n + 1 / ( 2 * n ) ) < sqrt ( 1 / n + 1 ) ; ( l `1 ) `1 = [ [ l , cod l ] , [ l , cod l ] ] ; y +* ( i , y ) in dom g & y . i in dom g ; let p be Element of QC-WFF ( Al ) ; X /\ X1 c= dom ( f1 - f2 ) & X /\ ( X /\ X1 ) c= dom ( f1 - f2 ) ; p2 in rng ( f /^ ( 1 + 1 ) ) ; 1 <= indx ( D2 , D1 , j1 ) ; assume x in K1 /\ ( ( ( - 1 ) / 2 ) \/ ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( ( 1 / 2 ) * - 1 <= ( f2 . O ) `2 & - 1 <= ( f2 . O ) `2 ; let f , g be Function of I[01] , TOP-REAL 2 , R^1 , a , b be Real ; k1 -' k2 = k1 - k2 .= ( k1 - k2 ) + k2 - k2 .= ( k2 - k2 ) + k2 - k2 ; rng ( seq ^\ k ) c= ]. x0 - r , x0 .[ ; g2 in ]. x0 - r , x0 + r .[ & g2 in ]. x0 - r , x0 + r .[ ; sgn ( p `1 , K ) = - - 1 .= - 1 ; consider u being Nat such that b = p |^ y * u ; there exists a normal form of A st a = Sum A & A c= B ; Cl ( Cl ( H ) ) = union ( Cl ( H ) ) .= Cl ( Cl ( H ) ) .= Cl ( Cl ( H ) ) ; len t = len t1 + len t2 .= len t1 + len t2 .= len t2 + len t1 ; vmax = v + w & w + ( v + w ) in A ; v <> DataLoc ( ( t . GBP ) , 3 ) ; g . s = sup ( d " { s } ) ; ( \dot y ) . s = s . ( y , s ) ; { s : s < t } in [: Q , S :] & t in [: Q , S :] & t in Q } = {} ; s ` \ s = s ` \ ( s \ t ) .= s ` \ ( s \ t ) .= ( s \ t ) \ ( s \ t ) ; defpred P [ Nat ] means B + $1 in A & B c= A ; ( 339 + 1 ) ! = 1239 * ( 139 + 1 ) ; U = [: A , B :] & U = [: A , B :] ; reconsider y = y as Element of COMPLEX n -tuples_on COMPLEX ; consider i2 being Integer such that y = p * i2 and i2 in A ; reconsider p = Y | Seg k as FinSequence of NAT ; set f = ( S , U ) \! \mathop , F = ( S , U ) \! \mathop , F = ( S , U ) \! \mathop , G = S , F = S \! \mathop { R } , G = S \! \mathop { R } , F = S \! consider Z be set such that lim s in Z and Z in F ; let f be Function of I[01] , TOP-REAL n , R^1 ; SAT M . [ n + i , 'not' A ] <> 1 ; ex r being Real st x = r & a <= r & r <= b & b <= 1 ; R1 , R2 be FinSequence of REAL n , R2 be FinSequence of REAL n ; reconsider l = 0. ( V ) as Linear_Combination of A ; set r = |. e .| + |. s .| + |. w .| + |. s .| ; consider y being Element of S such that z <= y and y in X ; a 'or' ( b 'or' c ) = 'not' ( ( a 'or' b ) 'or' 'not' c ) ) ; ||. x9 - y9 .|| < r2 / 2 & ||. x9 - y9 .|| < r ; b9 , c9 // b9 , c9 & b9 , c9 // c9 , a9 & c9 , c9 // b9 , c9 & c9 , c9 // c9 , a9 ; 1 <= k2 & k1 + 1 <= k2 & k2 + 1 = k2 + 1 & k2 + 1 = k2 + 1 & k2 + 1 = k2 + 1 & k2 + 1 = k2 + 1 ; sqrt ( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) ^2 >= 0 ; sqrt ( ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 ) < 0 ; E-max C in right_cell ( R , 1 ) & E-max L~ R in LSeg ( R , 1 ) ; consider e being Element of NAT such that a = 2 * e + 1 ; Re ( ( lim F ) | D ) = Re ( ( lim G ) | D ) ; LIN b , a , c or LIN b , c , a & LIN c , a , b ; p `2 , a `2 // a , b or p `2 = c ; g . n = a * Sum ( f | n ) .= f . n * ( g | n ) ; consider f being Subset of X such that e = f and f is 1-element ; F | ( N2 ) = ( CircleMap * ( F | [: N , S :] ) ) * ( F | [: N , S :] ) ; q in LSeg ( q , v ) \/ LSeg ( v , p ) ; Ball ( m , r ) c= Ball ( m , s ) ; the carrier of V = { 0. V } & the carrier of V = { 0. V } ; rng ( ( - 1 ) (#) ( cos * sin ) `| [. - 1 , 1 .] ) = [. - 1 , 1 .] ; assume that Re ( seq ) is summable and Im ( seq ) is summable ; ||. v . n - t . m .|| < e / 2 ; set g = O --> 1 ; reconsider t2 = t2 as 0 -started string of S2 , S2 , S2 be ( the connectives of S2 ) -valued FinSequence ; reconsider x9 = seq as sequence of REAL-NS n , REAL n ; assume that E-max C meets L~ Cage ( C , n ) and not E-max C in L~ Cage ( C , n ) and not E-max C in L~ Cage ( C , n ) ; - ( ( 1 / ( 2 * x ) ) * ( 1 / 2 ) ) < F . n ; set d1 = \bf \bf min ( ( x1 , y1 , y2 ) , d2 ) , d2 = dist ( x2 , y2 , z2 ) ; 2 to_power ( 100 -' 1 ) = 2 to_power ( 2 to_power 1 ) ; dom ( v | ( len ( m + 1 ) ) ) = Seg ( len ( m + 1 ) ) .= Seg ( len ( m + 1 ) ) ; set x1 = - ( ( k + 1 ) / 2 ) , x2 = ( k + 1 ) / 2 , x3 = ( k + 1 ) / 2 , x4 = ( k + 1 ) / 2 , x4 = ( k + 1 ) / 2 , x4 = ( k + 1 ) / assume for n being Element of X holds 0. <= F ( n ) & 0 <= F ( n ) ; assume that 0 <= T-32 . i and T-32 . i <= 1 ; for A being Subset of X holds c . ( c . A ) = c . A the support of ( L2 + L2 ) c= I & the support of ( L2 + L2 ) c= I ; 'not' All ( x , p ) => 'not' All ( x , p ) is valid ; ( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ; reconsider Z = { [ {} , {} ] } as Element of the normal w.r.t. over S ; Z c= dom ( ( - 1 ) (#) ( ( #Z 2 ) * ( f1 + f2 ) ) ) ; |. 0. TOP-REAL 2 - ( q `2 / |. q .| - sn ) ) .| < r / 2 ; ConsecutiveSet2 ( A , succ B ) c= ConsecutiveSet2 ( A , succ ( succ ( A , succ B ) ) ) ; E = dom ( L | E ) & L | E is measurable & L | E is measurable ; C |^ ( A + B ) = C |^ B * C |^ A ; the carrier of W2 c= the carrier of V & the carrier of V c= the carrier of V ; I . IC Comput ( P , s , m ) = P . IC Comput ( P , s , m ) .= s . IC Comput ( P , s , m ) ; attr x > 0 means : Def1 : x / ( x / ( x / ( x / ( x / ( x / ( x / ( x / ( x / ( x / ( x / ( x / ( x / ( x / ( x / ( x / ( x / ( x / ( LSeg ( f ^ g , i ) = LSeg ( f , k ) ; consider p being Point of T such that C = [. p , q .] and p in R ; b , c are_connected & - b , c are_connected & - a , b - c are_connected & - b , c are_connected ; assume f = id the carrier of O & f is Function of ( the carrier of O ) , ( the carrier of O ) ; consider v such that v <> 0. V and f . v = L . v ; let l be Linear_Combination of {} ( ( the carrier of V ) \/ { v } ) ; reconsider g = f " as Function of U1 , U2 , U1 , U2 ; A1 in the points of ( G . k ) & A2 = the point of ( G . k ) ; |. - x .| = - x .= - x .= - x .= - x .= - x .= - x ; set S = 1GateCircStr ( x , y , c ) ; Fib ( n ) * ( 5 * n ) >= 4 * n ; v /. ( k + 1 ) = v . ( k + 1 ) ; 0 mod i = - ( i * ( 0 qua Nat ) ) .= - ( i mod ( i -' 1 ) ) ; Indices M1 = [: Seg n , Seg n :] & M1 = [: Seg n , Seg n :] ; Line ( SIT , j ) = SIT . j .= SIT . j ; h . ( x1 , y1 ) = [ y1 , y2 ] ; |. f .| / ( Re ( f (#) h ) ) (#) ( ( Re ( b (#) h ) (#) ( ( Im ( b (#) h ) (#) ( ( Im ( b (#) h ) (#) ( ( Im ( b (#) h ) (#) ( ( Re ( b (#) h ) (#) ( ( Re ( b assume x = ( a1 ^ <* b1 *> ) ^ ( a2 ^ <* b2 *> ) ^ ( b1 ^ b2 ) ; M1 is closed on IExec ( I , P , s ) , P & I is_halting_on s , P ; DataLoc ( t1 . a , 4 ) = intpos ( 0 + 4 ) ; x + y < - x & |. x - y .| = - x + y ; LIN c , q , b & LIN c , b , q & LIN c , d , q ; f\rangle . ( 1 , t ) = f . ( 0 , t ) .= a ; x + ( y + z ) = x1 + ( y1 + z ) .= x1 + ( y1 + z ) ; fE . a = fE . a .= ( the InternalRel of S ) . a .= ( the InternalRel of S ) . a .= ( the InternalRel of S ) . a ; ( p `1 ) ^2 <= ( E-max C ) ^2 / ( ( E-max C ) `2 ) ^2 ; set R8 = Cage ( C , n ) , R8 = Cage ( C , n ) , R8 = Cage ( C , n ) ; ( p `1 ) ^2 >= ( E-max C ) ^2 / ( ( E-max C ) `2 ) ^2 ; consider p such that p = p1 and s1 < p and p < 1 and p in X ; |. ( f /* s ) . l - ( f /* s ) . l .| < r ; Segm ( M , p , q ) = Segm ( M , p , q ) ; len Line ( N , k + 1 ) = width N & width N = width N ; f1 /* s1 is convergent & f2 /* s1 is convergent & lim ( f1 /* s1 ) = x0 ; f . x1 = x1 & f . x2 = y1 & f . x1 = y2 & f . x2 = y2 ; len f <= len f + 1 & len f + 1 <> 0 & len f + 1 <> 0 ; dom ( Proj ( i , n ) * s ) = REAL m .= REAL m ; n = k * ( 2 * t ) + ( n mod 2 ) ; dom B = 2 -tuples_on the carrier of V \ { {} } .= the carrier of V ; consider r such that r , a _|_ x , y and r , x _|_ y , r ; reconsider B1 = the carrier of X1 , B2 = the carrier of X2 as Subset of X ; 1 in the carrier of [. 1 / 2 , 1 .] & [. 1 / 2 , 1 .] c= dom ( 1 / 2 ) ; for L being complete LATTICE holds lattice (# L , L #) , L #) is isomorphic ; [ gi , gj ] in [: I , I :] \ { i } ; set S2 = 1GateCircStr ( x , y , c ) , S2 = 1GateCircStr ( x , y , c ) ; assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and f1 . x0 = f2 . x0 ; reconsider y = ( a ` ) / ( F . n ) as Element of L ; dom s = { 1 , 2 , 3 } & s . 1 = d1 & s . 2 = d2 & s . 3 = d2 ; ( min ( g , min ( f , g ) ) ) . c <= h . c ; set G1 = the subgraph of G , G2 = the subgraph of G , G2 = the subgraph of G , e = the Vertex of G , e = the Vertex of G , e = the Vertex of G , e = the Vertex of G , e = the Vertex of G , e ; reconsider g = f as PartFunc of REAL , REAL-NS n ; |. s1 . m - 1 .| / ( p / ( p / ( m + 1 ) ) ) .| < d / ( p / ( m + 1 ) ) ; for x being element st x in B holds x in B & x in B & x in B & x in B & x in B P = the carrier of ( ( TOP-REAL n ) | K1 ) .= P ; assume that p2 in LSeg ( p1 , p2 ) /\ LSeg ( p2 , p3 ) and p2 in LSeg ( p3 , p2 ) /\ LSeg ( p2 , p4 ) ; ( 0. X \ x ) to_power ( m * k + 1 ) = 0. X ; let g be Element of Hom ( cod f , \square ) ; 2 * a * b + ( 2 * c ) <= 2 * ( a * b + c ) ; f , g be Point of X , h be Point of X , g be Point of X , h be Point of X , i be Nat ; set h = Hom ( a , g ) , f = Hom ( b , f ) ; then idseq ( n ) | Seg m = idseq ( m ) & m <= n ; H * ( g " * a ) in the right of H & ( g " * a ) * ( g " * a ) in the right of H ; x in dom ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) cell ( G , i1 , j1 -' 1 ) misses C ; LE q2 , q1 , P , p1 , p2 & LE q2 , q1 , P , p2 ; attr B is component means : Def1 : B c= BDD A & B c= BDD B ; deffunc D ( set , set ) = union rng $2 & $2 = union rng $2 & $2 = union rng $2 ; n + - n < len p + - n - n ; attr a <> 0. K means : Def1 : the_rank_of ( M ) = the_rank_of ( a * M ) ; consider j such that j in dom Seg m and I = Seg ( len B ) + j ; consider x1 such that z in x1 and x1 in ( P * R ) and x1 in ( P * R ) and x = ( P * R ) . x1 ; for n being Element of REAL ex r being Element of REAL st X [ n , r ] set C1 = Comput ( P2 , s2 , i + 1 ) , C2 = Comput ( P2 , s2 , i + 1 ) ; set \cal v = 3 / ( 2 |^ ( b , c ) ) , w = 2 / ( 2 |^ ( b , c ) ) , v = - 1 / ( 2 |^ ( b , c ) ) , w = - 1 / ( 2 |^ ( b , c ) ) , w = - 1 / conv ( F .: W ) c= union ( F .: ( E .: W ) ) ; 1 in [. - 1 , 1 .] /\ dom ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( r3 <= s2 + sqrt ( ( 1 - r ) * ( 1 - r ) ) ; dom ( f (#) ( f1 + f2 ) ) = dom f /\ dom ( f1 + f2 ) .= dom ( f1 + f2 ) ; dom ( f (#) G ) = dom ( l (#) F ) /\ Seg k .= Seg ( k + 1 ) ; rng ( s ^\ k ) c= dom f1 \ ( f2 /* s ) & rng ( f1 /* s ) c= dom f2 \ ( f2 /* s ) ; reconsider g9 = gp as Point of ( TOP-REAL n ) | K1 , R^1 ; ( T * h ) . x = T . ( h . ( s . x ) ) ; I . ( J . x ) = ( I * L ) . ( J . x ) ; y in dom ( the ObjectMap of ( ( ( the Sorts of A ) * ( the Arity of S ) ) * ( the Arity of S ) ) ) ; for I being non degenerated commutative Ring , R being commutative non empty doubleLoopStr holds R is commutative iff R is commutative set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) , P1 = P +* Initialize ( ( intloc 0 ) .--> 1 ) ; P1 /. IC s1 = P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 ; lim S1 in the carrier of [. a , b .] & for x st x in the carrier of [. a , b .] holds x in dom ( F * ( x , a ) ) v . i = ( v *' ) . i .= ( v *' ) . i ; consider n being element such that n in NAT and x = seq . n and n in dom ( seq . n ) ; consider x being Element of c such that F1 . x <> F2 . x and F1 . x <> F2 . x ; Segment ( X , 0 , x1 , x2 , x3 , x4 ) = { E , F } ; j + ( 2 * ( k + 1 ) ) > j + ( 2 * ( k + 1 ) ) ; { s , t } on Q & { s , t } on Q & { t , s } on Q & { t , t } on Q & { t , s } on Q & { t , t } on Q & { t , s } on Q & { t , s } on Q ; n1 > len crossover ( p2 , p1 , p2 , n1 , n1 , n2 , n3 , n2 , n3 , n4 , n4 ) ; g1 . HT ( g2 , T ) = 0. L ; then that H1 , H2 are_isomorphic and card H1 = card H2 and card H1 = k and card H1 = k ; ( E-max L~ f ) .. ( f /. ( len f ) ) > 1 ; ]. s , 1 .[ = ]. s , 2 .[ /\ [. s , 1 .] .= ]. s , 1 .] ; x1 in [#] ( ( TOP-REAL 2 ) | K1 ) & x2 in [#] ( ( TOP-REAL 2 ) | K1 ) ; let f1 , f2 be continuous PartFunc of REAL , REAL , c be Element of REAL , d be Element of REAL ; DigA ( t1 , z2 ) is Element of k -tuples_on ( the carrier of k ) -tuples_on ( the carrier of k ) ; I = d1 , I = d2 , k1 = d2 , k2 = d2 , k2 = d2 , k2 = d2 , k2 = d2 , i = d2 , k2 = d2 , i = d2 , k2 = d2 , i = d2 , i = d2 , i = d2 , i = d2 , i = d2 , i = d2 , i = d2 u9 [: { a } , { u } :] = { [ a , u ] } ; ( w | p ) | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w consider u2 such that u2 in W2 and x = v + u2 and x in W and u in W and v in W ; for y st y in rng F ex n st y = a |^ n & F is one-to-one dom ( ( g * ( > 1 ) --> ( V , C ) ) ) = K ; ex x being element st x in ( the Sorts of U0 ) . s & x in ( the Sorts of U0 ) . s ; ex x being element st x in ( [#] O ) \/ A & x in ( the Sorts of U1 ) . s ; f . x in the carrier of [. r , s .] & f . x in [. r , s .] ; ( the carrier of X1 ) \/ ( the carrier of X2 ) /\ ( the carrier of X1 ) <> {} ; L1 /\ LSeg ( p1 , p2 ) c= { p1 } /\ LSeg ( p2 , p2 ) ; sqrt ( b + ( b`1 ) ^2 ) in { r : a < r & r < b } ; ex_sup_of { x , y } , L & x "\/" y = sup { x , y } ; for x being element st x in X ex u being element st P [ x , u ] consider z being Point of G such that z = y and P [ z ] and P [ z ] ; ( the l1 of ( X ) ) . e <= e & ( the Y of ( X ) ) . e <= e ; len ( w ^ w ) + 1 = len w + ( len w + 1 ) ; assume q in the carrier of ( ( TOP-REAL 2 ) | K1 ) & q in the carrier of ( ( TOP-REAL 2 ) | K1 ) ; f | ( ( E-4 ) | D ) = g | ( E-4 ) .= g | ( E-4 ) .= g | ( EK ) ; reconsider i1 = x1 , i2 = x2 as Element of NAT ; ( a * A ) / ( b * B ) = ( a * B ) / ( b * B ) ; assume ex n1 being Element of NAT st f to_power n1 is eventually of n , R & f to_power n1 is eventually 5 ; Seg len ( ( Sum ( f2 ) ) | i ) = dom ( ( Sum ( f2 ) ) | i ) ) .= dom ( ( Sum ( f2 ) ) | i ) ; ( Complement ( A ) ) . m c= ( Complement ( A ) ) . n ; f1 . p = p8 & g1 . ( p , q ) = d & g1 . ( p , q ) = d ; FinS ( F , Y ) = FinS ( F , Y ) ^ ( F | Y ) ; ( x | y ) | z = z | ( y | x ) ; sqrt ( ( |. x .| / 2 ) ^2 ) <= sqrt ( ( |. x .| / 2 ) ^2 ) ; Sum ( F ) = Sum f & dom ( F | n ) = dom g & dom ( F | n ) = dom g & dom ( F | n ) = dom g ; assume for x , y being set st x in Y & y in Y holds x /\ y in Y ; assume that W1 is Subspace of W2 and W2 is Subspace of V and W1 is Subspace of W2 and W2 is Subspace of V ; ||. ( t . x ) - ( t . x ) .|| = lim ||. ( t . x ) - ( t . x ) .|| .= ||. ( ( t . x ) - ( t . x ) ) .|| ; assume that i in dom D and f | A is bounded and g | A is bounded ; sqrt ( ( ( p `2 ) ^2 - ( p `2 ) ^2 ) <= sqrt ( 1 + ( p `2 ) ^2 ) ; g | Ball ( p , r ) = id ( Ball ( p , r ) ) ; set N8 = ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) , 1 = ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ; for T being non empty TopSpace holds T is with_constant implies the TopStruct of T is with_T width B |-> 0. K = Line ( B , i ) .= width B .= width B .= width B .= width B .= width B .= width B .= width B .= width B .= width B .= width B .= width B .= width B .= width B .= width B .= width B .= width B .= width B .= width B .= width B ; attr a <> 0 means : Def2 : ( A *^ B ) = ( A *^ B ) *^ ( B *^ C ) ; then f is_partially differentiable u , 1 & pdiff1 ( f , u ) is_differentiable_in 1 ; assume that a > 0 and a > 1 and b > 0 and c > 0 and d > 0 and d > 0 and c > 0 and d > 0 and d > 0 and d > 0 and d > 0 and d > 0 and d > 0 and d > 0 and d > 0 and d > 0 and d > 0 and d w1 , w2 in Lin { ( { w1 , w2 } ) \/ { w2 , w1 } ; p2 /. IC Comput ( p2 , s2 , i ) = p2 . IC Comput ( p2 , s2 , i ) .= p2 . IC Comput ( p2 , s2 , i ) ; ind ( T-10 | b ) = ind b .= ind B .= ind B .= ind B .= ind B .= ind B ; [ a , A ] in the InternalRel of Line ( X , 1 ) & [ a , A ] in the InternalRel of Line ( X , 1 ) ; m in ( the Arrows of C ) . ( o1 , o2 ) ; ( Y. ( a , CompF ( PA , G ) ) ) . z = TRUE ; reconsider phi = phi , phi = phi , phi = phi , phi = ( l , {} ) , phi = ( l , {} ) , phi = ( S , {} ) , phi = ( S , {} ) , phi = S , phi = S , phi = S , phi = S , phi = S , phi = S , phi = S , phi = len s1 - ( len s2 - 1 ) > 0 + 1 - 1 ; \delta ( D ) * ( f . ( upper_bound A ) - f . ( lower_bound A ) ) ) < r ; [ f21 , f22 ] in the InternalRel of A & [ f21 , f22 ] in the InternalRel of A & [ f21 , f22 ] in the InternalRel of A ; the carrier of ( ( TOP-REAL 2 ) | K1 ) = K1 & the carrier of ( ( TOP-REAL 2 ) | K1 ) = K1 ; consider z being element such that z in dom g2 and p = g2 . z and z in dom g2 and p = g2 . z ; [#] ( V1 ) = { 0. V } .= the carrier of ( V1 ) \/ { 0. V } .= the carrier of ( V1 ) ; consider P2 being FinSequence such that rng P2 = M and P2 is one-to-one and P2 is one-to-one and P2 is one-to-one ; assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and ||. x1 - x0 .|| < s ; h1 = f ^ ( <* p3 *> ^ <* p3 *> ^ <* p3 *> ) .= h ^ ( <* p3 *> ^ <* p3 *> ^ <* p3 *> ) .= h ^ ( <* p3 *> ^ <* p3 *> ) .= h ; c = c .= c .= c .= ( a , c ) `1 .= c `1 .= c `1 .= c `1 ; reconsider t1 = p1 , t2 = p2 as Morphism of C , V , f be Morphism of C , V ; sqrt ( 1 - 2 ) in the carrier of [. 1 / 2 , 1 .] ; ex W being Subset of X st p in W & W is open & h .: W c= V & h .: W c= V ; ( h . p1 ) `2 = C * ( ( p1 `2 ) `2 ) + D * ( p1 `2 ) ) `2 .= D * ( ( p1 `2 ) `2 + D ) `2 ; R . b ` = 2 * r2 .= 2 * r2 .= r2 * r2 .= r2 ; consider L1 such that B = 1- L1 * C + ( 0 - 1 ) * A and 0 <= L1 and 0 <= L1 ; dom g = dom ( ( the Sorts of A ) * the Arity of S ) & dom ( ( the Sorts of A ) * the Arity of S ) = ( the Sorts of A ) * the Arity of S ; [ P . ( l1 + 1 ) , P . ( l1 + 1 ) ] in => ( T . ( l + 1 ) ) ; set s2 = Initialize s , P1 = P +* I ; reconsider M = mid ( z , i2 , i1 ) as Matrix of 2 , REAL ; y in product ( ( Carrier J ) +* ( { 1 } , { 1 } ) ) ; 1 / ( 0 , 1 ) = 1 / ( 0 , 1 / ( 1 + 1 ) ) & 1 / ( 1 + 1 ) = 0 ; assume x in the left options of g or x in the left of g & y in the right of g ; consider M being strict Subspace of A such that a = M and T is Subspace of M and T is Subspace of M ; for x st x in Z holds ( ( ( ( exp_R * f ) `| Z ) ) `| Z ) . x <> 0 ; len ( W1 + W2 ) = 1 + len ( W2 + W1 ) .= len ( W1 + W2 ) + len ( W2 + W2 ) ; reconsider h1 = ( v . n - t ) * t1 as Lipschitzian Function of X , Y ; ( i mod len ( p + q ) ) + 1 in dom ( p + q ) ; assume that s2 is negative and F in the |= of s1 and F is conjunctive and F is conjunctive and F is U ; ( ( gcd ( x , y ) ) , 3 ) = gcd ( x , y ) & ( gcd ( x , z ) , 3 ) = gcd ( x , z ) ; for u being element st u in Bags n holds ( p `2 ) . u = p . u for B being Subset of u st B in E & A = B holds A = B or A misses B or A misses B ; ex a being Point of X st a in A & A /\ Cl ( { y } ) = { a } ; set W2 = [: p , q :] , W1 = [: p , q :] , W2 = [: p , q :] , W2 = [: p , q :] ; x in { X where X is Subset of L : X is Subset of L & Y c= X } ; the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 /\ W2 c= the carrier of W2 & the carrier of W1 /\ W2 c= the carrier of W2 ; in ( 1 / a ) * id ( ( 1 / a ) * id ( b * a ) ) ) * id ( b * a ) = ( 1 / b ) * id ( b * a ) ; ( dom ( X --> f ) ) . x = ( X --> dom f ) . x .= ( X --> dom f ) . x ; set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ; p => ( q => r ) in TAUT ( A ) & ( p => ( p => ( q => r ) ) ) in TAUT ( A ) ; set cos = LSeg ( G * ( i1 , j ) , G * ( i2 , k ) ) ; set cos = LSeg ( G * ( i1 , j ) , G * ( i2 , k ) ) ; - 1 + 1 <= sqrt ( ( E-bound / 2 ) - ( m / 2 ) ) ^2 + 1 ; ( reproj ( 1 , z0 ) ) . x in dom ( f1 (#) f2 ) & ( f1 (#) f2 ) . x in dom ( f1 (#) f2 ) ; assume that b1 . r = { c1 . r } and b2 . r = c1 . ( c2 . r ) and b1 . r = c2 . ( c1 . r ) ; ex P st a1 on P & a2 on P & a3 on P & a4 on P & a4 on P & a4 on P & a4 on P & a4 on P & a4 on P & a4 on P & a4 on P & a4 on P & a4 on P & a4 on P & a4 on P & a4 on P & a4 on P & a4 on P reconsider gf = g * f as strict Subgroup of X , Y ; consider v1 being Element of T such that Q = ( \mathopen { \downarrow v1 ) ` and v1 in A ` and v1 in A ; n in { i where i is Nat : i < n & n < m + 1 } ; ( F /. ( i , j ) ) `2 >= ( F /. ( m , k ) ) `2 ; assume that K1 = { p : ( p `1 / |. p .| - sn ) / ( 1 + sn ) >= sn } and K1 = { p .| * ( |. p .| - sn ) ; ConsecutiveSet2 ( A , succ O1 ) = ( ConsecutiveSet2 ( A , O1 ) ) * ( succ O1 ) ; set I1 = Macro ( a , intloc 0 ) , I2 = P +* ( a , intloc 0 ) , I2 = P +* ( a , intloc 0 ) , I2 = P +* ( a , intloc 0 ) , P3 = P +* ( a , intloc 0 ) , P4 = P +* ( a , intloc 0 ) , P3 = P +* ( a , intloc 0 ) ; for i be Nat st 1 < i & i < len z holds z /. i <> z /. ( i + 1 ) X c= ( the carrier of L1 ) \/ ( the carrier of L2 ) & the carrier of L1 = the carrier of L2 ; consider x9 being Element of GF ( p ) such that x9 |^ 2 = a and x9 in A and x9 in A ; reconsider e1 = e1 , f = f as Element of D ( ) , f = f . ( e , f ) , g = f . ( g . ( g . e ) ) , h = f . ( g . e , f . ( g . e ) ) as Element of D ( ) ; ex O being set st O in S & C1 c= O & M . O = 0. ( X , L ) & M . ( ( X , L ) . O ) = 0. ( X , L ) ; consider n be Nat such that for m be Nat st n <= m holds S . m in U1 and S . m in U1 ; f * reproj ( i , x ) is_differentiable_in x0 & f * reproj ( i , x ) is_differentiable_in x0 ; defpred P [ Nat ] means A + $1 = succ ( A + $1 ) & A = succ ( A + $1 ) ; the left options of g = the left options of g & the right of g = the left ' of g & the right of g = the right ' of g ; reconsider p0 = x , p0 = y as Point of TOP-REAL 2 , p = p , q = q , r = p , s = q , t = p , s = q , t = p , t = q , s = p , t = q , s = p , t = q , s = q , t = p , q = q , s = p , s = q , t = p , consider g2 such that g2 = y and x <= g2 and x <= g2 and g2 <= y and g2 <= x and g2 <= y and g2 <= x and g2 <= y ; for n being Element of NAT ex r being Element of REAL st X [ n , r ] len ( x2 ^ y2 ) = len ( x2 ^ y2 ) + len ( y2 ^ y1 ) .= len ( x2 ^ y2 ) + len ( y2 ^ y1 ) ; for x being element st x in X holds x in the set of ( n -tuples_on the carrier of K ) & x in the carrier of ( n -tuples_on the carrier of K ) & x is set of ( n -tuples_on the carrier of K ) LSeg ( p1 , p2 ) /\ LSeg ( p2 , p3 ) = {} ; func ( ( X ) --> ( Y ) ) -> set equals ( ( the carrier of X ) --> ( id X ) ) +* ( id X , ( id X ) --> ( id X ) ) ; len ( ( C /. 1 ) * ( len ( C /. 1 ) ) ) <= len ( C /. 1 ) * ( len ( C /. 1 ) ) ; attr K is L means : Def1 : a <> 0. K & v . ( a |^ ( i + 1 ) ) = i * v . ( a |^ ( i + 1 ) ) ; consider o being OperSymbol of S such that t . {} = [ o , the carrier of S ] and o . {} = [ o , the carrier of S ] ; for x st x in X ex y st x c= y & y in X & x in X & y in X & x in X IC Comput ( P1 , s1 , k ) in dom ( Comput ( P2 , s2 , k ) ) ; attr q < s & r < s & s < q & q in ]. p , q .[ implies [. p , s .] c= ]. p , q .[ ; consider c being Element of Class ( f , c ) such that Y = ( F . c ) `1 and c in X ; func the ResultSort of S2 -> Function of [: the carrier of S2 , the carrier' of S2 :] , the carrier' of S2 equals id the carrier' of S2 ; set y9 = [ <* y , z *> , f2 ] , z9 = [ <* z , x *> , f2 ] , y9 = [ <* z , x *> , f3 ] ; assume x in dom ( ( ( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( r-7 in Int cell ( GoB f , i , j ) \ { LSeg ( GoB f , i ) : 1 <= i & i + 1 <= len GoB f } ; ( q `2 / |. q .| - sn ) / ( 1 + sn ) >= ( ( q `2 / |. q .| - sn ) ) / ( 1 + sn ) ; set Y = { a "/\" a : a in X } ; i - len f <= len f - len g + len g - 1 ; for n holds x in N & x in N1 & h . n = x- ( x0 - h / 2 ) & h . n = - h / 2 set s2 = ( \mathop { > 0 , I , p , s ) ) . i , s1 = ( ( IExec ( I , p , s ) ) ) . i ; ( p . k ) . 0 = 1 or ( p . k ) . 0 = 1 & ( p . k ) . 0 = 1 & ( p . k ) . 0 = 1 ; u + Sum ( L ) in ( U \ { u } ) \/ { u + Sum ( L ) } ; consider x9 being set such that x in x9 and x9 in V and x9 in V and x9 in V and x9 in V and x9 in V and x9 in V and x9 in V and x9 in V and x9 in V and x9 in V ; ( p ^ q ) . m = ( q | k ) . ( m + 1 ) .= ( q | ( len p + 1 ) ) . ( m + 1 ) ; g + h = gg + h + h1 & h + c = g + h + c ; L1 is distributive & L2 is distributive implies L1 "\/" L2 is distributive & L2 is distributive & L2 is distributive & L1 "\/" L2 is distributive attr x in rng f & y in rng ( f | x ) & x in rng ( f | y ) ; assume that 1 < p and sqrt ( 1 + ( p `2 / p `1 ) ^2 ) = 1 and 0 <= p `2 and p `2 <= 1 and p `2 <= 1 and p `2 <= 1 and p `2 <= 1 and p `2 <= 1 and p `2 <= 1 and p `2 <= 1 ; FM * ( f , sM ) = rpoly ( 1 , H ) * ( f , O ) + ( f , O ) * ( f , O ) ; for X being set , A being Subset of X holds A ` = {} implies A ` = {} & A = {} or A = {} ( E-max X ) `1 <= ( E-max X ) `1 & ( E-max X ) `2 <= ( E-max X ) `2 ; for c being Element of the generators of A , a , b being Element of the Sorts of A holds c <> a & c <> b s1 . intloc 0 = ( Exec ( i2 , s2 ) ) . intpos ( 0 + 1 ) .= Exec ( i2 , s2 ) . intpos ( 0 + 1 ) .= 0 ; for a , b being Real holds [ a , b ] in ( y >= 0 iff a >= 0 & b >= 0 & b >= 0 & a >= 0 & b >= 0 & b >= 0 & a >= 0 & b >= 0 implies a >= 0 for x , y being Element of X holds x ` \ y = ( x \ y ) ` & y \ x = ( x \ y ) ` mode BCK-algebra of i , j , m , n , k be Nat , i , j be Nat st i in dom m & j in dom m holds m * ( i , j ) = m * ( i , j ) set x2 = ( Re ( y - Im ( x , y ) ) ) / ( 2 * PI ) ) ; [ y , x ] in dom u & u . ( y , x ) = g . ( y , x ) ; ]. lower_bound ( divset ( D , k ) , upper_bound ( divset ( D , k ) ) ) , upper_bound ( divset ( D , k ) ) c= A ; 0 <= \delta ( S2 . n ) & |. \delta ( S2 . n ) - 0 .| < e / 2 ; ( - ( q `2 / |. q .| - sn ) ) / ( 1 + sn ) <= ( - ( q `2 / |. q .| - sn ) ) / ( 1 + sn ) ; set A = sqrt 2 / 2 , B = sqrt 2 ; for x , y being set st x in R1 " { x } & y in R2 " { y } holds x , y are_equipotent deffunc F ( Nat ) = b ( ( $1 ) * ( M * ( $1 , n ) ) ) * ( M * ( $1 , n ) ) ) ; for s being element holds s in ( |= f 'or' g ) iff s in ( ( f 'or' g ) ) \/ ( ( f 'or' g ) ) ; for S being non empty non void non empty non void ManySortedSign st S is connected for T being connected SubSpace of S st S is connected holds S is connected max ( ( degree ( K ) ) , degree ( K ) ) >= 0 & degree ( ( K ) ) >= 0 ; consider n1 be Nat such that for k be Nat holds seq . ( n + k ) < r + s ; Lin ( A /\ B ) is Subspace of Lin ( B ) & Lin ( B ) is Subspace of Lin ( B ) ; set n-15 = nnI-15 ( M . x , n ) , I-15 = ( M . x , n ) , InA = ( M . x , n ) , IB = ( M . x , n ) , IA = ( M . x , n ) , IB = ( M . x , n ) , NB = ( M . x , n ) , N f " V in ( [#] X ) & f " V in D & f " V in D & f " V in D & f " V in D & f " V in D & f " V in D & f " V in D & f " V in D ; rng ( ( a / c ) to_power ( 1 / 2 ) ) c= { a , c } ; consider y being subgraph of G1 such that y `2 = y and dom y = WG1 and y `2 = WG2 ; dom ( 1 / 2 ) /\ ]. - 1 , 1 / 2 .[ c= ]. - 1 , 1 / 2 .[ ; O is Matrix of i , j , K & O is Matrix of i , j , K implies ( f /. i ) * ( - ( f /. i ) ) = ( f /. i ) * ( - ( f /. i ) ) v ^ ( n |-> 0 ) in Lin ( ( B . ( ( n |-> 0 ) ) ) , ( B . ( n |-> 0 ) ) ) ; ex a , k1 , k2 st i = a & i = b & ( i = c & i = d ) & i = f . ( i + 1 ) ; t . ( [: { i1 } , { f } :] ) = ( [: { i1 } , { f } :] ) . ( t . ( t . ( t . ( t . ( t . ( t . ( t . ( t . ( t . ( t . ( t . ( t . ( t . ( t . ( t . ( t . ( t . ( t . ( t . ( t . ( assume that F is bbSubset-Family and rng p = F and rng p = Seg ( n + 1 ) and rng p = Seg ( n + 1 ) and rng p = Seg ( n + 1 ) ; not LIN b , a , c & not LIN b , a , c & not b , c // a , c & a , c // c , d ( L1 := L2 ) . O c= ( L1 . O ) := ( L2 . O ) & ( L1 . O ) . O = ( L1 . O ) . O ; consider F be ManySortedSet of E such that for d being Element of E holds F . d = F ( d ) ; consider a , b such that a * ( w - v ) = b * ( w - v ) and 0 < a and a < b and b < 1 ; defpred P [ FinSequence of D ] means |. $1 - 1 .| <= Sum ( |. $1 - 1 .| ) & |. $1 - 1 .| <= ( |. $1 - 1 .| ) * ( |. $1 - 1 .| ) ; u = cos / ( x , y ) * x + cos / ( x , y ) * y .= v ; dist ( ( seq . n ) , x ) + dist ( ( seq . n ) , x ) <= dist ( ( seq . n ) , x ) + ( dist ( ( seq . n ) , x ) ; P [ p , |. p .| : not contradiction } , id ( the carrier of A ) \/ id ( the carrier of A ) ] ; consider X be Subset of [: A , A :] such that X c= Y and X is finite and X is finite and X is finite ; |. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ; 1 < ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ; l in { l1 where l1 is Real : g <= l1 & l <= 1 } ; ( Partial_Sums ( ( G . n ) ) * vol ( A ) ) . m <= ( ( Partial_Sums ( G ) ) . m ) * vol ( A ) ) . m ; f . y = x * 1. L .= x * 1. L .= x * ( 0. L ) .= x * ( 0. L ) .= x * ( 0. L ) .= x * ( x * ( y * ( y * ( y * x ) ) ) .= x * ( y * ( y * x ) ) .= x * ( y * ( y * x ) ) ; NIC ( ( a , i ) , ( b , i ) ) = { ( a , i ) , ( b , i ) } ; LSeg ( p1 , p2 ) /\ LSeg ( p2 , p1 ) = { p1 , p2 } ; Product ( ( the support of I1 ) +* ( { i } , { { i } ) ) ) in Z ; Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) .= Following ( s1 , n ) ; ( W-min ( Q ) ) `1 <= ( q `1 ) `1 & ( E-max ( Q ) ) `2 <= ( E-max ( Q ) ) `2 ; f /. i2 <> f /. ( ( i1 + len g -' 1 ) + 1 ) ; M , v / ( ( ( x. ( x , y ) / ( x. 3 ) ) ) / ( x. 4 , x ) ) / ( x. 4 , x / ( x. 4 , x / ( x. 3 , x / ( x. 4 , x ) ) ) / ( x. 4 , x ) ) / ( x. 4 , x / ( x. 4 , x ) ) / ( x. 4 , x / ( x. 4 , x ) ) ) / ( x. 4 len ( ( P ^ Q ) ^ ( P ^ Q ) ) in dom ( ( P ^ Q ) ^ ( P ^ Q ) ) ; A |^ ( m , n ) c= A |^ ( m , n ) & A |^ ( m , n ) c= A |^ ( m , n ) ; ( for n holds |. q .| \ |. q .| ) . n < a & |. q .| >= a ; consider n1 being element such that n1 in dom p1 and y = p1 . n1 and p1 . n1 = p2 . n1 ; consider X be set such that X in Q and for Z being set st Z in Q & Z <> X holds Z c= Z ; CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA & CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA ; for v be VECTOR of l1 holds ||. v .|| = upper_bound ( rng ( |. v .| ) | A ) & ||. v .|| <= upper_bound ( rng |. v .| ) for phi , phi st phi in X & phi in X holds phi in X & phi in X & phi in X rng ( ( Sgm dom ( f | dom g ) ) | dom ( ( Sgm dom ( f | dom g ) ) ) ) c= dom ( ( Sgm dom ( f | dom g ) ) ; ex c being FinSequence of D st len c = k & P [ c ] & P [ c ] ; the_arity_of ( a , b ) = <* \underbrace ( b , c ) , {} , {} *> .= <* {} , {} , {} , {} *> ; consider f1 be Function of the carrier of X , REAL such that f1 = |. f1 .| and f1 is continuous and rng f1 = { f1 . ( f1 . ( f2 . ( f2 . ( f2 . ( f2 . ( f1 . ( f2 . ( f2 . ( f2 . ( ( f2 . ( f2 . ( f2 . ( f2 . ( ( f2 . ( ( f2 . ( f2 . ( ( f2 . ( f2 . ( f2 . ( f2 . ( f2 a1 = b1 & a2 = b2 & a3 = b3 & a4 = b3 & a4 = 6 & a4 = 6 & a4 = 6 & a4 = 6 & 8 = 6 & 8 = 6 & 8 = 6 & 6 = 6 & 8 = 6 & 8 = 6 & 6 = 8 & 8 = 8 & 6 = 6 & 8 = 8 & 8 = 6 & 8 = 8 & 8 = 8 & 8 = 8 & 8 = 8 & 8 = 8 D2 . indx ( D2 , D1 , n1 ) = D1 . indx ( D2 , D1 , n1 ) .= D1 . indx ( D2 , D1 , n1 ) ; f . ( |. r .| ) = |. |[ r , s ]| .| .= <* r , s ]| .= <* r , s *> .= <* r , s *> .= x ; consider n be Nat such that for m be Nat st n <= m holds C . m = C ( m ) ; consider d be Real such that for a , b be Real st a in X & b in Y holds a <= b & b <= d holds a <= b ; ||. L /. h .|| - ( K * |. h .| ) + K * ( K * |. h .| ) <= p1 + K * ( K * |. h .| ) ; attr F is commutative means : Def1 : for b being Element of X holds F . b = f . b ; p = 2 * ( p2 + p1 ) .= 1 * ( p2 + p3 ) .= 1 * ( p2 + p3 ) .= p2 + p3 + p4 .= p2 + p4 .= p2 + p3 + p4 .= p2 + p3 + p4 .= p2 + p3 + p4 .= p2 + p4 + p4 ; consider z1 such that b , x1 , z1 , z2 is_collinear and ( for x , y st x , y , z is_collinear & [ x , y , z ] in R & [ x , y , z ] in R & [ x , y , z ] in R & [ x , z , z ] in R & [ y , z , z ] in R & [ x , z , y ] in R ) ; consider i such that Arg ( ( Rotate ( s , q ) ) . q ) = s + Arg ( q ) ; consider g such that g is one-to-one and dom g = card ( f . x ) and rng g c= X and rng g c= Y ; assume that A = P2 \/ Q and ( for x st x in Q & x in Q holds x in Q and x in Q and Q . x = Q . x ) and P . x = Q . x and Q . x = Q . x and Q . x = Q . x and Q . x = Q . x and Q . x = Q . x and Q . x = Q . x ; attr F is associative means : Def1 : F .: ( F .: ( f , g ) ) = F .: ( f .: ( g , h ) ) ; ex x being Element of NAT st m = x `1 & x in z & m in { i } & n in { i } & m in { i } & n in { i } & m in { i } ; consider k2 being Nat such that k2 in dom ( P . ( k + 1 ) ) and l in dom ( P . ( k + 1 ) ) and l in dom ( P . ( k + 1 ) ) ; seq = r * seq implies for n holds seq . n = r * seq . ( n + 1 ) F1 . [ [ a , a ] , [ b , a ] ] = [ f * ( a , b ] ) .= [ a , b ] ; { p } "\/" D2 = { p "\/" y where y is Element of L : y in D2 & p in D2 } ; consider z being element such that z in dom ( ( dom F ) * ( F . z ) ) and ( dom F ) . z = y ; for x , y being element st x in dom f & y in dom f & f . x = f . y holds x = y ; cell ( G , i , j ) = { |[ r , s ]| : r <= G * ( 0 , 1 ) `1 } ; consider e being element such that e in dom ( T | ( E1 . e ) ) and ( T | ( E1 . e ) ) . e = v ; ( F `2 ) . x = ( Mx2Tran ( J , b1 , b2 ) ) . x .= ( Mx2Tran ( J , b1 , b2 , b3 ) ) . x ; - 1 _ { \mathbb R } = ( - D ) (#) D .= ( - D ) (#) D .= ( - D ) (#) D .= ( - D ) (#) D .= ( ( - D ) (#) D .= ( ( - D ) (#) D ) (#) D .= ( ( - D ) (#) D ) (#) D .= ( ( - D ) (#) D ) (#) D .= ( ( - D ) (#) D .= ( ( - D ) (#) D ) attr x in dom f /\ dom g & g in dom f /\ dom g & g in dom g & f . x <= g . x ; len ( f1 . j ) = len ( f2 . j ) .= len ( f1 . j ) .= len ( f1 . j ) .= len ( f1 . j ) .= len ( f1 . j ) .= len ( f1 . j ) .= len ( f1 . j ) .= len ( f1 . j ) .= len ( f1 . j ) .= len ( f1 . j ) .= len ( f1 . j ) .= len ( f1 . j ) .= len ( All ( All ( 'not' a , A , G ) , G ) |= All ( All ( 'not' a , B , G ) , G ) ; LSeg ( E . ( k + 1 ) , F . ( k + 1 ) ) c= Cl RightComp Cage ( C , n ) ; x \ ( a |^ m ) = x \ ( a |^ ( k + 1 ) ) .= ( x \ ( a |^ ( k + 1 ) ) ) \ ( x |^ ( k + 1 ) ) .= x \ ( x |^ ( k + 1 ) ) .= x ; k -func func func ( ( the Sorts of U1 ) * the Arity of U2 ) -> strict MSAlgebra over S equals ( ( the Sorts of U1 ) * the Arity of S ) . k .= ( the Arity of S ) . k .= ( the Arity of S ) . k .= ( the Arity of S ) . k .= ( the Arity of S ) . k .= ( the Arity of S ) . k .= ( the Arity of S ) . k .= ( the Arity of S ) . for s being State of A1 holds Following ( s , 0 ) . ( n + 1 ) is stable ; for x st x in Z holds ( f1 . x ) ^2 = a / ( x ^2 - a ^2 ) & ( f1 . x ) ^2 - ( f1 . x ) ^2 <> 0 & ( f1 . x ) ^2 - ( f1 . x ) ^2 > 0 support ( support ( n ) ) \/ support ( ( support ( m ) ) ) c= support ( ( n ) ) \/ support ( ( m ) ) ; reconsider t = u as Function of ( the carrier of A ) , ( the carrier of B ) , the carrier of C ; - ( a * sqrt ( 1 + ( b ^2 ) / 2 ) ) <= - ( ( b ^2 + ( b ^2 ) / 2 ) ) ; phi /. ( succ a ) = g . a & phi . ( a ) = f . ( g . a ) & phi . ( a ) = f . ( g . a ) ; assume that i in dom ( F ^ <* p *> ) and j in dom ( F ^ <* p *> ) and i in dom ( F ^ <* p *> ) and j in dom ( F ^ <* p *> ) and i in dom ( F ^ <* p *> ) and j in dom ( F ^ <* p *> ) ; { x1 , x2 , x3 , x4 , x5 , x5 , x5 , M } = { x1 , x2 , x3 , x4 , x5 , x5 , M , N } \/ { x2 , x3 , x4 , x5 , M } .= { x1 , x2 , x3 , x4 , x4 , x5 , x5 , M } ; the Sorts of U1 /\ ( the Sorts of U2 ) c= the Sorts of U1 & the Sorts of U2 /\ ( the Sorts of U2 ) c= the Sorts of U2 ; ( - ( 2 * a ) / 2 ) * ( ( 2 * a ) / 2 ) + ( - ( 2 * a ) / 2 ) * ( ( 2 * a ) / 2 ) ) > 0 ; consider W such that for z being element holds z in W iff z in N & z in N & P [ z ] ; assume that ( the Arity of S ) . o = <* a , b *> and ( the Arity of S ) . o = r and ( the Arity of S ) . o = s and ( the Arity of S ) . o = s ; Z = dom ( ( exp_R / ( arccot * f1 ) ) ^2 ) /\ dom ( ( exp_R / ( arccot * f1 ) ^2 ) ) ; sum ( f , S ) is convergent & lim ( f , S ) = lim ( f , S ) & lim ( f , S ) = lim ( f , S ) ; ( X \ ( f . x ) ) => ( ( g . x ) => ( ( g . x ) => ( ( g . x ) => ( ( g . x ) => ( ( g . x ) => ( ( g . x ) => ( ( g . x ) => ( ( g . x ) => ( ( g . x ) => ( ( g . x ) => ( ( g . x ) => ( ( g . x ) ) ) ) ) ) ) len ( M2 * M1 ) = n & width ( M2 * M1 ) = n & width ( M2 * M1 ) = n & width ( M2 * M1 ) = n & width ( M2 * M1 ) = n ; attr X1 \/ X2 is open means : Def1 : X1 , X2 are_separated & X1 , X2 are_separated & X1 , X2 are_separated & X2 , X1 are_separated & X1 , X2 are_separated & X2 , X2 are_separated & X1 , X2 are_separated & X2 , X2 are_separated & X1 , X2 are_separated & X2 , X2 are_separated & X1 , X2 are_separated & X2 , X1 are_separated ; for L being lower-bounded antisymmetric antisymmetric RelStr for X being non empty Subset of L for s being non empty Subset of L holds X "\/" { Bottom L } = { Bottom L } reconsider f29 = ( F . b ) * ( F . c ) as Function of [: X , Y :] , Y ; consider w being FinSequence of I such that the InitS of M , the InitS of M , s , t being Element of I such that the InitS of M , s ^ t ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ t ^ w ^ w ^ w ^ w ^ w ^ t ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w g . ( a |^ 0 ) = g . ( 1_ G ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a assume for i being Nat st i in dom f ex z being Element of L st f . i = rpoly ( 1 , z ) . ( i + 1 ) ; ex L being Subset of X st L = L & for K being Subset of X st K in C & K <> {} & K is open & L /\ K <> {} & K is open & L /\ K <> {} & L /\ K <> {} & K <> {} & L /\ K <> {} & L /\ K <> {} & K /\ ( K /\ L ) <> {} implies K /\ ( K /\ L ) <> {} ( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C2 & the carrier' of C2 c= the carrier' of C2 & the carrier' of C2 c= the carrier' of C2 & the carrier' of C2 c= the carrier' of C2 & the carrier' of C2 c= the carrier' of C2 & the carrier' of C2 c= the carrier' of C2 ; reconsider o1 = o `2 as Element of TS ( ( the Sorts of A ) . o , ( the Sorts of A ) . o ) ; 1 * ( x1 + x2 ) + ( 0 * ( x1 + x2 ) ) = x1 + ( 0 * ( x1 + x2 ) ) .= x1 + x2 .= x1 + x2 .= x1 + x2 ; E " . 1 = ( ( E qua Function ) " ) . 1 .= ( E " ) . 1 .= ( E " ) . 1 .= ( E " ) . 1 .= ( E " ) . 1 .= ( E " ) . 1 .= ( E " ) . 1 .= ( E " ) . 1 .= ( E " ) . 1 .= ( E " ) . 1 .= ( E " ) . 1 .= ( E " ) . 1 .= ( E " .= ( E " ) . 1 .= E reconsider u1 = the carrier of U1 /\ ( the carrier of U2 ) , u2 = the carrier of U2 /\ ( the carrier of U2 ) as non empty Subset of ( the carrier of U1 ) ; ( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( ( x "/\" z ) "\/" ( ( x "/\" z ) "\/" ( x "/\" y ) ) ) <= ( x "/\" ( x "/\" ( x "/\" z ) ) "\/" ( ( x "/\" ( x "/\" z ) ) ) ; |. f . ( s1 . ( l1 + 1 ) ) - ( f . ( l1 + 1 ) ) .| < \frac { 1 / 2 * ( M . ( l + 1 ) ) - ( M . ( l + 1 ) ) } ; LSeg ( ( for n holds ( for C being Subset of TOP-REAL 2 ) /. ( i + 1 ) ) , ( ( GoB Cage ( C , n ) ) /. ( i + 1 ) ) ) is vertical ; ( f | Z ) /. x - ( f | Z ) /. x = L /. ( x - x ) + R /. ( x - x ) ; g . c * ( - ( g . c ) * f ) + f . c <= h . c * ( - ( f . c ) * f ) + f . c ; ( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) ; assume that width ( GoB f ) in the carrier of A and width ( GoB f ) = width A and width ( GoB f ) = width A and width ( GoB f ) = width A and width ( GoB f ) = width A and width ( GoB f ) = width A and width ( GoB f ) = width A and width ( GoB f ) = width A and width ( GoB f ) = width A ; len ( - M1 ) = len M1 & width ( - M1 ) = width M1 & width ( - M1 ) = width M1 & width ( - M1 ) = width M1 & width ( - M1 ) = width M1 & width ( - M1 ) = width M1 & width ( - M1 ) = width M1 ; for n , i being Nat st i + 1 < n & i < n holds [ i , i ] in the InternalRel of n & [ i , j ] in the InternalRel of n pdiff1 ( f1 , 2 ) is_partial_differentiable_in z , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z , 2 & pdiff1 ( f1 , 2 ) is_partial_differentiable_in z , 2 & f1 , 2 One c= dom f2 & f2 , 2 \rangle c= dom f1 & f2 , 2 \rangle c= dom f2 implies f1 , 2 are_fiberwise_equipotent z , 2 attr a <> 0 & b <> Arg ( a ) & Arg ( b ) = Arg ( b ) & Arg ( a ) = Arg ( b ) & Arg ( a ) = Arg ( - ( - ( - b ) ) ) ; for c being set st not c in [. a , b .[ holds not c in Intersection ( the topology of f , b ) & not c in Intersection ( the topology of f , c ) assume that V1 is linearly closed and V is linearly closed and V is linearly-independent and V is linearly-independent and V is linearly-independent and V is linearly-independent ; z * ( x1 - x2 ) in M & z * ( x1 - x2 ) in M & z * ( x1 - x2 ) in N & z * ( x1 - x2 ) in N & z * ( x1 - x2 ) in N & z * ( x1 - x2 ) in N & z * ( x1 - x2 ) in N & z * ( x1 - x2 ) in N ; rng ( ( ( P qua Function ) " ) * ( S * ( S * ( T * ( T * S ) ) ) ) ) = Seg ( card ( ( T * ( T * S ) ) ) ) .= Seg ( ( ( T * S ) * ( T * S ) ) ) .= Seg ( ( T * ( T * S ) ) ) ; consider s2 being Real_Sequence such that s2 is convergent and b = lim s2 and for n being Nat holds s2 . n <= lim s2 . n ; h2 " . n = h2 " . n & 0 < h2 . n & 0 < h2 . n & for x st x in dom h2 holds 0 < x & x < 1 / 2 * ( ( h " ) . n ) & 0 < ( h " ) . x ; ( Partial_Sums ( |. r .| ) ) . m = |. ( r ) . m .| .= ( |. r .| ) . m .= ( |. r .| ) . m .= ( |. r .| ) . m .= ( |. r .| ) . m .= ( |. r .| ) . m .= ( |. r .| ) . m ; ( Comput ( P1 , s1 , 1 ) ) . b = 0 .= Comput ( P2 , s2 , 1 ) . b .= Comput ( P2 , s2 , 1 ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b ; - v = ( - 1 ) * v & - 1 * v = - 1 * v & - 1 * v = ( - 1 ) * v & ( - 1 ) * v = ( - 1 ) * v ; upper_bound ( ( k .: D ) .: D ) = upper_bound ( ( k .: D ) .: D ) .= sup ( ( k .: D ) .: D ) .= sup ( ( k .: D ) .: D ) .= ( sup ( ( k .: D ) .: D ) ) .: D .= ( sup D ) .: D .= ( sup D ) .: D .= ( sup D ) .: D .= ( sup D ) .: D .= ( sup D ) .: D .= ( sup D ) .: D .= ( sup D ) .: D A |^ ( k , l ) = ( A |^ ( n , l ) ) * ( A |^ ( k , l ) ) .= A |^ ( ( n , l ) * ( A |^ ( k , l ) ) ; for R being add-associative right_zeroed right_complementable distributive non empty doubleLoopStr , I , J being Subset of R holds I + ( J + K ) = ( I + K ) + ( J + K ) ( f . p ) `1 = sqrt ( 1 + ( p `2 / p `1 ) ^2 ) .= sqrt ( 1 + ( p `2 / p `1 ) ^2 ) .= sqrt ( 1 + ( p `2 / p `2 ) ^2 ) ; for a , b being non zero Nat st a , b are_relative_prime & a , b are_relative_prime holds ( e * a ) * ( e * b ) = e * ( e * ( a * b ) ) + e * ( e * ( a * b ) ) consider A5 being Nat such that r is Element of k and r is Element of k -tuples_on Al and ( A = { [ S , x ] } ) & ( A = { [ S , x ] } ) & ( A = { [ S , x ] } ) & ( A = { [ S , x ] } ) ; for X being non empty addLoopStr , M being Subset of X , x being Point of X st x in M holds x + M in M + M { [ x1 , x2 ] } c= { [ x1 , y1 ] } & { [ x1 , x2 ] } c= { [ x1 , y1 ] } ; h . O = |[ A * ( f . O ) + B , C * ( f . O ) + D ]| .= |[ A * ( f . O ) + B , D * ( f . I ) + D ]| ; ( Gauge ( C , n ) * ( i , j ) ) * ( i , j ) in L~ Cage ( C , n ) /\ L~ Cage ( C , n ) & ( G * ( i , j ) ) `2 <= ( G * ( i , j ) ) `2 ; cluster m , n are_relative_prime for Nat ; ( f * F ) . x1 = f . ( F . x1 ) & ( f * F ) . x2 = f . ( F . x2 ) ; for L being LATTICE , a , b , c being Element of L st a \ b = c & a \ b = c holds a \ b = c \ b consider b being element such that b in dom ( H / ( ( ( x , y ) / ( ( x , z ) / ( ( x , y ) / ( ( x , z ) / ( ( x , y ) / ( x , z ) ) ) ) ) ) and z = H / ( ( x , y ) / ( ( x , z ) / ( x , z ) ) ) ) ; assume that x in dom ( F (#) g ) and y in dom ( F (#) g ) and x in dom ( F (#) g ) and y in dom ( F (#) g ) and z = ( F (#) g ) . x ; assume ex e being element st e Joins W . 1 , W & e in G & e in G & e in G & e in G & e in G & e in G & e in G ; ( L (#) f ) . x = ( L (#) f ) . x .= ( L (#) f ) . x ; j + 1 = j + 1 + 1 .= i + 1 - len ( h | ( len h + 1 ) ) .= i + ( len h + 1 ) - len ( h | ( len h + 1 ) ) .= i + ( len h + 1 - len ( h | ( len h + 1 ) ) ) ; ^ ( S *' f ) = S *' ( S *' f ) .= S *' ( S *' f ) .= S *' ( S *' f ) .= S *' ( S *' f ) .= S *' ( S *' f ) .= S *' ( S *' f ) .= S *' ( S *' f ) ; consider H such that H is one-to-one and rng H = the support of L2 and Sum ( L2 ) = Sum ( L ) and Sum ( L ) = Sum ( L ) & Sum ( ( L ) ) = Sum ( L ) ; attr R is Al1 \ means : Def1 : for p , q st p in R & q in R & p in R holds p in R ; dom ( Product ( X --> f ) ) = meet ( dom ( X --> f ) ) .= meet ( dom ( X --> f ) ) .= meet ( dom ( X --> f ) ) .= meet ( dom ( X --> f ) ) .= meet ( dom ( X --> f ) ) .= meet ( dom ( X --> f ) ) .= meet ( dom ( X --> f ) ) .= dom ( ( X --> f ) ) .= dom ( ( X --> f ) ) .= dom ( ( X --> f upper_bound ( proj2 .: ( ( proj2 .: ( ( C /\ L ) /\ Vertical_Line w ) ) ) ) <= upper_bound ( proj2 .: ( ( C /\ L ) /\ Vertical_Line w ) ) ; for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - 0 .| < r i * ( f - g ) = i * ( f - g ) .= i * ( f - g ) .= i * ( f - g ) .= i * ( f - g ) .= i * ( f - g ) .= i * ( f - g ) .= i * ( f - g ) ; consider f being Function such that dom f = 2 -tuples_on X and for Y being set st Y in 2 -tuples_on X holds f . Y = F ( Y ) and for Y being set st Y in 2 -tuples_on X holds f . Y = F ( Y ) ; consider g1 , g2 being element such that g1 in [#] Y and g2 in C and g2 in C and [ g1 , g2 ] in the InternalRel of Y and [ g1 , g2 ] in the InternalRel of Y and [ g2 , g2 ] in the InternalRel of Y and [ g2 , g1 ] in the InternalRel of Y ; func d \! > n -> Nat means : Def1 : d |^ ( n + 1 ) divides n & d |^ ( n + 1 ) divides n |^ ( n + 1 ) & d |^ ( n + 1 ) divides n |^ ( n + 1 ) ; fy1 . [ 0 , t ] = f . [ 0 , t ] .= ( - P ) . [ 0 , t ] .= a .= a ; t = h . D or t = h . E or t = h . F or t = h . J or t = F . M ; consider m1 be Nat such that for n be Nat st n >= m1 holds dist ( ( ( seq . n ) - ( seq . n ) ) , ( ( seq . n ) - ( seq . n ) ) ) < 1 / ( ( ( seq . n ) - ( seq . m ) ) ) ; sqrt ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 <= sqrt ( ( ( q `2 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ) ; h1 . ( i + 1 ) = h1 . ( i + 1 ) .= h1 . ( i + 1 ) .= h1 . ( i + 1 ) ; consider o being Element of the carrier' of S , x2 being Element of { [ o , x2 ] } such that a = [ o , x2 ] and [ o , y2 ] in the carrier' of S and [ o , x2 ] in the carrier' of S ; for L being RelStr , a , b being Element of L holds a <= b iff a <= b & b <= c & c <= d & d <= e & e <= e & e <= f . ( a , b ) ||. h1 . n - g .|| = ||. h1 . n - g .|| .= ||. ( h1 . n - g ) .|| .= ||. ( ( h1 . n ) - g ) .|| .= ||. ( ( h1 . n ) - g ) .|| .= ||. ( ( h1 . n ) - g ) .|| .= ||. ( ( h1 . n ) - g ) .|| ; ( - ( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( ( - 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( ( ( 1 ) * ( ( ( 1 ) * ( ( 1 + 2 ) * ( ( 1 ) * attr r = F .: ( p , q ) means : Def1 : len r = len ( p ^ q ) ; sqrt ( ( r / 2 ) ^2 + ( r / 2 ) ^2 ) <= sqrt ( ( r / 2 ) ^2 + ( r / 2 ) ^2 ) + ( r / 2 ) ^2 ) ; for i being Nat , M being Matrix of K st i in Seg n & i in Seg n holds Det ( M @ ) = Sum ( M @ ) then a " * ( a * v ) = 1 * v " & a " * ( a * v ) = 1 * v " * v ; p . ( j -' 1 ) * ( q . ( i -' 1 ) ) = Sum ( p ) - ( q . ( i -' 1 ) ) * ( p . ( j -' 1 ) ) ; deffunc F ( Nat ) = L . 1 + ( ( R /* h ) " ) * ( ( R /* h ) " ) . $1 ; assume that the carrier of H = f .: ( the carrier of H1 ) and the carrier of H2 = f .: ( the carrier of H2 ) and the carrier of H1 = f .: ( the carrier of H2 ) and the carrier of H2 = f .: ( the carrier of H1 ) and the carrier of H2 = the carrier of H2 ; Args ( o , Free ( X , Free ( X , Y ) ) ) = ( ( the Sorts of Free ( S , X ) ) * the Arity of S ) * the Arity of S ) . o .= ( the Arity of S ) . o ; H1 = n + 1 / ( ( 2 to_power ( n + 1 ) ) * ( ( 2 to_power ( n + 1 ) ) * ( ( 2 to_power ( n + 1 ) ) * ( ( 2 to_power ( n + 1 ) ) ) ) ) .= n + 1 / ( 2 to_power ( n + 1 ) ) .= n + 1 / ( 2 to_power ( n + 1 ) ) .= n ; ( O ) `1 = 0 & ( O ) `1 = 0 & ( O ) `2 = 1 & ( O ) `2 = 0 & ( O ) `2 = 1 & ( O ) `2 = 0 & O `2 = 1 & O `2 = 1 & O `2 = 1 & O `2 = 0 & O `2 = 1 & O `2 = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 0 & O = 1 & O = 1 & O = 1 & O = 0 F1 .: ( dom ( F1 | ( dom ( F1 | ( dom ( F1 | ( dom ( F1 | ( dom ( F1 | ( dom ( F1 | ( dom ( F1 | ( dom ( F1 | ( dom ( F1 | ( dom ( F1 | ( dom ( F1 | ( dom ( F1 | ( dom ( F1 | ( dom ( F1 | ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( R | ( R | ( R | ( R | ( R | ( R | ( R | ( R | ( R attr b <> 0 & d <> 0 & b <> 0 & d <> 0 & d <> 0 & d <> 0 & d <> 0 & d <> 0 & d <> 0 & e = ( - e ) / ( 2 * b ) ) ; dom ( ( f +* g ) | D ) = dom ( ( f +* g ) | D ) .= ( dom ( f +* g ) | D ) \/ D .= D .= D \/ D .= D .= ( dom ( f +* g ) | D ) .= D .= D \/ ( ( dom ( f +* g ) | D ) ) .= D ; for i being set st i in dom g ex u being Element of L st g /. i = u * v & ex a being Element of B st g /. i = a * v g `2 * P `2 = g `2 * P `2 .= g `2 * P `2 .= g `2 * P `2 .= g `2 * P `2 .= g `2 * P `2 .= g `2 * ( g `2 * P `2 ) `2 .= g `2 * ( g `2 * P `2 ) `2 .= g `2 * ( g `2 * P `2 ) `2 .= g `2 * ( g `2 * P `2 ) `2 .= g `2 * ( g `2 * P `2 ) `2 ; consider i , s1 such that f . i = s1 and ( not i in dom s1 & not i in dom s1 & s1 . i in dom s2 & s1 . i = s2 . i ) & ( not i in dom s1 & s1 . i in dom s2 ) & not i in dom s2 & s1 . i in dom s2 ) & not i in dom s2 & j . i in dom s2 & not i in dom s2 & i in dom s2 & j in dom s2 & j in dom s2 & j in dom s2 & j in dom s2 & j in dom s2 & j in dom s2 h5 | ]. a , b .[ = ( g | ]. a , b .[ ) | ]. a , b .[ .= g | ]. a , b .[ .= g | [. a , b .[ .= g | [. a , b .] .= g | [. a , b .] .= g | [. a , b .] .= g | [. a , b .] .= g | [. a , b .] .= g | [. a , b .] .= g | [. a , b .] .= g | [. a , b .] .= g | [. a , b .] .= g | [. a , b .] .= g | [ s1 , t1 ] in R & [ s2 , t2 ] in R & [ s1 , t2 ] in R & [ s2 , t2 ] in R & [ s1 , t2 ] in R & [ s1 , t2 ] in R & [ s1 , t2 ] in R & [ s2 , t2 ] in R & [ s1 , t2 ] in R & [ s1 , t2 ] in R & [ s2 , t2 ] in R & [ s2 , t2 ] in R & [ s1 , t2 ] in R & [ s1 , t2 ] in R & [ s2 , t2 ] in R & [ s2 , t2 ] in then H is negative means : Def1 : H is negative & H is negative & H is negative & H is negative & H is negative & H is negative ; attr f1 is total means : Def1 : ( f1 - f2 ) /* c is total & ( f1 - f2 ) /* c is convergent & lim ( f1 - f2 ) = f1 . ( ( f1 - f2 ) /* c ) ; z1 in W2 ` & z2 in W2 & z1 in W2 & z2 in W1 & z in W2 & z in W1 & z in W2 & z in W1 & z in W2 & z in W2 & z in W1 & z in W2 & z in V & z in V & z in V & z in V & z in V & z in V & z in V & z in V & z in V & z in V & z in V & z in V & z in V & z in V & z in V & z in V & z in V & z in V & z in V & z p = 1 * p .= a " * a * p .= a " * p .= a " * p .= a " * p .= a * p " * p .= a * p " * p .= a * p " * p .= a * p * p .= a * p * p .= a * ( p * q ) .= a * ( a * p ) .= a * ( a * p ) .= a * ( a * p ) .= a * p * p .= p * ( a * ( a * ( a * ( a * ( a * ( a * ( a * ( a * ( for rK be Real_Sequence , K be add-associative right_zeroed right_complementable distributive non empty doubleLoopStr st for n be Nat st n <= K holds upper_bound ( rng ( K | n ) ) <= K . n holds upper_bound ( rng ( K | n ) ) <= K . n Index ( E-max C , C ) meets L~ Cage ( C , n ) or Index ( E-max C , C ) in L~ Cage ( C , n ) & Index ( E-max C , C ) in L~ Cage ( C , n ) ; ||. f . ( g . ( k + 1 ) ) - g . ( g . ( k + 1 ) ) .|| <= ||. g . ( g . ( k + 1 ) ) - g . ( g . ( k + 1 ) ) .|| * K ; assume h = ( ( B .--> ( C .--> D ) ) +* ( D .--> ( E .--> F ) ) +* ( F .--> ( J .--> F ) ) ) +* ( J .--> ( F .--> ( J .--> D ) ) ) +* ( F .--> ( J .--> E ) ) ) ; |. ( ( lower . n ) || ( A , T ) ) . k - ( ( lower . n ) || ( A , T ) ) . k .| <= e * ( e / ( 2 * n ) ) ; ( ( the Sorts of Free ( S , X ) ) . v ) . e = [ ( the connectives of S ) . v , ( the connectives of S ) . v ] .= [ ( the connectives of S ) . v , ( the connectives of S ) . v ] ; { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x9 , y9 , x9 , y9 } = { x1 , x2 , x3 , x4 , x5 , x9 } .= { x1 , x2 , x3 , x4 , x5 , x5 , x9 , y9 } .= { x1 , x2 , x3 , x4 , x5 , x9 , x9 } ; assume that A = [. 0 , PI / 2 * PI , PI / 2 .] and integral ( f , A ) = 0 ; p `2 is permutation of dom f1 & p `2 = ( Sgm Y ) * ( Sgm X ) ) * ( Sgm X ) = ( Sgm Y ) * ( Sgm X ) ; for x , y st x in A holds |. ( 1 - x ) * ( 1 - x ) .| <= 1 * |. ( 1 - x ) * ( 1 - x ) .| ( p2 `2 ) ^2 = |. q2 .| * sqrt ( 1 + ( p2 `2 ) ^2 ) .= ( p2 `2 ) ^2 * ( ( p2 `2 ) ^2 ) .= ( p2 `2 ) ^2 * ( ( p2 `2 ) ^2 ) .= ( p2 `2 ) ^2 ; for f being PartFunc of the carrier of C , REAL st dom f = the carrier of C & dom f = the carrier of C & rng f c= the carrier of C holds f is compact assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( All ( z , CompF ( B , G ) ) ) . x = TRUE ; consider F3 such that dom ( F . n ) = n1 and for k be Nat st k in dom ( F . n ) holds Q [ k , F . ( n + 1 ) ] ; ex u , u1 st u <> u1 & u in v & u1 , u1 .: ( v , u1 ) .: ( v , u1 ) > u1 & u1 , v1 .: ( v , v1 ) > u1 & u1 , v1 .: ( v , u1 ) > u1 & u1 , v1 > u1 , u1 .: ( v , v1 ) & u1 , v1 // u1 , v1 & u1 , v1 // u1 , u1 , u1 implies ( u1 , v1 // v1 , u1 ) & ( u1 , v1 , u1 , v1 ) , v1 , u1 , u1 , u1 , u1 ) & ( u1 , v1 , u1 , v1 for G being Group , A , B being strict Subgroup of G , N being strict Subgroup of G holds ( N meets A ) iff N = N & ( N meets B ) . ( N . ( N . ( N . B ) ) ) = N . ( N . B ) ) & N . ( N . B ) = N . ( N . B ) ) for s be Real st s in dom F holds F . s = Integral ( M , ( f . 0 ) (#) ( g . 0 ) (#) ( g . 0 ) (#) ( h . 0 ) (#) ( h . 0 ) (#) ( g . 0 ) ) width ( AutMt ( f1 , b1 , b2 ) ) = len ( f1 ^ f2 ) .= len ( f1 ^ f2 ) .= len ( f1 ^ f2 ) .= len ( f1 ^ f2 ) .= len ( f1 ^ f2 ) .= len ( f1 ^ f2 ) .= len ( f1 ^ f2 ) .= len ( f1 ^ f2 ) .= len ( f1 ^ f2 ) .= len ( f1 ^ f2 ) .= len ( f1 ^ f2 ) .= len ( f1 ^ f2 ) .= len ( f1 ^ f2 ) .= len ( f1 ^ f2 ) .= len ( f1 ^ f2 ) .= len ( f1 ^ f2 ) .= len ( f | ]. - PI / 2 , PI / 2 .[ = f & f | [. - PI / 2 , PI / 2 .[ = f | [. - PI / 2 , PI / 2 .[ & f | [. - PI / 2 , PI / 2 .] = f | [. - PI / 2 , PI / 2 .[ & f | [. - PI / 2 , PI / 2 .] = f | [. - PI / 2 , PI / 2 .] ; assume that X is closed and a in X and a in X and y in a & x in X & y in X & x in X & y in X & x in X ; Z = dom ( ( ( ( - 1 / 2 ) (#) ( ( #Z 2 ) * ( f1 + #Z 2 ) ) * ( f1 + #Z 2 ) ) ) ) /\ dom ( ( ( - 1 ) (#) ( f1 + #Z 2 ) ) ) ; func VERUM ( V ) -> Subset of V means : Def1 : for k being Nat holds it . k = { l . k : 1 <= k & k <= len l & l . k in V } ; for L being non empty TopSpace , N being net over L , M being net of L , N being net of L st N is net & M is net holds N is net of N for s being Element of NAT holds ( ( ( for v being Element of NAT ) holds ( ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v + ( v then z /. 1 = E-max ( L~ z ) & ( E-max L~ z ) .. z < ( E-max L~ z ) .. z ; len ( p ^ <* 0 *> ) = len p + len <* 0 *> .= len p + len <* 1 *> .= len p + 1 .= len p + 1 .= len p + 1 .= len p + 1 .= len p + 1 .= len p + 1 .= len p + 1 + 1 .= len p + 1 .= len p + 1 .= len p + 1 .= len p + 1 ; assume that Z c= dom ( - 1 ) and for x st x in Z holds f . x = a - x and f . x > 0 and f . x > 0 ; for R being add-associative right_zeroed right_complementable distributive non empty doubleLoopStr , I being Subset of R , J being Subset of R , I being Subset of R , J being Subset of R , K being Subset of R st K c= I & K c= J & K c= L & L /\ K c= I /\ K holds K is closed consider f being Function of B1 , B2 such that for x being Element of B1 holds f . x = F ( x ) ; dom ( x2 + y2 ) = Seg len x .= dom ( x ^ ( y ^ z ) ) .= Seg len ( x ^ ( y ^ z ) ) .= dom ( x ^ ( y ^ z ) ) .= Seg len ( x ^ ( y ^ z ) ) .= dom ( x ^ ( y ^ z ) ) .= dom ( x ^ ( y ^ z ) ) .= dom ( x ^ ( y ^ z ) ) .= dom ( x ^ ( y ^ z ) .= dom ( x ^ ( y ^ ( y ^ z ) ) .= dom ( x ^ ( y ^ ( y ^ ( y ^ z ) ) .= dom ( x ^ ( y for S being -1 Functor of C , B for c being object of C holds ( id S ) . ( id c ) = id ( ( the carrier of C ) . c ) & ( id ( ( the carrier of C ) . c ) = id ( ( the carrier of C ) . c ) ex a st a = a2 & a in dom f & f . a = f . ( a , a ) & f . ( a , a ) = f . ( a , a ) & f . ( a , b ) = f . ( a , b ) ; a in Free ( H / ( ( ( ( ( ( ( x , y ) / ( ( x , p ) / ( ( x , p ) / ( ( x , y ) / ( ( x , p ) / ( x , y ) ) ) ) ) ) ) ) ) ) ; for C1 , C2 being stable non empty set , f being Function of C1 , C2 st f is stable & g is stable holds f = g & f = g & g is stable & f = g holds f = g ( W-min ( P ) ) `1 = ( W-min ( P ) ) `1 .= ( W-min ( P ) ) `1 .= ( W-min ( P ) ) `1 .= ( E-max ( P ) ) `1 .= ( E-max ( P ) ) `1 .= ( E-max ( P ) ) `1 .= ( E-max ( P ) ) `1 .= ( E-max ( P ) ) `1 .= ( E-max ( P ) ) `1 .= ( E-max ( P ) ) `1 .= ( E-max ( P ) ) `1 ; assume that u = <* x0 , y0 , z0 *> and f is PartFunc of 3 , REAL 1 and u = <* x0 , y0 , z0 *> & u = <* y0 , y0 , z0 *> ; then ( t . {} ) `1 in Vars & ex x being Element of Vars st x = ( t . {} ) `1 & t . {} = x & t . {} = x & t . {} = x & t . {} = x & t . {} = y ; Valid ( p '&' p , J ) . v = Valid ( p , J ) . v .= Valid ( p , J ) . v .= Valid ( p , J ) . v .= Valid ( p , J ) . v .= Valid ( p , J ) . v ; assume for x , y being Element of S st x <= y & x = f . x holds x >= y & y >= z & z >= x & z >= y & z >= y & x >= y ; func Class R -> Subset-Family of R means : Def1 : for A being Subset of R holds A in it iff ex a being Element of R st a in A & A c= A & a in A & A c= B ; defpred P [ Nat ] means ( ( ( ( G ) . $1 ) `1 ) `1 ) `1 c= G ( ( ( G . $1 ) `2 ) `1 ) `1 & ( G . $1 ) `2 c= G ( ( G . $1 ) `2 ) `2 ) `2 ; assume that dim ( W1 ) = 0 and dim ( W2 ) = 0 and dim ( W1 ) = 0 and dim ( W2 ) = 0 and dim ( W1 ) = 0 & dim ( W2 ) = 0 & dim ( W1 ) = 0 & dim ( W2 ) = 0 & dim ( W1 ) = 0 ; ma-\hbox { - } ( m , t ) = ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) d1 = x9 ^ <* d1 *> .= f . ( y9 ^ <* d2 *> ) .= f . ( y9 ^ <* d2 *> ) .= f . ( y9 ^ <* d2 *> ) .= f . ( y9 ^ <* d2 *> ) .= f . ( y9 ^ <* d2 *> ) .= f . ( y9 ^ <* d2 *> ) .= f . ( y9 ^ <* d2 *> ) .= f . ( y9 ^ <* d2 *> ) .= f . ( y9 ^ <* d2 *> .= f . ( y9 ^ <* d2 *> .= f . ( y9 ^ <* d2 *> .= f . ( y9 ^ <* d2 *> ) .= f . ( y9 ^ <* d2 , d2 , d2 , f . ( d2 ^ <* d2 *> ) .= f . ( y9 ^ <* d2 , d2 *> ) .= consider g such that x = g and dom g = dom f and for x being element st x in dom f holds g . x = f . x and g . x = f . x ; x + 0. F_Complex = x + len x .= x + len x .= x + len x .= x + len x .= x + len x .= x + len x .= x + len x .= x + len x .= x + len x .= x + len x + len x .= x + len x + len x .= x + len x + len x .= x + len x + len x .= x + len x ; k11 + 1 in dom ( f | ( ( k -' 1 ) + 1 ) ) ) & ( f | ( ( k -' 1 ) + 1 ) ) . ( ( f | ( k -' 1 ) ) . ( ( f | ( k -' 1 ) + 1 ) ) . ( ( f | ( k -' 1 ) + 1 ) ) . ( ( f | ( k -' 1 ) + 1 ) ) . ( ( f | ( k -' 1 ) ) . ( ( f | ( k -' 1 ) ) . ( ( ( k -' 1 ) + 1 ) ) . ( ( f | ( k -' 1 ) ) . ( ( f | ( k -' 1 ) ) ) . ( ( k -' 1 ) ) ) = ( ( f | ( assume that P1 is an arc and P2 is an arc of TOP-REAL 2 and P is_an_arc_of p1 , p2 and P is_an_arc_of p2 , p1 and P is_an_arc_of p2 , p1 and P is_an_arc_of p1 , p2 and Q is_an_arc_of p2 , p2 and P is_an_arc_of p1 , p2 and Q is_an_arc_of p2 , p1 and Q is_an_arc_of p2 , p2 and P is_an_arc_of p1 , p2 and P is_an_arc_of p2 , p1 and Q is_an_arc_of p2 , p1 , p2 and Q is_an_arc_of p2 , p1 and Q is_an_arc_of p2 , p1 and Q meets p2 and P \/ Q and Q \/ Q and Q \/ Q and Q \/ Q and Q \/ Q and Q \/ Q and Q \/ Q and Q and Q \/ Q and Q \/ Q and Q \/ P \/ Q and Q \/ Q and Q \/ P \/ Q and Q \/ Q and Q \/ reconsider a1 = a , b1 = b , c1 = c , c2 = d , c2 = d , c2 = c , c2 = d , c2 = d , c2 = c , c2 = d , c2 = d , c2 = d , c2 = c , c2 = d , 6 = d , 6 , 7 = d , 6 , 8 , 8 , 8 , 7 = 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 7 8 , 8 = 8 reconsider G1 = G1 . t , G1 = G1 . t as Morphism of ( G1 * F1 ) . t , ( G1 * F2 ) . t as Morphism of ( G1 * F2 ) . t , ( G1 * F2 ) . t ; LSeg ( f , i + 1 ) = LSeg ( f /. ( i + 1 ) , f /. ( i + 1 ) ) .= LSeg ( f /. ( i + 1 ) , f /. ( i + 1 ) ) ; Integral ( M , P . m ) | dom ( P . n ) <= Integral ( M , P . m ) | dom ( P . n ) & Integral ( M , P . m ) = Integral ( M , P . n ) ; assume that dom f1 = dom f2 and for x , y being element st [ x , y ] in dom f1 holds f1 . [ x , y ] = f2 . [ x , y ] and [ y , z ] in dom f2 and [ x , z ] in dom f1 and [ y , z ] in dom f2 and [ x , z ] in dom f2 and [ y , z ] in dom f1 and x , z ] in dom f2 and assume assume f1 is element ; consider v such that v = y and dist ( u , v ) < min ( ( 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 ) for G being Group , H being Subgroup of G , a , b being Element of H st a = b holds a |^ b = b |^ ( a |^ b ) & b |^ ( a |^ b ) = b |^ ( a |^ b ) consider B being Function of [: S , L :] , the carrier of V such that for x being element st x in the carrier of V holds P [ x , B . x ] ; reconsider K1 = { p1 where p1 is Point of TOP-REAL 2 : P [ p1 ] & p1 `2 <= p2 `2 & p2 `2 <= p3 `2 } as Subset of TOP-REAL 2 ; sqrt ( ( W-bound C - W-bound C ) / 2 ) <= sqrt ( ( W-bound C - W-bound C ) / 2 ) ^2 + ( N-bound C ) ^2 ) ; for x be Element of X , n be Nat st x in E holds |. ( Re F ) . n .| <= P . x & |. ( Im F ) . n .| <= P . x & |. ( Im F ) . x .| <= P . x len ( @ @ ( @ p ^ @ q ) ) = len ( @ ( @ p ^ @ q ) ) + len @ ( @ q ^ @ p ) .= len ( @ ( @ q ^ @ @ p ) ) + len @ ( @ q ^ @ @ q ) .= len @ ( @ ( @ p ^ @ q ) ) + len @ ( @ q ^ @ p ) .= len @ ( @ q ^ @ p ) .= len @ ( @ q ^ @ p ) .= len @ ( @ q ) .= len @ ( @ p ^ @ q ) .= len @ ( @ q ) + len @ ( @ p ^ @ q ) ; v / ( ( ( x , m ) / ( ( x , m ) / ( x , m ) ) / ( x , m ) ) ) . ( ( ( x , m ) / ( x , m ) / ( x , m ) ) ) . ( ( x , m ) / ( x , m ) / ( x , m ) ) ) . ( ( x , m ) / ( x , m ) ) . ( ( x , m ) / ( x , m ) ) . ( x , m ) ) = m / ( x , m ) / ( x , m ) ) . ( x , m ) ) . ( x , m ) . ( x , m ) ) consider r being Element of M such that M , v / ( ( ( x , m ) / ( ( x , m ) / ( ( x , m ) / ( ( x , m ) / ( x , m ) ) ) ) / ( ( x , m ) / ( ( x , m ) / ( x , m ) / ( x , m ) ) ) ) / ( x , m ) ) / ( x , m ) ) / ( x , m ) / ( x , m ) ) |= r / ( x , m ) ) iff m / ( x , m ) ) = r / ( x , m ) ; func w \ ( w , ( y , w ) ) -> Element of Union ( G , ( y , w ) ) equals ( ( the Sorts of G ) . ( ( ( the Sorts of G ) . ( ( the Sorts of G ) . ( ( the Sorts of G ) . ( ( the Sorts of G ) . ( ( the Sorts of G ) . ( ( the Sorts of G ) . ( ( the Sorts of G ) . ( y , w ) ) ) ) ) ) . ( ( ( ( ( y , w ) ) ) ) ) . ( ( ( the Sorts of G ) . ( ( ( the Sorts of G ) . ( ( ( the Sorts of G ) . ( s2 . b = ( Exec ( n , s1 ) ) . b .= s2 . b .= n . b .= n . b .= n . b .= n . b .= ( n + 1 ) .= ( n + 1 ) .= ( n + 1 ) .= ( n + 1 ) * ( ( n + 1 ) * ( ( n + 1 ) * ( ( n + 1 ) * ( ( n + 1 ) * ( ( n + 1 ) * ( ( n + 1 ) ) ) .= ( n + 1 ) ) * ( ( n + 1 ) * ( ( n + 1 ) * ( ( n + 1 ) * ( ( ( ( n + 1 ) * ( ( n + for n , k be Nat holds 0 <= ( Partial_Sums ( |. seq .| ) ) . n + ( Partial_Sums ( |. seq .| ) ) . n - ( Partial_Sums ( |. seq .| ) ) . ( n + k ) ; set F = S \! \mathop { 0 } , F = S \! \mathop { 1 } ; ( Partial_Sums ( s ) ) . ( n + 1 ) + Partial_Sums ( s ) . ( n + 1 ) >= ( Partial_Sums ( s ) ) . ( n + 1 ) + ( Partial_Sums ( s ) ) . ( n + 1 ) ; consider L , R such that for x st x in N holds ( f | Z ) . x - f . x0 = L . ( x - x0 ) + R . ( x - x0 ) ; func \HM { a , b \HM { , c } : \HM { a , b \HM { , c } = ( \HM { the } \HM { component ( a , b , c ) ) ` & c in ( \HM { the } \HM { component ( b , c , d ) ) ` & d in ( the distance of \HM { a , b } ) ` } ; a * b / ( c / ( a / b ) + c / ( a / b ) ) + ( a / ( c / ( a / b ) + c / ( c / ( a / b ) + c / ( a / b ) ) ) >= 6 ; v / ( ( x1 , m1 ) / ( x2 , m1 ) ) . ( ( x1 , m1 ) / ( x2 , m1 ) ) . ( ( x2 , m1 ) / ( x2 , m1 ) ) = v / ( x2 , m1 ) / ( x2 , m1 ) . ( ( x1 , m1 ) / ( x2 , m1 ) ) . ( ( x1 , m1 ) / ( x2 , m1 ) ) . ( x2 , m1 ) ) ; mid ( Q ^ <* x *> , M ) = ( mid ( Q ^ <* x *> , M ) ) ^ ( ( Q ^ <* x *> ^ <* x *> ) ) ^ ( ( Q ^ <* x *> ^ <* x *> ) ) .= ( Q ^ <* x *> ) ^ ( ( Q ^ <* x *> ^ ( Q ^ <* x *> ) ) .= ( ( Q ^ <* x *> ) ^ ( ( Q ^ <* x *> ) ) ; Sum ( F |^ ( n1 + 1 ) ) = ( r |^ ( n1 + 1 ) ) * ( F |^ ( n1 + 1 ) ) .= ( ( r |^ ( n1 + 1 ) ) * ( F |^ ( n1 + 1 ) ) .= ( r |^ ( n1 + 1 ) ) * ( F |^ ( n1 + 1 ) ) .= ( r |^ ( n1 + 1 ) ) * ( F |^ ( n1 + 1 ) ) .= ( r |^ ( n1 + 1 ) * ( F |^ ( n1 + 1 ) ) .= ( r |^ ( n1 + 1 ) ) * ( F |^ ( n1 + 1 ) ) * ( F |^ ( n1 + 1 ) ) * ( F |^ ( n1 + 1 ) ( ( GoB f ) * ( len GoB f , 1 ) ) `1 = ( GoB f ) * ( 1 , 1 ) `1 .= ( GoB f ) * ( 1 , 1 ) `1 .= ( GoB f ) * ( 1 , 1 ) `1 .= ( GoB f ) * ( 1 , 1 ) `1 ; defpred X [ Element of NAT ] means ( Partial_Sums ( s ) ) . $1 = ( Partial_Sums ( s ) ) . $1 + ( Partial_Sums ( s ) ) . $1 * ( Partial_Sums ( s ) ) . $1 ; the_arity_of g = ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g ; ( X \times Y ) \/ Z misses X & card ( X \times Y ) c= card ( X \times Y ) & card ( X \times Z ) = card ( X \times Y ) & card ( X \times Y ) = card ( X \times Y ) & card ( X \times Y ) = card ( X \times Y ) & card ( X \times Y ) = card ( X \times Y ) & card ( X \times Y ) = card ( X /\ Y ) ; for a , b being Element of S , s being Element of NAT st s = F ( n ) & a = F ( n ) & b = F ( n ) holds s = G ( n ) \ G ( n ) & a = G ( n ) \ G ( n ) & b = G ( n ) \ G ( n ) \ G ( n ) \ G ( n ) \ G ( n ) \ G ( n ) ) E , f |= All ( x , H ) => ( ( All ( x , H ) '&' ( ( ( ( ( ( x , H ) '&' ( ( x , H ) '&' ( ( x , H ) '&' ( ( ( ( ( ( ( ( ( ( ( ( x , H ) '&' ( ( x , H ) '&' ( ( x , H ) '&' ( ( x , H ) ) ) ) ) / ( ( ( ( ( ( ( ( ( ( ( ( ( ( x , H ) '&' ( ( x , H ) '&' ( ( ( x , H ) '&' ( ( ( ( ( ( ( ( ( ( x , H ) '&' ( x , H ) '&' ( x , H ) '&' ( ( x , ex R2 being 1-sorted st R2 = ( p | ( n + 1 ) ) . i & ( the InternalRel of ( n + 1 ) ) . i = the carrier of ( n + 1 ) & ( the InternalRel of ( n + 1 ) ) . i = the InternalRel of ( n + 1 ) . i & ( the InternalRel of ( n + 1 ) ) . i = the InternalRel of ( n + 1 ) . i ; [. a , b + sqrt ( 1 - ( k + 1 ) ) / ( 1 - ( k + 1 ) ) / ( 1 - ( k + 1 ) ) ) is Element of the partial of S , REAL ; Comput ( P , s , 2 + 1 ) = Exec ( P , Comput ( P , s , 2 ) ) .= Exec ( i , Comput ( P , s , 2 ) ) .= Exec ( i , Comput ( P , s , 2 ) ) .= Exec ( i , Comput ( P , s , 2 ) ) .= Exec ( i , Comput ( P , s , 2 ) ) ; card ( h1 . k ) = power ( F_Complex , z ) * u .= ( - ( - ( z * u ) ) ) * u .= ( - ( - ( z * u ) ) ) * u .= ( - ( - ( z * u ) ) * u .= ( - ( - ( z * u ) ) ) * u .= ( - ( - ( z * u ) ) * u .= ( - ( - ( z * u ) ) * u ) .= ( - ( - ( z * v ) ) * u .= ( - ( - ( z * v ) ) * u .= ( - ( - z ) ) ) * u .= ( - ( - ( z * v ) ) * u .= ( - ( z sqrt ( ( f /. c ) " ) = f /. c * ( g /. c ) .= ( ( f /. c ) " ) * ( g /. c ) .= ( ( f /. c ) " ) * ( g /. c ) .= ( ( f /. c ) " ) * ( g /. c ) .= ( ( f /. c ) " ) * ( g /. c ) .= ( ( f /. c ) " * ( g /. c ) ) ; len ( C - len ( D , E ) ) - len ( ( D , E ) - 1 ) = len ( ( D , E ) - 1 ) - 1 .= len ( D , E ) - 1 .= len ( D , E ) - 1 .= len ( D , E ) - 1 .= len ( D , E ) - 1 .= len ( D , E ) - 1 .= len ( D , E ) - 1 .= len ( D , E ) - 1 .= len ( D , E ) - 1 .= len ( D , E ) - 1 .= len ( D , E ) - 1 .= len ( D , E ) - 1 .= len ( D , E ) - 1 .= len ( D , E ) - 1 .= len ( D , E ) - 1 .= len ( D , E dom ( r (#) f ) = dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) ( f | X ) ) .= dom ( r (#) ( f | X ) ) .= X /\ dom ( ( r (#) f ) | X ) .= X /\ dom ( ( r (#) f ) | X ) .= X /\ dom ( ( r (#) f ) | X ) .= X .= X /\ X .= dom ( ( r (#) ( ( r (#) ( f | X ) ) .= X /\ X ) .= dom ( ( r (#) ( ( r (#) ( r (#) ( f | X ) ) .= X /\ X ) .= X /\ X ) .= X /\ X ) .= X /\ X ) .= X /\ X ) .= X defpred P [ Nat ] means for n being Nat holds 2 * $1 + 2 * $1 = Fib ( n ) * ( 2 * $1 + 1 ) + ( 2 * n + 1 ) * ( 2 * n + 1 ) * ( 2 * n + 1 ) * ( 2 * n + 1 ) ) ; consider f being Function of [: { n } , { k } :] , INT such that f = f and f is onto and f is onto and for n being Nat st n in dom f holds f " { n } = { n } ; consider c9 being Function of S , BOOLEAN such that c9 = IExec ( A , B , D ) and c9 = IExec ( B , C , D ) . ( A \/ B ) and E = chi ( A , B , D ) . ( A \/ B ) ; consider y being Element of [: Y , { x } :] such that a = "\/" ( { F ( x ) where x is Element of Y : P [ x ] } , L ) and for y being Element of Y st y in { F ( x ) } holds Q [ y , x ] ; assume that A c= dom f and f = ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( ( ( GoB f ) * ( i , j ) ) `2 = ( ( GoB f ) * ( i , j ) ) `2 .= ( ( GoB f ) * ( i , j ) ) `2 .= ( ( GoB f ) * ( i , j ) ) `2 .= ( GoB f ) * ( i , j ) `2 .= ( GoB f ) * ( i , j ) `2 .= ( GoB f ) * ( i , j ) `2 ; dom Shift ( q , len ( q + 1 ) ) = { j + 1 where j is Nat : j in dom ( q + 1 ) & j in dom ( q + 1 ) & len ( q + 1 ) = len q + 1 } ; consider G1 , G2 being Morphism of V such that G1 <= G2 and G1 <= G2 and G2 <= G2 and G1 = G2 and G1 = G2 and G2 = G2 and G1 = G2 and G2 = G2 and G2 = G2 and G2 is open and G2 is open and G2 is open & G2 is open & G2 is open & G2 is open & G2 is open & G2 is open & G2 is open & G2 is open & G2 is open & G1 is open & G2 is open & G2 is open & G2 is open & G2 is open & G2 is open & G2 is open & G2 is open & G2 is open & G2 is open & G2 is open & G2 is open & G2 is open & G2 is open & G2 is open & G2 is open & G2 is open & G2 is open & G2 is open func - f -> PartFunc of C , V means : Def1 : dom it = dom f & for c st c in dom it holds it . c = - f /. c & it . c = - ( f /. c ) * ( f /. c ) + ( f /. c ) * ( f /. c ) * ( f /. c ) ) ; consider phi such that phi is increasing and for a st phi . a = a & phi . a = a and for v st v in dom ( L * ( a , v ) ) holds L . v = a . v and L . v = a . v ; consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) ; consider i , n such that n <> 0 and len ( p | n ) = n and for i1 , i2 being Nat st i1 in dom p & i2 in dom p & i1 < i2 & i1 < i2 holds len ( p | n ) = len ( p | n ) & len ( p | n ) = len ( p | n ) ; assume that not 0 in Z and Z c= dom ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( ( 1 / 2 ) (#) ( ( ( 1 / 2 ) (#) ( ( ( ( cell ( G1 , i1 -' 1 , j1 -' 1 ) \ ( ( L~ f ) \ ( L~ f ) \ ( L~ f ) ) c= ( BDD ( L~ f ) \ ( L~ f ) \/ ( L~ f ) ) & ( L~ f ) \ ( L~ f ) \ ( L~ f ) ) c= ( BDD ( L~ f ) \ ( L~ f ) \ ( L~ f ) ) ; ex q1 being open Subset of X st s = q1 & ex F being Subset-Family of X st F c= ( X ) & ( for x being set st x in F ex y being Subset of X st 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 & x in F & y in F & x in F & x in F & y in F & x in F & y in F & x in F & y in F & x in F & x in F & y in F & x in F & y in F & x in F & y in F & y in F & x in F & x in F & x gcd ( A , ( 1 , 1 ) , ( 2 , 1 ) ) = 1 / ( 2 * ( 1 , 2 ) ) & gcd ( A , ( 2 * ( 1 , 2 ) ) , ( 2 * ( 1 , 2 ) ) ) = 1 / ( 2 * ( 1 , 2 ) ) ; R8 = ( the { s2 } ) . ( m2 , 1 ) .= ( the { m2 } ) . ( m2 , 1 ) .= ( the InternalRel of ( s , 1 ) ) . ( m2 , 1 ) .= ( the InternalRel of ( s , 1 ) ) . ( m2 , 1 ) .= ( the InternalRel of ( s , 1 ) ) . ( m2 , 1 ) .= ( the InternalRel of ( s , 1 ) ) . ( m2 , 1 ) .= ( the InternalRel of ( s , 1 ) .= ( the InternalRel of ( s , 1 ) ) . ( m2 , 1 ) .= ( the InternalRel of ( s , 1 ) .= ( the InternalRel of ( s , 1 ) . ( m2 , 1 ) .= ( the InternalRel of ( s , 1 ) .= ( the InternalRel of ( s , 1 ) . ( CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) .= CurInstr ( P3 , Comput ( P3 , Comput P1 /\ P2 = ( { p1 } \/ LSeg ( p2 , p1 ) ) \/ LSeg ( p2 , p1 ) .= { p2 } \/ LSeg ( p1 , p2 ) \/ LSeg ( p2 , p2 ) .= { p2 } \/ LSeg ( p1 , p2 ) \/ LSeg ( p2 , p2 ) ; func -> Element of the bound ( A ) means : Def1 : for a , b st a in dom f & b in dom f & a in dom f & b in dom f & f . a = f . b holds a = b & b = f . ( a , b ) ; for a , b being Element of COMPLEX st |. a .| > 0 & |. b .| > 0 & |. a .| >= 0 holds |. a * ( f . ( len f ) ) .| >= 1 defpred P [ Nat ] means 1 <= $1 & $1 <= len g implies for i , j st [ i , j ] in Indices G & [ i , j ] in Indices G & [ i , j ] in Indices G & [ i , j ] in Indices G & [ i , j ] in Indices G & [ i , j ] in Indices G & [ i , j ] in Indices G & [ i , j ] in Indices G & [ i , j ] in G & [ i , j ] in G & [ i , j ] in G & [ i , j ] in G * ( i , j ] in G * ( i , j ] in G * ( i , j ] in G * ( i , j ] in G * ( i , j ] in G * ( i , j assume that C1 , C2 are_separated and g in dom f and f . ( g . ( h . ( g . ( h . ( g . ( h . ( h . ( g . ( h . ( g . ( h . ( g . ( h . ( h . ( g . ( h . ( g . ( h . ( h . ( g . ( h . ( g . ( h . ( g . ( h . ( h . ( g . ( ( , , ( ( , , ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) & ( ( h . ( ( ( ( ( h . ( ( h . ( ( h . ( ( h . ( ( h . ( ( . ( ( ( h . ( ( ( ( ||. f .|| ) . c = ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c .= ( ||. f .|| ) . c .= ( ||. f .|| ) . c .= ( ||. f .|| ) . c .= ( ||. f .|| ) . c .= ( ||. f .|| ) . c .= ( ||. f .|| ) . c .= ( ||. f .|| ) . c .= ( ||. f .|| ) . c .= ( ||. f .|| ) . c .= ( ||. f .|| ) . c .= ( ||. f .|| ) . c .= ( ||. f .|| ) . c .= ( ||. f .|| ) . c .= ( ||. f .|| ) . c .= ( ||. f .|| ) . c .= ( ||. f .|| ) . c .= ( ||. f .|| .= ( ||. f |. q .| ^2 = ( ( q `1 ) ^2 + ( q `2 ) ^2 ) + ( ( q `2 ) ^2 ) ^2 + ( ( q `2 ) ^2 ) + ( ( q `2 ) ^2 ) ^2 + ( ( q `2 ) ^2 ) ^2 + ( ( q `2 ) ^2 ) ^2 ) ^2 + ( ( q `2 ) ^2 ) ^2 ) ^2 + ( ( q `2 ) ^2 ) ^2 ) + ( ( q `2 ) ^2 ) ^2 ) ^2 ) + ( ( ( q `2 ) ^2 ) + ( ( ( q `2 ) ^2 ) ^2 ) + ( ( ( q `2 ) ^2 ) ^2 ) + ( ( q `2 ) ^2 ) + ( ( ( q `2 ) ^2 ) + ( ( q `2 ) ^2 ) + ( ( q `2 ) ^2 ) + ( ( q `2 ) ^2 for F being Subset-Family of T st F is open & not {} in F & for A , B being Subset of T st A in F & B in F & A c= B & A c= B holds A c= B & B c= B & A c= B & B c= B & A c= B & B c= B & A c= B & B c= B & B c= B & A c= B & B c= B & B c= B assume that len F >= 1 and len F = k + 1 and len F = k and for k st 1 <= k & k < len F holds F . k = g . ( k + 1 ) and F . ( k + 1 ) = g . ( k + 1 ) and F . ( k + 1 ) = g . ( k + 1 ) ; i |^ ( mod n ) = i |^ ( s * ( s |^ n ) ) .= i |^ ( s * ( s |^ n ) ) .= i |^ ( s * ( s |^ n ) ) .= i |^ ( s * ( s |^ n ) ) .= i |^ ( s * ( s |^ n ) ) .= i |^ ( s |^ n ) .= i |^ ( s |^ n ) ; consider q being oriented Chain of G , L such that r = q and q <> {} and q in rng ( p ^ q ) and rng ( p ^ q ) = { v } and rng ( p ^ q ) = { v } and rng ( p ^ q ) = { v } and rng ( p ^ q ) = { v } ; defpred P [ Element of NAT ] means $1 <= len ( ( f , Z ) ^ <* ( g , Z ) ^ <* ( f , Z ) . $1 *> ) . $1 = ( ( f , Z ) ^ <* ( g , Z ) . ( $1 + 1 ) *> ) . $1 ; for A , B being Matrix of n , REAL holds len ( A * B ) = len A & width ( A * B ) = width B & width ( A * B ) = width A & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B consider s being FinSequence of the carrier of R such that Sum s = u and for i being Element of NAT st 1 <= i & i < len s ex a being Element of R st s . i = a * ( i , a ) & a in I & s . i = a * ( i , a ) ; func Re ( x , y ) -> Element of COMPLEX equals ( Re ( x , y ) ) * ( Im ( x , y ) ) + ( Im ( x , y ) ) * ( Im ( x , y ) ) ; consider g1 be FinSequence of ( F . 1 ) such that g1 is continuous and rng g1 = A and g1 is continuous and rng g1 c= A and rng g1 c= A and rng g1 c= A and rng g1 c= A and f . 0 = f . 1 and g . 1 = g . ( ( F . 1 ) * g ) . ( ( F . 1 ) * g ) . ( ( F . 1 ) * g ) ; then that n1 >= len p1 and n2 >= len p2 and n1 <= len p2 and len p2 = len p1 and len p2 = len p1 and len p1 = len p2 and len p2 = len p2 and len p1 = len p2 and len p2 = len p2 and len p1 = len p2 and len p2 = len p2 and len p1 = len p2 and len p2 = len p2 and len p1 = len p2 ; ( q `1 ) * a <= ( q `1 ) * a & ( - q `1 ) * a <= ( - q `1 ) * a & ( - q `1 ) * a <= ( - q `1 ) * a & ( - q `1 ) * a <= ( - q `1 ) * a & ( - q `1 ) * a <= ( - q `1 ) * a & ( - q `1 ) * a <= ( - q `1 ) * a & ( - q `1 ) * a <= ( - q `1 ) * a & ( - ( - q `1 ) * a <= ( - q `1 ) * a & ( - ( - q `1 ) * a & ( - ( - q `1 ) * a <= ( - ( - q `1 ) * a & ( - ( - ( - q `1 ) * a <= ( - ( - ( - q `1 ) * a ) * a <= ( F . ( len ( p . ( len p ) ) ) ) `2 = F . ( ( len p ) + 1 ) .= v . ( len p + 1 ) .= v . ( len p + 1 ) .= v . ( len p + 1 ) .= v . ( len p + 1 ) .= v . ( len p + 1 ) .= v . ( len p + 1 ) .= v . ( len p + 1 ) .= v . ( len v + 1 ) .= v . ( len v + 1 ) .= v . ( len v + 1 ) .= v . ( len v + 1 ) .= v . ( len v + 1 ) .= v . ( len v + 1 ) .= v . ( len v + 1 ) .= v . ( len v + 1 ) .= v . ( len v + 1 ) .= v . ( len v + 1 ) .= v . consider k1 being Nat such that k1 + k = 1 and a = ( <* a *> := k1 ) . ( k + 1 ) and ( a = ( a := k1 ) . ( k + 1 ) ) ; consider B1 being Subset of B1 , B2 being Subset of ( A * ) such that B1 is finite and B1 is finite and ( A is finite & B = { x , y } implies B is finite ) ; v2 . b2 = ( curry ( F2 , g ) ) * ( ( curry ( F2 , g ) ) * ( ( curry ( F2 , g ) ) * ( ( curry ( F1 , g ) ) ) ) .= ( ( curry ( F2 , g ) ) * ( ( curry ( F2 , g ) ) * ( ( curry ( F1 , g ) ) * ( ( curry ( F2 , g ) ) ) ) .= ( ( ( curry ( F1 , g ) ) * ( ( ( ( F2 , g ) * ( ( ( F2 , g ) ) * ( ( ( F2 , g ) * ( ( F2 , g ) ) ) ) ) ) * ( ( ( ( ( F2 , g ) ) ) ) ) ) * ( ( ( ( ( ( F2 , g ) * ( ( ( ( F2 , g ) * ( ( ( F2 , g ) ) ) ) ) . dom IExec ( I , P , Initialize s ) = the carrier of SCMPDS .= dom ( ( Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( ( card I + 2 ) ) .= ( card I + 2 ) .= ( card I + 2 ) + 2 .= card ( ( card I + 2 ) + 2 ) .= ( card I + 2 ) + 2 .= ( card I + 2 ) + 2 .= ( card I + 2 ) + 2 .= ( card I + 2 ) + 2 .= ( card I + 2 ) + 2 ) + 2 .= ( card I + 2 ) + 2 .= ( card I + 2 ) + 2 .= ( card I + 2 ) .= ( card I + 2 ) + 2 .= ( card I + 2 ) + 2 .= ( card I + 2 ) + 2 + 2 + 2 ) + 2 .= ( card ex d1 be Real st d1 > 0 & for h be Real st h > 0 & h < 1 holds |. h .| * ||. ( h + c ) .|| < e * ( ( h + c ) * ( h + c ) ) ) LSeg ( G * ( len G , 1 ) , G * ( len G , 1 ) ) c= Int LSeg ( G * ( len G , 1 ) , G * ( len G , 1 ) ) \/ LSeg ( G * ( len G , 1 ) , G * ( len G , 1 ) ) ; LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( len h + 1 ) , i ) .= LSeg ( h /. ( len h + 1 ) , h /. ( len h + 1 ) ) .= LSeg ( h /. ( len h + 1 ) , h /. ( len h + 1 ) ) .= LSeg ( h /. ( len h + 1 ) , i ) .= LSeg ( h /. ( len h + 1 ) , h /. ( len h + 1 ) , h /. ( len h + 1 ) , h /. ( len h + 1 ) , h /. ( len h + 1 ) ) .= LSeg ( h /. ( len h + 1 ) , h /. ( len h + 1 ) ) .= LSeg ( h /. ( len h + 1 ) .= LSeg ( h + 1 ) .= LSeg ( h /. ( len h + 1 ) .= LSeg ( h + 1 ) .= LSeg ( h /. ( len h + A = { q where q is Point of TOP-REAL 2 : LE q , q , P & LE q , p , P & LE q , p , P } ; ( ( - x ) (#) y ) | ( ( - x ) (#) y ) = - ( ( - x ) (#) y ) .= ( - x ) (#) ( - x ) (#) ( - y ) .= - ( - x ) (#) ( - y ) .= - ( - x ) (#) ( - y ) .= - ( - x ) * ( - y ) .= - ( - x ) * ( - y ) .= - ( - x ) * ( - y ) .= - ( - x ) * ( - x ) .= - ( - x ) * ( - x ) * ( - x ) * ( - x ) * ( - x ) * ( - x ) * ( - x ) .= - x * ( - x ) * ( - x ) * ( - x ) * ( - x ) * ( - x ) * ( - x ) * ( - x ) .= - x ) * ( - x ) * ( - x ) * 0 * sqrt ( 1 + ( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) ^2 ) = sqrt ( ( p `2 / |. p .| - sn ) ) ^2 + ( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) ^2 ) ; sqrt ( ( W7 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 ) A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) = ( ( ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 * ( U1 func Shift ( f , h ) -> PartFunc of REAL , REAL means : Def1 : for x be Element of REAL m st x in dom it holds it . x = - ( f /. x ) + ( - ( - h /. x ) ) & for x be Element of REAL m st x in dom it holds it . x = - ( f /. x ) + ( - ( - h /. x ) ) ; assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices G and [ i , j ] in Indices G and [ i , j ] in Indices G and [ i , j ] in Indices G and [ i , j ] in G and [ i , j ] in G * ( i , j ) and [ i , j ] in G * ( i , j ) and 1 <= i and i = j and 1 <= j and i = j and 1 <= j and i = k and j = k and i = k and k = k and k = k and k = k and k = k and k = k and k = k and k = k and k = k and k = k and k = k and k = k and k = k and k = k and k = k and k = k and k = k and k = k and k = k and k = k and k = k and assume that not y in Free ( H ) and not x in Free ( H ) and not y in Free ( H ) and not x in Free ( H ) and y in Free ( H ) and not y in Free ( H ) and not x in Free ( H ) and y in Free ( H ) and not y in Free ( H ) and x in Free ( H ) and y in Free ( H ) and y in Free ( H ) and x in Free ( H ) and y in Free ( H ) and y in Free ( H ) and y in Free ( H ) and y in Free ( H ) and y in Free ( H ) and y in Free ( H ) and y in Free ( H ) and y in Free ( H ) and y in Free ( H ) and y in Free ( H ) and y in Free ( H ) and y in Free ( H ) and y in Free ( H ) and y in Free ( H ) and y in Free ( H ) and y in Free ( H ) and y in Free defpred P [ Element of NAT ] means $1 |^ ( p - 1 ) < ( p |^ ( $1 - 1 ) ) |^ ( $1 - 1 ) & ( $1 |^ ( $1 - 1 ) ) |^ ( $1 - 1 ) = ( ( p |^ ( $1 - 1 ) ) |^ ( $1 - 1 ) ) |^ ( $1 - 1 ) ) |^ ( $1 - 1 ) ; func natural number equals ( X \ C ) & for A being Subset of X st A in X & A in X holds A c= X \ C & A c= X \ C & A c= X \ C & A c= X \ C & B c= X implies A c= X \ C [#] ( ( dist ( ( dist ( 0 ) ) .: Q ) ) .: Q ) = ( dist ( ( dist ( 0 ) ) .: Q ) .: Q ) .: Q & lower_bound ( ( dist ( 0 , Q ) .: Q ) = lower_bound ( ( dist ( 0 , Q ) .: Q ) ) .: Q ) ; rng ( F | ( [: S , T :] ) ) = {} or rng ( F | [: S , T :] ) = { 1 } or rng ( F | [: S , T :] ) = { 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 .= ( 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 .= ( f " ) . i .= consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 /\ P2 = { p1 } and P1 is closed and P2 is closed and P1 /\ P2 = { p2 } and P1 is closed and P2 is closed and P1 /\ P2 = { p1 } and P1 is closed and P2 is closed and P1 \/ P2 = { p2 } ; f . p2 = |[ ( ( p2 `2 ) ) ^2 / ( ( p2 `2 ) ) ^2 ) .= ( p2 `2 ) ^2 / ( ( p2 `2 ) ^2 ) .= ( p2 `2 ) ^2 / ( ( p2 `2 ) ^2 ) .= ( p2 `2 ) ^2 / ( p2 `2 ) ^2 ; ( \HM { the } \HM { carrier } \HM { of X ) " . x = ( \HM { the } \HM { carrier } \HM { of X ) . x .= ( \HM { the } \HM { carrier } \HM { of X } ) . x .= ( the carrier of X ) . x .= ( the carrier of X ) . x .= ( the carrier of X ) . x .= ( the carrier of X ) . x .= ( the carrier of X .= x .= x ; for T being non empty TopSpace , A being Subset of T , B being Subset of T st A <> {} & B is closed holds A is closed & B is closed & B c= B & A c= B & B c= B & A c= B & B c= B & B c= B & A c= B & B c= B & B c= B & B c= B & B c= B & A c= B for i , j being strict Subgroup of G st i + 1 in dom F for G1 being strict Subgroup of G st G1 = F . i & G1 = F . j & G2 = F . i holds G1 = F . j & G1 is strict Subgroup of G & G2 is strict & G1 is strict & G2 is strict & G1 is strict & G2 is strict & G2 is strict & G2 is strict & G1 is strict & G2 is strict & G1 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is strict & G2 is for x st x in Z holds ( ( ( exp_R * ( exp_R * ( f1 + f2 ) ) ) `| Z ) . x = ( ( exp_R * ( f1 + f2 ) ) `| Z ) . x / ( exp_R . x ) ^2 pred f /* a is convergent means : Def1 : for x st x in dom ( f /* a ) & x in dom ( f /* a ) & x in dom ( f /* a ) & for x st x in dom ( f /* a ) holds ( f /* a ) . x = ( f /* a ) . x ; then that X1 , X2 are_separated and ( X1 , X2 are_separated & ( X1 , X2 are_separated ) misses ( X1 , X2 ) & ( X1 , X2 are_separated ) misses ( X2 , X1 ) & ( X1 , X2 are_separated ) misses ( X2 union X1 ) & ( X1 , X2 are_separated implies X1 , X2 are_separated ) misses ( X2 union X1 ) ; ex N be Neighbourhood of x0 st N c= dom SVF1 ( 1 , f , u ) & ex L be Neighbourhood of x0 st for x st x in N holds SVF1 ( 1 , f , u ) . x - L . x = L . x - R . x sqrt ( ( ( p2 `2 ) ^2 + ( p2 `2 ) ^2 ) + ( p2 `2 ) ^2 ) >= sqrt ( ( p2 `2 ) ^2 + ( p2 `2 ) ^2 ) ; ( ( 1 / t ) * ( ( 1 / t ) |^ n ) ) / ( 1 / t ) ) |^ ( ( 1 / t ) |^ ( ( 1 / t ) |^ n ) ) = ( 1 / t ) |^ ( ( 1 / t ) |^ n ) / ( 1 / t ) |^ ( ( 1 / t ) |^ n ) & ( 1 / t ) |^ ( ( 1 / t ) |^ n ) = 1 / t |^ n ; assume that for x holds f . x = ( - 1 ) * ( sin . x ) and for x st x in dom ( ( - 1 ) (#) ( sin ) ) holds ( ( - 1 ) (#) ( sin ) ) . x = ( - 1 ) * ( cos . x ) - ( cos . x ) * ( cos . x ) ) and ( - 1 ) (#) ( cos ) . x = ( - 1 ) * ( cos . x ) ; consider X1 being Subset of Y , Y1 being Subset of X such that t = X1 and Y1 in Y1 and Y1 = Y1 and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open ; card ( S . n ) = card ( { d , ( a |^ n ) } ) .= ( d |^ ( n + 1 ) ) * ( ( a |^ ( n + 1 ) ) ) .= ( d |^ ( n + 1 ) ) * ( ( a |^ ( n + 1 ) ) ) .= ( d |^ ( n + 1 ) ) * ( ( a |^ ( n + 1 ) ) ) ; sqrt ( ( E-bound D ) - ( W-bound D ) / ( 2 |^ ( m + 1 ) ) / ( 2 |^ ( m + 1 ) ) ) = sqrt ( ( E-bound D ) / ( 2 |^ ( m + 1 ) ) - ( E-bound D ) / ( 2 |^ ( m + 1 ) ) ) / ( 2 |^ ( m + 1 ) ) ) .= sqrt ( ( E-bound D ) / ( 2 |^ ( m + 1 ) ) ) .= sqrt ( ( ( ( ( m + 1 ) ) / ( 2 |^ ( m + 1 ) ) ) ^2 ) .= sqrt ( ( 2 |^ ( m + 1 ) ) ) .= sqrt ( ( 2 |^ ( m + 1 ) ) ^2 ) .= sqrt ( ( ( ( 2 |^ ( m + 1 ) ) ^2 ) .= sqrt ( ( 2 |^ ( 2 |^ ( m + 1 ) ) ^2 ) .= sqrt ( ( 2 |^ ( m + 1 ) ) ^2 ) .= sqrt ( ( ( ( 2 |^ ( m + 1 ) ) ^2 ) .= sqrt ( ( 2 |^ ( m + 1 ) ) ^2 ) .= sqrt ( ( 2 |^ ( m + 1