:: LIMFUNC3 semantic presentation begin Lm1: for g, r, r1 being real number st 0 < g & r <= r1 holds ( r - g < r1 & r < r1 + g ) proof let g, r, r1 be real number ; ::_thesis: ( 0 < g & r <= r1 implies ( r - g < r1 & r < r1 + g ) ) assume that A1: 0 < g and A2: r <= r1 ; ::_thesis: ( r - g < r1 & r < r1 + g ) r - g < r1 - 0 by A1, A2, XREAL_1:15; hence r - g < r1 ; ::_thesis: r < r1 + g r + 0 < r1 + g by A1, A2, XREAL_1:8; hence r < r1 + g ; ::_thesis: verum end; Lm2: for seq being Real_Sequence for f1, f2 being PartFunc of REAL,REAL for X being set st rng seq c= (dom (f1 (#) f2)) \ X holds ( rng seq c= dom (f1 (#) f2) & dom (f1 (#) f2) = (dom f1) /\ (dom f2) & rng seq c= dom f1 & rng seq c= dom f2 & rng seq c= (dom f1) \ X & rng seq c= (dom f2) \ X ) proof let seq be Real_Sequence; ::_thesis: for f1, f2 being PartFunc of REAL,REAL for X being set st rng seq c= (dom (f1 (#) f2)) \ X holds ( rng seq c= dom (f1 (#) f2) & dom (f1 (#) f2) = (dom f1) /\ (dom f2) & rng seq c= dom f1 & rng seq c= dom f2 & rng seq c= (dom f1) \ X & rng seq c= (dom f2) \ X ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: for X being set st rng seq c= (dom (f1 (#) f2)) \ X holds ( rng seq c= dom (f1 (#) f2) & dom (f1 (#) f2) = (dom f1) /\ (dom f2) & rng seq c= dom f1 & rng seq c= dom f2 & rng seq c= (dom f1) \ X & rng seq c= (dom f2) \ X ) let X be set ; ::_thesis: ( rng seq c= (dom (f1 (#) f2)) \ X implies ( rng seq c= dom (f1 (#) f2) & dom (f1 (#) f2) = (dom f1) /\ (dom f2) & rng seq c= dom f1 & rng seq c= dom f2 & rng seq c= (dom f1) \ X & rng seq c= (dom f2) \ X ) ) assume A1: rng seq c= (dom (f1 (#) f2)) \ X ; ::_thesis: ( rng seq c= dom (f1 (#) f2) & dom (f1 (#) f2) = (dom f1) /\ (dom f2) & rng seq c= dom f1 & rng seq c= dom f2 & rng seq c= (dom f1) \ X & rng seq c= (dom f2) \ X ) hence A2: rng seq c= dom (f1 (#) f2) by XBOOLE_1:1; ::_thesis: ( dom (f1 (#) f2) = (dom f1) /\ (dom f2) & rng seq c= dom f1 & rng seq c= dom f2 & rng seq c= (dom f1) \ X & rng seq c= (dom f2) \ X ) thus A3: dom (f1 (#) f2) = (dom f1) /\ (dom f2) by VALUED_1:def_4; ::_thesis: ( rng seq c= dom f1 & rng seq c= dom f2 & rng seq c= (dom f1) \ X & rng seq c= (dom f2) \ X ) then A4: dom (f1 (#) f2) c= dom f2 by XBOOLE_1:17; dom (f1 (#) f2) c= dom f1 by A3, XBOOLE_1:17; hence ( rng seq c= dom f1 & rng seq c= dom f2 ) by A2, A4, XBOOLE_1:1; ::_thesis: ( rng seq c= (dom f1) \ X & rng seq c= (dom f2) \ X ) A5: (dom (f1 (#) f2)) \ X c= (dom f2) \ X by A3, XBOOLE_1:17, XBOOLE_1:33; (dom (f1 (#) f2)) \ X c= (dom f1) \ X by A3, XBOOLE_1:17, XBOOLE_1:33; hence ( rng seq c= (dom f1) \ X & rng seq c= (dom f2) \ X ) by A1, A5, XBOOLE_1:1; ::_thesis: verum end; Lm3: for r being Real for n being Element of NAT holds ( r - (1 / (n + 1)) < r & r < r + (1 / (n + 1)) ) proof let r be Real; ::_thesis: for n being Element of NAT holds ( r - (1 / (n + 1)) < r & r < r + (1 / (n + 1)) ) let n be Element of NAT ; ::_thesis: ( r - (1 / (n + 1)) < r & r < r + (1 / (n + 1)) ) 0 < 1 / (n + 1) by XREAL_1:139; hence ( r - (1 / (n + 1)) < r & r < r + (1 / (n + 1)) ) by Lm1; ::_thesis: verum end; Lm4: for seq being Real_Sequence for f1, f2 being PartFunc of REAL,REAL for X being set st rng seq c= (dom (f1 + f2)) \ X holds ( rng seq c= dom (f1 + f2) & dom (f1 + f2) = (dom f1) /\ (dom f2) & rng seq c= dom f1 & rng seq c= dom f2 & rng seq c= (dom f1) \ X & rng seq c= (dom f2) \ X ) proof let seq be Real_Sequence; ::_thesis: for f1, f2 being PartFunc of REAL,REAL for X being set st rng seq c= (dom (f1 + f2)) \ X holds ( rng seq c= dom (f1 + f2) & dom (f1 + f2) = (dom f1) /\ (dom f2) & rng seq c= dom f1 & rng seq c= dom f2 & rng seq c= (dom f1) \ X & rng seq c= (dom f2) \ X ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: for X being set st rng seq c= (dom (f1 + f2)) \ X holds ( rng seq c= dom (f1 + f2) & dom (f1 + f2) = (dom f1) /\ (dom f2) & rng seq c= dom f1 & rng seq c= dom f2 & rng seq c= (dom f1) \ X & rng seq c= (dom f2) \ X ) let X be set ; ::_thesis: ( rng seq c= (dom (f1 + f2)) \ X implies ( rng seq c= dom (f1 + f2) & dom (f1 + f2) = (dom f1) /\ (dom f2) & rng seq c= dom f1 & rng seq c= dom f2 & rng seq c= (dom f1) \ X & rng seq c= (dom f2) \ X ) ) assume A1: rng seq c= (dom (f1 + f2)) \ X ; ::_thesis: ( rng seq c= dom (f1 + f2) & dom (f1 + f2) = (dom f1) /\ (dom f2) & rng seq c= dom f1 & rng seq c= dom f2 & rng seq c= (dom f1) \ X & rng seq c= (dom f2) \ X ) hence A2: rng seq c= dom (f1 + f2) by XBOOLE_1:1; ::_thesis: ( dom (f1 + f2) = (dom f1) /\ (dom f2) & rng seq c= dom f1 & rng seq c= dom f2 & rng seq c= (dom f1) \ X & rng seq c= (dom f2) \ X ) thus A3: dom (f1 + f2) = (dom f1) /\ (dom f2) by VALUED_1:def_1; ::_thesis: ( rng seq c= dom f1 & rng seq c= dom f2 & rng seq c= (dom f1) \ X & rng seq c= (dom f2) \ X ) then A4: dom (f1 + f2) c= dom f2 by XBOOLE_1:17; dom (f1 + f2) c= dom f1 by A3, XBOOLE_1:17; hence ( rng seq c= dom f1 & rng seq c= dom f2 ) by A2, A4, XBOOLE_1:1; ::_thesis: ( rng seq c= (dom f1) \ X & rng seq c= (dom f2) \ X ) A5: (dom (f1 + f2)) \ X c= (dom f2) \ X by A3, XBOOLE_1:17, XBOOLE_1:33; (dom (f1 + f2)) \ X c= (dom f1) \ X by A3, XBOOLE_1:17, XBOOLE_1:33; hence ( rng seq c= (dom f1) \ X & rng seq c= (dom f2) \ X ) by A1, A5, XBOOLE_1:1; ::_thesis: verum end; theorem Th1: :: LIMFUNC3:1 for x0 being Real for seq being Real_Sequence for f being PartFunc of REAL,REAL st ( rng seq c= (dom f) /\ (left_open_halfline x0) or rng seq c= (dom f) /\ (right_open_halfline x0) ) holds rng seq c= (dom f) \ {x0} proof let x0 be Real; ::_thesis: for seq being Real_Sequence for f being PartFunc of REAL,REAL st ( rng seq c= (dom f) /\ (left_open_halfline x0) or rng seq c= (dom f) /\ (right_open_halfline x0) ) holds rng seq c= (dom f) \ {x0} let seq be Real_Sequence; ::_thesis: for f being PartFunc of REAL,REAL st ( rng seq c= (dom f) /\ (left_open_halfline x0) or rng seq c= (dom f) /\ (right_open_halfline x0) ) holds rng seq c= (dom f) \ {x0} let f be PartFunc of REAL,REAL; ::_thesis: ( ( rng seq c= (dom f) /\ (left_open_halfline x0) or rng seq c= (dom f) /\ (right_open_halfline x0) ) implies rng seq c= (dom f) \ {x0} ) assume A1: ( rng seq c= (dom f) /\ (left_open_halfline x0) or rng seq c= (dom f) /\ (right_open_halfline x0) ) ; ::_thesis: rng seq c= (dom f) \ {x0} let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng seq or x in (dom f) \ {x0} ) assume A2: x in rng seq ; ::_thesis: x in (dom f) \ {x0} then consider n being Element of NAT such that A3: seq . n = x by FUNCT_2:113; now__::_thesis:_x_in_(dom_f)_\_{x0} percases ( rng seq c= (dom f) /\ (left_open_halfline x0) or rng seq c= (dom f) /\ (right_open_halfline x0) ) by A1; supposeA4: rng seq c= (dom f) /\ (left_open_halfline x0) ; ::_thesis: x in (dom f) \ {x0} then seq . n in left_open_halfline x0 by A2, A3, XBOOLE_0:def_4; then seq . n in { g1 where g1 is Real : g1 < x0 } by XXREAL_1:229; then ex g1 being Real st ( g1 = seq . n & g1 < x0 ) ; then A5: not x in {x0} by A3, TARSKI:def_1; seq . n in dom f by A2, A3, A4, XBOOLE_0:def_4; hence x in (dom f) \ {x0} by A3, A5, XBOOLE_0:def_5; ::_thesis: verum end; supposeA6: rng seq c= (dom f) /\ (right_open_halfline x0) ; ::_thesis: x in (dom f) \ {x0} then seq . n in right_open_halfline x0 by A2, A3, XBOOLE_0:def_4; then seq . n in { g1 where g1 is Real : x0 < g1 } by XXREAL_1:230; then ex g1 being Real st ( g1 = seq . n & x0 < g1 ) ; then A7: not x in {x0} by A3, TARSKI:def_1; seq . n in dom f by A2, A3, A6, XBOOLE_0:def_4; hence x in (dom f) \ {x0} by A3, A7, XBOOLE_0:def_5; ::_thesis: verum end; end; end; hence x in (dom f) \ {x0} ; ::_thesis: verum end; theorem Th2: :: LIMFUNC3:2 for x0 being Real for seq being Real_Sequence for f being PartFunc of REAL,REAL st ( for n being Element of NAT holds ( 0 < abs (x0 - (seq . n)) & abs (x0 - (seq . n)) < 1 / (n + 1) & seq . n in dom f ) ) holds ( seq is convergent & lim seq = x0 & rng seq c= dom f & rng seq c= (dom f) \ {x0} ) proof let x0 be Real; ::_thesis: for seq being Real_Sequence for f being PartFunc of REAL,REAL st ( for n being Element of NAT holds ( 0 < abs (x0 - (seq . n)) & abs (x0 - (seq . n)) < 1 / (n + 1) & seq . n in dom f ) ) holds ( seq is convergent & lim seq = x0 & rng seq c= dom f & rng seq c= (dom f) \ {x0} ) let seq be Real_Sequence; ::_thesis: for f being PartFunc of REAL,REAL st ( for n being Element of NAT holds ( 0 < abs (x0 - (seq . n)) & abs (x0 - (seq . n)) < 1 / (n + 1) & seq . n in dom f ) ) holds ( seq is convergent & lim seq = x0 & rng seq c= dom f & rng seq c= (dom f) \ {x0} ) let f be PartFunc of REAL,REAL; ::_thesis: ( ( for n being Element of NAT holds ( 0 < abs (x0 - (seq . n)) & abs (x0 - (seq . n)) < 1 / (n + 1) & seq . n in dom f ) ) implies ( seq is convergent & lim seq = x0 & rng seq c= dom f & rng seq c= (dom f) \ {x0} ) ) assume A1: for n being Element of NAT holds ( 0 < abs (x0 - (seq . n)) & abs (x0 - (seq . n)) < 1 / (n + 1) & seq . n in dom f ) ; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= dom f & rng seq c= (dom f) \ {x0} ) A2: now__::_thesis:_for_r_being_real_number_st_0_<_r_holds_ ex_n_being_Element_of_NAT_st_ for_k_being_Element_of_NAT_st_n_<=_k_holds_ abs_((seq_._k)_-_x0)_<_r let r be real number ; ::_thesis: ( 0 < r implies ex n being Element of NAT st for k being Element of NAT st n <= k holds abs ((seq . k) - x0) < r ) assume A3: 0 < r ; ::_thesis: ex n being Element of NAT st for k being Element of NAT st n <= k holds abs ((seq . k) - x0) < r consider n being Element of NAT such that A4: r " < n by SEQ_4:3; take n = n; ::_thesis: for k being Element of NAT st n <= k holds abs ((seq . k) - x0) < r let k be Element of NAT ; ::_thesis: ( n <= k implies abs ((seq . k) - x0) < r ) assume n <= k ; ::_thesis: abs ((seq . k) - x0) < r then n + 1 <= k + 1 by XREAL_1:6; then A5: 1 / (k + 1) <= 1 / (n + 1) by XREAL_1:118; n <= n + 1 by NAT_1:12; then r " < n + 1 by A4, XXREAL_0:2; then 1 / (n + 1) < 1 / (r ") by A3, XREAL_1:76; then 1 / (k + 1) < 1 / (r ") by A5, XXREAL_0:2; then A6: 1 / (k + 1) < r by XCMPLX_1:216; abs (x0 - (seq . k)) < 1 / (k + 1) by A1; then abs (- ((seq . k) - x0)) < r by A6, XXREAL_0:2; hence abs ((seq . k) - x0) < r by COMPLEX1:52; ::_thesis: verum end; hence seq is convergent by SEQ_2:def_6; ::_thesis: ( lim seq = x0 & rng seq c= dom f & rng seq c= (dom f) \ {x0} ) hence lim seq = x0 by A2, SEQ_2:def_7; ::_thesis: ( rng seq c= dom f & rng seq c= (dom f) \ {x0} ) thus A7: rng seq c= dom f ::_thesis: rng seq c= (dom f) \ {x0} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng seq or x in dom f ) assume x in rng seq ; ::_thesis: x in dom f then ex n being Element of NAT st seq . n = x by FUNCT_2:113; hence x in dom f by A1; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng seq or x in (dom f) \ {x0} ) assume A8: x in rng seq ; ::_thesis: x in (dom f) \ {x0} then consider n being Element of NAT such that A9: seq . n = x by FUNCT_2:113; 0 <> abs (x0 - (seq . n)) by A1; then (x0 - (seq . n)) + (seq . n) <> 0 + (seq . n) by ABSVALUE:2; then not x in {x0} by A9, TARSKI:def_1; hence x in (dom f) \ {x0} by A7, A8, XBOOLE_0:def_5; ::_thesis: verum end; theorem Th3: :: LIMFUNC3:3 for x0 being Real for seq being Real_Sequence for f being PartFunc of REAL,REAL st seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} holds for r being Real st 0 < r holds ex n being Element of NAT st for k being Element of NAT st n <= k holds ( 0 < abs (x0 - (seq . k)) & abs (x0 - (seq . k)) < r & seq . k in dom f ) proof let x0 be Real; ::_thesis: for seq being Real_Sequence for f being PartFunc of REAL,REAL st seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} holds for r being Real st 0 < r holds ex n being Element of NAT st for k being Element of NAT st n <= k holds ( 0 < abs (x0 - (seq . k)) & abs (x0 - (seq . k)) < r & seq . k in dom f ) let seq be Real_Sequence; ::_thesis: for f being PartFunc of REAL,REAL st seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} holds for r being Real st 0 < r holds ex n being Element of NAT st for k being Element of NAT st n <= k holds ( 0 < abs (x0 - (seq . k)) & abs (x0 - (seq . k)) < r & seq . k in dom f ) let f be PartFunc of REAL,REAL; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} implies for r being Real st 0 < r holds ex n being Element of NAT st for k being Element of NAT st n <= k holds ( 0 < abs (x0 - (seq . k)) & abs (x0 - (seq . k)) < r & seq . k in dom f ) ) assume that A1: seq is convergent and A2: lim seq = x0 and A3: rng seq c= (dom f) \ {x0} ; ::_thesis: for r being Real st 0 < r holds ex n being Element of NAT st for k being Element of NAT st n <= k holds ( 0 < abs (x0 - (seq . k)) & abs (x0 - (seq . k)) < r & seq . k in dom f ) let r be Real; ::_thesis: ( 0 < r implies ex n being Element of NAT st for k being Element of NAT st n <= k holds ( 0 < abs (x0 - (seq . k)) & abs (x0 - (seq . k)) < r & seq . k in dom f ) ) assume 0 < r ; ::_thesis: ex n being Element of NAT st for k being Element of NAT st n <= k holds ( 0 < abs (x0 - (seq . k)) & abs (x0 - (seq . k)) < r & seq . k in dom f ) then consider n being Element of NAT such that A4: for k being Element of NAT st n <= k holds abs ((seq . k) - x0) < r by A1, A2, SEQ_2:def_7; take n ; ::_thesis: for k being Element of NAT st n <= k holds ( 0 < abs (x0 - (seq . k)) & abs (x0 - (seq . k)) < r & seq . k in dom f ) let k be Element of NAT ; ::_thesis: ( n <= k implies ( 0 < abs (x0 - (seq . k)) & abs (x0 - (seq . k)) < r & seq . k in dom f ) ) assume n <= k ; ::_thesis: ( 0 < abs (x0 - (seq . k)) & abs (x0 - (seq . k)) < r & seq . k in dom f ) then abs ((seq . k) - x0) < r by A4; then A5: abs (- (x0 - (seq . k))) < r ; now__::_thesis:_for_n_being_Element_of_NAT_holds_(seq_._n)_-_x0_<>_0 let n be Element of NAT ; ::_thesis: (seq . n) - x0 <> 0 seq . n in rng seq by VALUED_0:28; then not seq . n in {x0} by A3, XBOOLE_0:def_5; hence (seq . n) - x0 <> 0 by TARSKI:def_1; ::_thesis: verum end; then (seq . k) - x0 <> 0 ; then 0 < abs (- (x0 - (seq . k))) by COMPLEX1:47; hence 0 < abs (x0 - (seq . k)) by COMPLEX1:52; ::_thesis: ( abs (x0 - (seq . k)) < r & seq . k in dom f ) thus abs (x0 - (seq . k)) < r by A5, COMPLEX1:52; ::_thesis: seq . k in dom f seq . k in rng seq by VALUED_0:28; hence seq . k in dom f by A3, XBOOLE_0:def_5; ::_thesis: verum end; theorem Th4: :: LIMFUNC3:4 for r, x0 being Real st 0 < r holds ].(x0 - r),(x0 + r).[ \ {x0} = ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ proof let r, x0 be Real; ::_thesis: ( 0 < r implies ].(x0 - r),(x0 + r).[ \ {x0} = ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ ) assume A1: 0 < r ; ::_thesis: ].(x0 - r),(x0 + r).[ \ {x0} = ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ thus ].(x0 - r),(x0 + r).[ \ {x0} c= ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ :: according to XBOOLE_0:def_10 ::_thesis: ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ c= ].(x0 - r),(x0 + r).[ \ {x0} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ].(x0 - r),(x0 + r).[ \ {x0} or x in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ ) assume A2: x in ].(x0 - r),(x0 + r).[ \ {x0} ; ::_thesis: x in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ then consider r1 being Real such that A3: r1 = x ; x in ].(x0 - r),(x0 + r).[ by A2, XBOOLE_0:def_5; then x in { g2 where g2 is Real : ( x0 - r < g2 & g2 < x0 + r ) } by RCOMP_1:def_2; then A4: ex g2 being Real st ( g2 = x & x0 - r < g2 & g2 < x0 + r ) ; not x in {x0} by A2, XBOOLE_0:def_5; then A5: r1 <> x0 by A3, TARSKI:def_1; now__::_thesis:_x_in_].(x0_-_r),x0.[_\/_].x0,(x0_+_r).[ percases ( r1 < x0 or x0 < r1 ) by A5, XXREAL_0:1; suppose r1 < x0 ; ::_thesis: x in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ then r1 in { g1 where g1 is Real : ( x0 - r < g1 & g1 < x0 ) } by A3, A4; then x in ].(x0 - r),x0.[ by A3, RCOMP_1:def_2; hence x in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ by XBOOLE_0:def_3; ::_thesis: verum end; suppose x0 < r1 ; ::_thesis: x in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ then r1 in { g1 where g1 is Real : ( x0 < g1 & g1 < x0 + r ) } by A3, A4; then x in ].x0,(x0 + r).[ by A3, RCOMP_1:def_2; hence x in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ by XBOOLE_0:def_3; ::_thesis: verum end; end; end; hence x in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ ; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ or x in ].(x0 - r),(x0 + r).[ \ {x0} ) assume A6: x in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ ; ::_thesis: x in ].(x0 - r),(x0 + r).[ \ {x0} now__::_thesis:_x_in_].(x0_-_r),(x0_+_r).[_\_{x0} percases ( x in ].(x0 - r),x0.[ or x in ].x0,(x0 + r).[ ) by A6, XBOOLE_0:def_3; suppose x in ].(x0 - r),x0.[ ; ::_thesis: x in ].(x0 - r),(x0 + r).[ \ {x0} then x in { g1 where g1 is Real : ( x0 - r < g1 & g1 < x0 ) } by RCOMP_1:def_2; then consider g1 being Real such that A7: g1 = x and A8: x0 - r < g1 and A9: g1 < x0 ; g1 < x0 + r by A1, A9, Lm1; then x in { g2 where g2 is Real : ( x0 - r < g2 & g2 < x0 + r ) } by A7, A8; then A10: x in ].(x0 - r),(x0 + r).[ by RCOMP_1:def_2; not x in {x0} by A7, A9, TARSKI:def_1; hence x in ].(x0 - r),(x0 + r).[ \ {x0} by A10, XBOOLE_0:def_5; ::_thesis: verum end; suppose x in ].x0,(x0 + r).[ ; ::_thesis: x in ].(x0 - r),(x0 + r).[ \ {x0} then x in { g1 where g1 is Real : ( x0 < g1 & g1 < x0 + r ) } by RCOMP_1:def_2; then consider g1 being Real such that A11: g1 = x and A12: x0 < g1 and A13: g1 < x0 + r ; x0 - r < g1 by A1, A12, Lm1; then x in { g2 where g2 is Real : ( x0 - r < g2 & g2 < x0 + r ) } by A11, A13; then A14: x in ].(x0 - r),(x0 + r).[ by RCOMP_1:def_2; not x in {x0} by A11, A12, TARSKI:def_1; hence x in ].(x0 - r),(x0 + r).[ \ {x0} by A14, XBOOLE_0:def_5; ::_thesis: verum end; end; end; hence x in ].(x0 - r),(x0 + r).[ \ {x0} ; ::_thesis: verum end; theorem Th5: :: LIMFUNC3:5 for r2, x0 being Real for f being PartFunc of REAL,REAL st 0 < r2 & ].(x0 - r2),x0.[ \/ ].x0,(x0 + r2).[ c= dom f holds for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) proof let r2, x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st 0 < r2 & ].(x0 - r2),x0.[ \/ ].x0,(x0 + r2).[ c= dom f holds for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) let f be PartFunc of REAL,REAL; ::_thesis: ( 0 < r2 & ].(x0 - r2),x0.[ \/ ].x0,(x0 + r2).[ c= dom f implies for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) assume that A1: 0 < r2 and A2: ].(x0 - r2),x0.[ \/ ].x0,(x0 + r2).[ c= dom f ; ::_thesis: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) A3: ].(x0 - r2),x0.[ c= ].(x0 - r2),x0.[ \/ ].x0,(x0 + r2).[ by XBOOLE_1:7; A4: ].x0,(x0 + r2).[ c= ].(x0 - r2),x0.[ \/ ].x0,(x0 + r2).[ by XBOOLE_1:7; let r1, r2 be Real; ::_thesis: ( r1 < x0 & x0 < r2 implies ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) assume that A5: r1 < x0 and A6: x0 < r2 ; ::_thesis: ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) consider g1 being Real such that A7: r1 < g1 and A8: g1 < x0 and A9: g1 in dom f by A1, A2, A3, A5, LIMFUNC2:3, XBOOLE_1:1; consider g2 being Real such that A10: g2 < r2 and A11: x0 < g2 and A12: g2 in dom f by A1, A2, A4, A6, LIMFUNC2:4, XBOOLE_1:1; take g1 ; ::_thesis: ex g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) take g2 ; ::_thesis: ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) thus ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) by A7, A8, A9, A10, A11, A12; ::_thesis: verum end; theorem Th6: :: LIMFUNC3:6 for x0 being Real for seq being Real_Sequence for f being PartFunc of REAL,REAL st ( for n being Element of NAT holds ( x0 - (1 / (n + 1)) < seq . n & seq . n < x0 & seq . n in dom f ) ) holds ( seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} ) proof let x0 be Real; ::_thesis: for seq being Real_Sequence for f being PartFunc of REAL,REAL st ( for n being Element of NAT holds ( x0 - (1 / (n + 1)) < seq . n & seq . n < x0 & seq . n in dom f ) ) holds ( seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} ) let seq be Real_Sequence; ::_thesis: for f being PartFunc of REAL,REAL st ( for n being Element of NAT holds ( x0 - (1 / (n + 1)) < seq . n & seq . n < x0 & seq . n in dom f ) ) holds ( seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} ) let f be PartFunc of REAL,REAL; ::_thesis: ( ( for n being Element of NAT holds ( x0 - (1 / (n + 1)) < seq . n & seq . n < x0 & seq . n in dom f ) ) implies ( seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} ) ) assume A1: for n being Element of NAT holds ( x0 - (1 / (n + 1)) < seq . n & seq . n < x0 & seq . n in dom f ) ; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} ) hence ( seq is convergent & lim seq = x0 ) by LIMFUNC2:5; ::_thesis: rng seq c= (dom f) \ {x0} rng seq c= (dom f) /\ (left_open_halfline x0) by A1, LIMFUNC2:5; hence rng seq c= (dom f) \ {x0} by Th1; ::_thesis: verum end; theorem Th7: :: LIMFUNC3:7 for x0, g being Real for seq being Real_Sequence st seq is convergent & lim seq = x0 & 0 < g holds ex k being Element of NAT st for n being Element of NAT st k <= n holds ( x0 - g < seq . n & seq . n < x0 + g ) proof let x0, g be Real; ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & 0 < g holds ex k being Element of NAT st for n being Element of NAT st k <= n holds ( x0 - g < seq . n & seq . n < x0 + g ) let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & 0 < g implies ex k being Element of NAT st for n being Element of NAT st k <= n holds ( x0 - g < seq . n & seq . n < x0 + g ) ) assume that A1: seq is convergent and A2: lim seq = x0 and A3: 0 < g ; ::_thesis: ex k being Element of NAT st for n being Element of NAT st k <= n holds ( x0 - g < seq . n & seq . n < x0 + g ) x0 - g < lim seq by A2, A3, Lm1; then consider k1 being Element of NAT such that A4: for n being Element of NAT st k1 <= n holds x0 - g < seq . n by A1, LIMFUNC2:1; lim seq < x0 + g by A2, A3, Lm1; then consider k2 being Element of NAT such that A5: for n being Element of NAT st k2 <= n holds seq . n < x0 + g by A1, LIMFUNC2:2; take k = max (k1,k2); ::_thesis: for n being Element of NAT st k <= n holds ( x0 - g < seq . n & seq . n < x0 + g ) let n be Element of NAT ; ::_thesis: ( k <= n implies ( x0 - g < seq . n & seq . n < x0 + g ) ) assume A6: k <= n ; ::_thesis: ( x0 - g < seq . n & seq . n < x0 + g ) k1 <= k by XXREAL_0:25; then k1 <= n by A6, XXREAL_0:2; hence x0 - g < seq . n by A4; ::_thesis: seq . n < x0 + g k2 <= k by XXREAL_0:25; then k2 <= n by A6, XXREAL_0:2; hence seq . n < x0 + g by A5; ::_thesis: verum end; theorem Th8: :: LIMFUNC3:8 for x0 being Real for f being PartFunc of REAL,REAL holds ( ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) iff ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) ) ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL holds ( ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) iff ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) iff ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) ) ) thus ( ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) implies ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) ) ) ::_thesis: ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) implies for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) proof assume A1: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ; ::_thesis: ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) ) thus for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ::_thesis: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) proof A2: x0 < x0 + 1 by Lm1; let r be Real; ::_thesis: ( r < x0 implies ex g being Real st ( r < g & g < x0 & g in dom f ) ) assume r < x0 ; ::_thesis: ex g being Real st ( r < g & g < x0 & g in dom f ) then consider g1, g2 being Real such that A3: r < g1 and A4: g1 < x0 and A5: g1 in dom f and g2 < x0 + 1 and x0 < g2 and g2 in dom f by A1, A2; take g1 ; ::_thesis: ( r < g1 & g1 < x0 & g1 in dom f ) thus ( r < g1 & g1 < x0 & g1 in dom f ) by A3, A4, A5; ::_thesis: verum end; A6: x0 - 1 < x0 by Lm1; let r be Real; ::_thesis: ( x0 < r implies ex g being Real st ( g < r & x0 < g & g in dom f ) ) assume x0 < r ; ::_thesis: ex g being Real st ( g < r & x0 < g & g in dom f ) then consider g1, g2 being Real such that x0 - 1 < g1 and g1 < x0 and g1 in dom f and A7: g2 < r and A8: x0 < g2 and A9: g2 in dom f by A1, A6; take g2 ; ::_thesis: ( g2 < r & x0 < g2 & g2 in dom f ) thus ( g2 < r & x0 < g2 & g2 in dom f ) by A7, A8, A9; ::_thesis: verum end; assume that A10: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) and A11: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ; ::_thesis: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) let r1, r2 be Real; ::_thesis: ( r1 < x0 & x0 < r2 implies ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) assume that A12: r1 < x0 and A13: x0 < r2 ; ::_thesis: ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) consider g2 being Real such that A14: g2 < r2 and A15: x0 < g2 and A16: g2 in dom f by A11, A13; consider g1 being Real such that A17: r1 < g1 and A18: g1 < x0 and A19: g1 in dom f by A10, A12; take g1 ; ::_thesis: ex g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) take g2 ; ::_thesis: ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) thus ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) by A17, A18, A19, A14, A15, A16; ::_thesis: verum end; definition let f be PartFunc of REAL,REAL; let x0 be Real; predf is_convergent_in x0 means :Def1: :: LIMFUNC3:def 1 ( ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ex g being Real st for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} holds ( f /* seq is convergent & lim (f /* seq) = g ) ); predf is_divergent_to+infty_in x0 means :Def2: :: LIMFUNC3:def 2 ( ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} holds f /* seq is divergent_to+infty ) ); predf is_divergent_to-infty_in x0 means :Def3: :: LIMFUNC3:def 3 ( ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} holds f /* seq is divergent_to-infty ) ); end; :: deftheorem Def1 defines is_convergent_in LIMFUNC3:def_1_:_ for f being PartFunc of REAL,REAL for x0 being Real holds ( f is_convergent_in x0 iff ( ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ex g being Real st for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} holds ( f /* seq is convergent & lim (f /* seq) = g ) ) ); :: deftheorem Def2 defines is_divergent_to+infty_in LIMFUNC3:def_2_:_ for f being PartFunc of REAL,REAL for x0 being Real holds ( f is_divergent_to+infty_in x0 iff ( ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} holds f /* seq is divergent_to+infty ) ) ); :: deftheorem Def3 defines is_divergent_to-infty_in LIMFUNC3:def_3_:_ for f being PartFunc of REAL,REAL for x0 being Real holds ( f is_divergent_to-infty_in x0 iff ( ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} holds f /* seq is divergent_to-infty ) ) ); theorem :: LIMFUNC3:9 for x0 being Real for f being PartFunc of REAL,REAL holds ( f is_convergent_in x0 iff ( ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ex g being Real st for g1 being Real st 0 < g1 holds ex g2 being Real st ( 0 < g2 & ( for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ) ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL holds ( f is_convergent_in x0 iff ( ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ex g being Real st for g1 being Real st 0 < g1 holds ex g2 being Real st ( 0 < g2 & ( for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_convergent_in x0 iff ( ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ex g being Real st for g1 being Real st 0 < g1 holds ex g2 being Real st ( 0 < g2 & ( for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ) ) thus ( f is_convergent_in x0 implies ( ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ex g being Real st for g1 being Real st 0 < g1 holds ex g2 being Real st ( 0 < g2 & ( for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ) ) ::_thesis: ( ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ex g being Real st for g1 being Real st 0 < g1 holds ex g2 being Real st ( 0 < g2 & ( for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) implies f is_convergent_in x0 ) proof assume that A1: f is_convergent_in x0 and A2: ( ex r1, r2 being Real st ( r1 < x0 & x0 < r2 & ( for g1, g2 being Real holds ( not r1 < g1 or not g1 < x0 or not g1 in dom f or not g2 < r2 or not x0 < g2 or not g2 in dom f ) ) ) or for g being Real ex g1 being Real st ( 0 < g1 & ( for g2 being Real st 0 < g2 holds ex r1 being Real st ( 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f & abs ((f . r1) - g) >= g1 ) ) ) ) ; ::_thesis: contradiction consider g being Real such that A3: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} holds ( f /* seq is convergent & lim (f /* seq) = g ) by A1, Def1; consider g1 being Real such that A4: 0 < g1 and A5: for g2 being Real st 0 < g2 holds ex r1 being Real st ( 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f & abs ((f . r1) - g) >= g1 ) by A1, A2, Def1; defpred S1[ Element of NAT , real number ] means ( 0 < abs (x0 - $2) & abs (x0 - $2) < 1 / ($1 + 1) & $2 in dom f & abs ((f . $2) - g) >= g1 ); A6: for n being Element of NAT ex r1 being Real st S1[n,r1] by A5, XREAL_1:139; consider s being Real_Sequence such that A7: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A6); A8: rng s c= (dom f) \ {x0} by A7, Th2; A9: lim s = x0 by A7, Th2; A10: s is convergent by A7, Th2; then A11: lim (f /* s) = g by A3, A9, A8; f /* s is convergent by A3, A10, A9, A8; then consider n being Element of NAT such that A12: for k being Element of NAT st n <= k holds abs (((f /* s) . k) - g) < g1 by A4, A11, SEQ_2:def_7; A13: abs (((f /* s) . n) - g) < g1 by A12; rng s c= dom f by A7, Th2; then abs ((f . (s . n)) - g) < g1 by A13, FUNCT_2:108; hence contradiction by A7; ::_thesis: verum end; assume A14: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ; ::_thesis: ( for g being Real ex g1 being Real st ( 0 < g1 & ( for g2 being Real holds ( not 0 < g2 or ex r1 being Real st ( 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f & not abs ((f . r1) - g) < g1 ) ) ) ) or f is_convergent_in x0 ) given g being Real such that A15: for g1 being Real st 0 < g1 holds ex g2 being Real st ( 0 < g2 & ( for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ; ::_thesis: f is_convergent_in x0 now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_f)_\_{x0}_holds_ (_f_/*_s_is_convergent_&_lim_(f_/*_s)_=_g_) let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom f) \ {x0} implies ( f /* s is convergent & lim (f /* s) = g ) ) assume that A16: s is convergent and A17: lim s = x0 and A18: rng s c= (dom f) \ {x0} ; ::_thesis: ( f /* s is convergent & lim (f /* s) = g ) A19: now__::_thesis:_for_g1_being_real_number_st_0_<_g1_holds_ ex_n_being_Element_of_NAT_st_ for_k_being_Element_of_NAT_st_n_<=_k_holds_ abs_(((f_/*_s)_._k)_-_g)_<_g1 let g1 be real number ; ::_thesis: ( 0 < g1 implies ex n being Element of NAT st for k being Element of NAT st n <= k holds abs (((f /* s) . k) - g) < g1 ) assume A20: 0 < g1 ; ::_thesis: ex n being Element of NAT st for k being Element of NAT st n <= k holds abs (((f /* s) . k) - g) < g1 g1 is Real by XREAL_0:def_1; then consider g2 being Real such that A21: 0 < g2 and A22: for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds abs ((f . r1) - g) < g1 by A15, A20; consider n being Element of NAT such that A23: for k being Element of NAT st n <= k holds ( 0 < abs (x0 - (s . k)) & abs (x0 - (s . k)) < g2 & s . k in dom f ) by A16, A17, A18, A21, Th3; take n = n; ::_thesis: for k being Element of NAT st n <= k holds abs (((f /* s) . k) - g) < g1 let k be Element of NAT ; ::_thesis: ( n <= k implies abs (((f /* s) . k) - g) < g1 ) assume A24: n <= k ; ::_thesis: abs (((f /* s) . k) - g) < g1 then A25: abs (x0 - (s . k)) < g2 by A23; A26: s . k in dom f by A23, A24; 0 < abs (x0 - (s . k)) by A23, A24; then abs ((f . (s . k)) - g) < g1 by A22, A25, A26; hence abs (((f /* s) . k) - g) < g1 by A18, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; hence f /* s is convergent by SEQ_2:def_6; ::_thesis: lim (f /* s) = g hence lim (f /* s) = g by A19, SEQ_2:def_7; ::_thesis: verum end; hence f is_convergent_in x0 by A14, Def1; ::_thesis: verum end; theorem :: LIMFUNC3:10 for x0 being Real for f being PartFunc of REAL,REAL holds ( f is_divergent_to+infty_in x0 iff ( ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ( for g1 being Real ex g2 being Real st ( 0 < g2 & ( for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds g1 < f . r1 ) ) ) ) ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL holds ( f is_divergent_to+infty_in x0 iff ( ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ( for g1 being Real ex g2 being Real st ( 0 < g2 & ( for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds g1 < f . r1 ) ) ) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_divergent_to+infty_in x0 iff ( ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ( for g1 being Real ex g2 being Real st ( 0 < g2 & ( for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds g1 < f . r1 ) ) ) ) ) thus ( f is_divergent_to+infty_in x0 implies ( ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ( for g1 being Real ex g2 being Real st ( 0 < g2 & ( for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds g1 < f . r1 ) ) ) ) ) ::_thesis: ( ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ( for g1 being Real ex g2 being Real st ( 0 < g2 & ( for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds g1 < f . r1 ) ) ) implies f is_divergent_to+infty_in x0 ) proof assume that A1: f is_divergent_to+infty_in x0 and A2: ( ex r1, r2 being Real st ( r1 < x0 & x0 < r2 & ( for g1, g2 being Real holds ( not r1 < g1 or not g1 < x0 or not g1 in dom f or not g2 < r2 or not x0 < g2 or not g2 in dom f ) ) ) or ex g1 being Real st for g2 being Real st 0 < g2 holds ex r1 being Real st ( 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f & f . r1 <= g1 ) ) ; ::_thesis: contradiction consider g1 being Real such that A3: for g2 being Real st 0 < g2 holds ex r1 being Real st ( 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f & f . r1 <= g1 ) by A1, A2, Def2; defpred S1[ Element of NAT , real number ] means ( 0 < abs (x0 - $2) & abs (x0 - $2) < 1 / ($1 + 1) & $2 in dom f & f . $2 <= g1 ); A4: for n being Element of NAT ex r1 being Real st S1[n,r1] by A3, XREAL_1:139; consider s being Real_Sequence such that A5: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A4); A6: rng s c= (dom f) \ {x0} by A5, Th2; A7: lim s = x0 by A5, Th2; s is convergent by A5, Th2; then f /* s is divergent_to+infty by A1, A7, A6, Def2; then consider n being Element of NAT such that A8: for k being Element of NAT st n <= k holds g1 < (f /* s) . k by LIMFUNC1:def_4; A9: g1 < (f /* s) . n by A8; rng s c= dom f by A5, Th2; then g1 < f . (s . n) by A9, FUNCT_2:108; hence contradiction by A5; ::_thesis: verum end; assume that A10: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) and A11: for g1 being Real ex g2 being Real st ( 0 < g2 & ( for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds g1 < f . r1 ) ) ; ::_thesis: f is_divergent_to+infty_in x0 now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_f)_\_{x0}_holds_ f_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom f) \ {x0} implies f /* s is divergent_to+infty ) assume that A12: s is convergent and A13: lim s = x0 and A14: rng s c= (dom f) \ {x0} ; ::_thesis: f /* s is divergent_to+infty now__::_thesis:_for_g1_being_Real_ex_n_being_Element_of_NAT_st_ for_k_being_Element_of_NAT_st_n_<=_k_holds_ g1_<_(f_/*_s)_._k let g1 be Real; ::_thesis: ex n being Element of NAT st for k being Element of NAT st n <= k holds g1 < (f /* s) . k consider g2 being Real such that A15: 0 < g2 and A16: for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds g1 < f . r1 by A11; consider n being Element of NAT such that A17: for k being Element of NAT st n <= k holds ( 0 < abs (x0 - (s . k)) & abs (x0 - (s . k)) < g2 & s . k in dom f ) by A12, A13, A14, A15, Th3; take n = n; ::_thesis: for k being Element of NAT st n <= k holds g1 < (f /* s) . k let k be Element of NAT ; ::_thesis: ( n <= k implies g1 < (f /* s) . k ) assume A18: n <= k ; ::_thesis: g1 < (f /* s) . k then A19: abs (x0 - (s . k)) < g2 by A17; A20: s . k in dom f by A17, A18; 0 < abs (x0 - (s . k)) by A17, A18; then g1 < f . (s . k) by A16, A19, A20; hence g1 < (f /* s) . k by A14, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; hence f /* s is divergent_to+infty by LIMFUNC1:def_4; ::_thesis: verum end; hence f is_divergent_to+infty_in x0 by A10, Def2; ::_thesis: verum end; theorem :: LIMFUNC3:11 for x0 being Real for f being PartFunc of REAL,REAL holds ( f is_divergent_to-infty_in x0 iff ( ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ( for g1 being Real ex g2 being Real st ( 0 < g2 & ( for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds f . r1 < g1 ) ) ) ) ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL holds ( f is_divergent_to-infty_in x0 iff ( ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ( for g1 being Real ex g2 being Real st ( 0 < g2 & ( for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds f . r1 < g1 ) ) ) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_divergent_to-infty_in x0 iff ( ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ( for g1 being Real ex g2 being Real st ( 0 < g2 & ( for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds f . r1 < g1 ) ) ) ) ) thus ( f is_divergent_to-infty_in x0 implies ( ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ( for g1 being Real ex g2 being Real st ( 0 < g2 & ( for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds f . r1 < g1 ) ) ) ) ) ::_thesis: ( ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ( for g1 being Real ex g2 being Real st ( 0 < g2 & ( for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds f . r1 < g1 ) ) ) implies f is_divergent_to-infty_in x0 ) proof assume that A1: f is_divergent_to-infty_in x0 and A2: ( ex r1, r2 being Real st ( r1 < x0 & x0 < r2 & ( for g1, g2 being Real holds ( not r1 < g1 or not g1 < x0 or not g1 in dom f or not g2 < r2 or not x0 < g2 or not g2 in dom f ) ) ) or ex g1 being Real st for g2 being Real st 0 < g2 holds ex r1 being Real st ( 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f & g1 <= f . r1 ) ) ; ::_thesis: contradiction consider g1 being Real such that A3: for g2 being Real st 0 < g2 holds ex r1 being Real st ( 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f & g1 <= f . r1 ) by A1, A2, Def3; defpred S1[ Element of NAT , real number ] means ( 0 < abs (x0 - $2) & abs (x0 - $2) < 1 / ($1 + 1) & $2 in dom f & g1 <= f . $2 ); A4: for n being Element of NAT ex r1 being Real st S1[n,r1] by A3, XREAL_1:139; consider s being Real_Sequence such that A5: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A4); A6: rng s c= (dom f) \ {x0} by A5, Th2; A7: lim s = x0 by A5, Th2; s is convergent by A5, Th2; then f /* s is divergent_to-infty by A1, A7, A6, Def3; then consider n being Element of NAT such that A8: for k being Element of NAT st n <= k holds (f /* s) . k < g1 by LIMFUNC1:def_5; A9: (f /* s) . n < g1 by A8; rng s c= dom f by A5, Th2; then f . (s . n) < g1 by A9, FUNCT_2:108; hence contradiction by A5; ::_thesis: verum end; assume that A10: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) and A11: for g1 being Real ex g2 being Real st ( 0 < g2 & ( for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds f . r1 < g1 ) ) ; ::_thesis: f is_divergent_to-infty_in x0 now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_f)_\_{x0}_holds_ f_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom f) \ {x0} implies f /* s is divergent_to-infty ) assume that A12: s is convergent and A13: lim s = x0 and A14: rng s c= (dom f) \ {x0} ; ::_thesis: f /* s is divergent_to-infty now__::_thesis:_for_g1_being_Real_ex_n_being_Element_of_NAT_st_ for_k_being_Element_of_NAT_st_n_<=_k_holds_ (f_/*_s)_._k_<_g1 let g1 be Real; ::_thesis: ex n being Element of NAT st for k being Element of NAT st n <= k holds (f /* s) . k < g1 consider g2 being Real such that A15: 0 < g2 and A16: for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds f . r1 < g1 by A11; consider n being Element of NAT such that A17: for k being Element of NAT st n <= k holds ( 0 < abs (x0 - (s . k)) & abs (x0 - (s . k)) < g2 & s . k in dom f ) by A12, A13, A14, A15, Th3; take n = n; ::_thesis: for k being Element of NAT st n <= k holds (f /* s) . k < g1 let k be Element of NAT ; ::_thesis: ( n <= k implies (f /* s) . k < g1 ) assume A18: n <= k ; ::_thesis: (f /* s) . k < g1 then A19: abs (x0 - (s . k)) < g2 by A17; A20: s . k in dom f by A17, A18; 0 < abs (x0 - (s . k)) by A17, A18; then f . (s . k) < g1 by A16, A19, A20; hence (f /* s) . k < g1 by A14, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; hence f /* s is divergent_to-infty by LIMFUNC1:def_5; ::_thesis: verum end; hence f is_divergent_to-infty_in x0 by A10, Def3; ::_thesis: verum end; theorem Th12: :: LIMFUNC3:12 for x0 being Real for f being PartFunc of REAL,REAL holds ( f is_divergent_to+infty_in x0 iff ( f is_left_divergent_to+infty_in x0 & f is_right_divergent_to+infty_in x0 ) ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL holds ( f is_divergent_to+infty_in x0 iff ( f is_left_divergent_to+infty_in x0 & f is_right_divergent_to+infty_in x0 ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_divergent_to+infty_in x0 iff ( f is_left_divergent_to+infty_in x0 & f is_right_divergent_to+infty_in x0 ) ) thus ( f is_divergent_to+infty_in x0 implies ( f is_left_divergent_to+infty_in x0 & f is_right_divergent_to+infty_in x0 ) ) ::_thesis: ( f is_left_divergent_to+infty_in x0 & f is_right_divergent_to+infty_in x0 implies f is_divergent_to+infty_in x0 ) proof assume A1: f is_divergent_to+infty_in x0 ; ::_thesis: ( f is_left_divergent_to+infty_in x0 & f is_right_divergent_to+infty_in x0 ) A2: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_f)_/\_(left_open_halfline_x0)_holds_ f_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom f) /\ (left_open_halfline x0) implies f /* s is divergent_to+infty ) assume that A3: s is convergent and A4: lim s = x0 and A5: rng s c= (dom f) /\ (left_open_halfline x0) ; ::_thesis: f /* s is divergent_to+infty rng s c= (dom f) \ {x0} by A5, Th1; hence f /* s is divergent_to+infty by A1, A3, A4, Def2; ::_thesis: verum end; A6: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_f)_/\_(right_open_halfline_x0)_holds_ f_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom f) /\ (right_open_halfline x0) implies f /* s is divergent_to+infty ) assume that A7: s is convergent and A8: lim s = x0 and A9: rng s c= (dom f) /\ (right_open_halfline x0) ; ::_thesis: f /* s is divergent_to+infty rng s c= (dom f) \ {x0} by A9, Th1; hence f /* s is divergent_to+infty by A1, A7, A8, Def2; ::_thesis: verum end; A10: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) by A1, Def2; then for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) by Th8; hence f is_left_divergent_to+infty_in x0 by A2, LIMFUNC2:def_2; ::_thesis: f is_right_divergent_to+infty_in x0 for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) by A10, Th8; hence f is_right_divergent_to+infty_in x0 by A6, LIMFUNC2:def_5; ::_thesis: verum end; assume that A11: f is_left_divergent_to+infty_in x0 and A12: f is_right_divergent_to+infty_in x0 ; ::_thesis: f is_divergent_to+infty_in x0 A13: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) by A12, LIMFUNC2:def_5; A14: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_f)_\_{x0}_holds_ f_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom f) \ {x0} implies f /* s is divergent_to+infty ) assume that A15: s is convergent and A16: lim s = x0 and A17: rng s c= (dom f) \ {x0} ; ::_thesis: f /* s is divergent_to+infty now__::_thesis:_f_/*_s_is_divergent_to+infty percases ( ex k being Element of NAT st for n being Element of NAT st k <= n holds s . n < x0 or for k being Element of NAT ex n being Element of NAT st ( k <= n & s . n >= x0 ) ) ; suppose ex k being Element of NAT st for n being Element of NAT st k <= n holds s . n < x0 ; ::_thesis: f /* s is divergent_to+infty then consider k being Element of NAT such that A18: for n being Element of NAT st k <= n holds s . n < x0 ; A19: rng s c= dom f by A17, XBOOLE_1:1; A20: rng (s ^\ k) c= (dom f) /\ (left_open_halfline x0) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (s ^\ k) or x in (dom f) /\ (left_open_halfline x0) ) assume x in rng (s ^\ k) ; ::_thesis: x in (dom f) /\ (left_open_halfline x0) then consider n being Element of NAT such that A21: (s ^\ k) . n = x by FUNCT_2:113; s . (n + k) < x0 by A18, NAT_1:12; then s . (n + k) in { g1 where g1 is Real : g1 < x0 } ; then s . (n + k) in left_open_halfline x0 by XXREAL_1:229; then A22: x in left_open_halfline x0 by A21, NAT_1:def_3; s . (n + k) in rng s by VALUED_0:28; then x in rng s by A21, NAT_1:def_3; hence x in (dom f) /\ (left_open_halfline x0) by A19, A22, XBOOLE_0:def_4; ::_thesis: verum end; A23: f /* (s ^\ k) = (f /* s) ^\ k by A17, VALUED_0:27, XBOOLE_1:1; lim (s ^\ k) = x0 by A15, A16, SEQ_4:20; then f /* (s ^\ k) is divergent_to+infty by A11, A15, A20, LIMFUNC2:def_2; hence f /* s is divergent_to+infty by A23, LIMFUNC1:7; ::_thesis: verum end; supposeA24: for k being Element of NAT ex n being Element of NAT st ( k <= n & s . n >= x0 ) ; ::_thesis: f /* s is divergent_to+infty now__::_thesis:_f_/*_s_is_divergent_to+infty percases ( ex k being Element of NAT st for n being Element of NAT st k <= n holds x0 < s . n or for k being Element of NAT ex n being Element of NAT st ( k <= n & x0 >= s . n ) ) ; suppose ex k being Element of NAT st for n being Element of NAT st k <= n holds x0 < s . n ; ::_thesis: f /* s is divergent_to+infty then consider k being Element of NAT such that A25: for n being Element of NAT st k <= n holds s . n > x0 ; A26: rng s c= dom f by A17, XBOOLE_1:1; A27: rng (s ^\ k) c= (dom f) /\ (right_open_halfline x0) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (s ^\ k) or x in (dom f) /\ (right_open_halfline x0) ) assume x in rng (s ^\ k) ; ::_thesis: x in (dom f) /\ (right_open_halfline x0) then consider n being Element of NAT such that A28: (s ^\ k) . n = x by FUNCT_2:113; x0 < s . (n + k) by A25, NAT_1:12; then s . (n + k) in { g1 where g1 is Real : x0 < g1 } ; then s . (n + k) in right_open_halfline x0 by XXREAL_1:230; then A29: x in right_open_halfline x0 by A28, NAT_1:def_3; s . (n + k) in rng s by VALUED_0:28; then x in rng s by A28, NAT_1:def_3; hence x in (dom f) /\ (right_open_halfline x0) by A26, A29, XBOOLE_0:def_4; ::_thesis: verum end; A30: f /* (s ^\ k) = (f /* s) ^\ k by A17, VALUED_0:27, XBOOLE_1:1; lim (s ^\ k) = x0 by A15, A16, SEQ_4:20; then f /* (s ^\ k) is divergent_to+infty by A12, A15, A27, LIMFUNC2:def_5; hence f /* s is divergent_to+infty by A30, LIMFUNC1:7; ::_thesis: verum end; supposeA31: for k being Element of NAT ex n being Element of NAT st ( k <= n & x0 >= s . n ) ; ::_thesis: f /* s is divergent_to+infty defpred S1[ Nat] means s . $1 < x0; A32: now__::_thesis:_for_k_being_Element_of_NAT_ex_n_being_Element_of_NAT_st_ (_k_<=_n_&_s_._n_<_x0_) let k be Element of NAT ; ::_thesis: ex n being Element of NAT st ( k <= n & s . n < x0 ) consider n being Element of NAT such that A33: k <= n and A34: s . n <= x0 by A31; take n = n; ::_thesis: ( k <= n & s . n < x0 ) thus k <= n by A33; ::_thesis: s . n < x0 s . n in rng s by VALUED_0:28; then not s . n in {x0} by A17, XBOOLE_0:def_5; then s . n <> x0 by TARSKI:def_1; hence s . n < x0 by A34, XXREAL_0:1; ::_thesis: verum end; then ex m1 being Element of NAT st ( 0 <= m1 & s . m1 < x0 ) ; then A35: ex m being Nat st S1[m] ; consider M being Nat such that A36: ( S1[M] & ( for n being Nat st S1[n] holds M <= n ) ) from NAT_1:sch_5(A35); defpred S2[ Nat] means s . $1 > x0; defpred S3[ set , set ] means for n, m being Element of NAT st $1 = n & $2 = m holds ( n < m & s . m < x0 & ( for k being Element of NAT st n < k & s . k < x0 holds m <= k ) ); defpred S4[ Element of NAT , set , set ] means S3[$2,$3]; reconsider M9 = M as Element of NAT by ORDINAL1:def_12; A37: now__::_thesis:_for_n_being_Element_of_NAT_ex_m_being_Element_of_NAT_st_ (_n_<_m_&_s_._m_<_x0_) let n be Element of NAT ; ::_thesis: ex m being Element of NAT st ( n < m & s . m < x0 ) consider m being Element of NAT such that A38: n + 1 <= m and A39: s . m < x0 by A32; take m = m; ::_thesis: ( n < m & s . m < x0 ) thus ( n < m & s . m < x0 ) by A38, A39, NAT_1:13; ::_thesis: verum end; A40: for n, x being Element of NAT ex y being Element of NAT st S4[n,x,y] proof let n, x be Element of NAT ; ::_thesis: ex y being Element of NAT st S4[n,x,y] defpred S5[ Nat] means ( x < $1 & s . $1 < x0 ); ex m being Element of NAT st S5[m] by A37; then A41: ex m being Nat st S5[m] ; consider l being Nat such that A42: ( S5[l] & ( for k being Nat st S5[k] holds l <= k ) ) from NAT_1:sch_5(A41); take l ; ::_thesis: ( l is Element of REAL & l is Element of NAT & S4[n,x,l] ) l in NAT by ORDINAL1:def_12; hence ( l is Element of REAL & l is Element of NAT & S4[n,x,l] ) by A42; ::_thesis: verum end; consider F being Function of NAT,NAT such that A43: ( F . 0 = M9 & ( for n being Element of NAT holds S4[n,F . n,F . (n + 1)] ) ) from RECDEF_1:sch_2(A40); A44: rng F c= NAT by RELAT_1:def_19; then A45: rng F c= REAL by XBOOLE_1:1; A46: dom F = NAT by FUNCT_2:def_1; then reconsider F = F as Real_Sequence by A45, RELSET_1:4; A47: now__::_thesis:_for_n_being_Element_of_NAT_holds_F_._n_is_Element_of_NAT let n be Element of NAT ; ::_thesis: F . n is Element of NAT F . n in rng F by A46, FUNCT_1:def_3; hence F . n is Element of NAT by A44; ::_thesis: verum end; now__::_thesis:_for_n_being_Element_of_NAT_holds_F_._n_<_F_._(n_+_1) let n be Element of NAT ; ::_thesis: F . n < F . (n + 1) A48: F . (n + 1) is Element of NAT by A47; F . n is Element of NAT by A47; hence F . n < F . (n + 1) by A43, A48; ::_thesis: verum end; then reconsider F = F as V37() sequence of NAT by SEQM_3:def_6; A49: s * F is subsequence of s by VALUED_0:def_17; then rng (s * F) c= rng s by VALUED_0:21; then A50: rng (s * F) c= (dom f) \ {x0} by A17, XBOOLE_1:1; A51: for n being Element of NAT st s . n < x0 holds ex m being Element of NAT st F . m = n proof defpred S5[ Nat] means ( s . $1 < x0 & ( for m being Element of NAT holds F . m <> $1 ) ); assume ex n being Element of NAT st S5[n] ; ::_thesis: contradiction then A52: ex n being Nat st S5[n] ; consider M1 being Nat such that A53: ( S5[M1] & ( for n being Nat st S5[n] holds M1 <= n ) ) from NAT_1:sch_5(A52); defpred S6[ Nat] means ( $1 < M1 & s . $1 < x0 & ex m being Element of NAT st F . m = $1 ); A54: ex n being Nat st S6[n] proof take M ; ::_thesis: S6[M] A55: M <> M1 by A43, A53; M <= M1 by A36, A53; hence M < M1 by A55, XXREAL_0:1; ::_thesis: ( s . M < x0 & ex m being Element of NAT st F . m = M ) thus s . M < x0 by A36; ::_thesis: ex m being Element of NAT st F . m = M take 0 ; ::_thesis: F . 0 = M thus F . 0 = M by A43; ::_thesis: verum end; A56: for n being Nat st S6[n] holds n <= M1 ; consider MX being Nat such that A57: ( S6[MX] & ( for n being Nat st S6[n] holds n <= MX ) ) from NAT_1:sch_6(A56, A54); A58: for k being Element of NAT st MX < k & k < M1 holds s . k >= x0 proof given k being Element of NAT such that A59: MX < k and A60: k < M1 and A61: s . k < x0 ; ::_thesis: contradiction now__::_thesis:_contradiction percases ( ex m being Element of NAT st F . m = k or for m being Element of NAT holds F . m <> k ) ; suppose ex m being Element of NAT st F . m = k ; ::_thesis: contradiction hence contradiction by A57, A59, A60, A61; ::_thesis: verum end; suppose for m being Element of NAT holds F . m <> k ; ::_thesis: contradiction hence contradiction by A53, A60, A61; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; consider m being Element of NAT such that A62: F . m = MX by A57; M1 in NAT by ORDINAL1:def_12; then A63: F . (m + 1) <= M1 by A43, A53, A57, A62; A64: s . (F . (m + 1)) < x0 by A43, A62; A65: MX < F . (m + 1) by A43, A62; now__::_thesis:_not_F_._(m_+_1)_<>_M1 assume F . (m + 1) <> M1 ; ::_thesis: contradiction then F . (m + 1) < M1 by A63, XXREAL_0:1; hence contradiction by A58, A65, A64; ::_thesis: verum end; hence contradiction by A53; ::_thesis: verum end; A66: now__::_thesis:_for_k_being_Element_of_NAT_ex_n_being_Element_of_NAT_st_ (_k_<=_n_&_s_._n_>_x0_) let k be Element of NAT ; ::_thesis: ex n being Element of NAT st ( k <= n & s . n > x0 ) consider n being Element of NAT such that A67: k <= n and A68: s . n >= x0 by A24; take n = n; ::_thesis: ( k <= n & s . n > x0 ) thus k <= n by A67; ::_thesis: s . n > x0 s . n in rng s by VALUED_0:28; then not s . n in {x0} by A17, XBOOLE_0:def_5; then s . n <> x0 by TARSKI:def_1; hence s . n > x0 by A68, XXREAL_0:1; ::_thesis: verum end; then ex mn being Element of NAT st ( 0 <= mn & s . mn > x0 ) ; then A69: ex m being Nat st S2[m] ; consider N being Nat such that A70: ( S2[N] & ( for n being Nat st S2[n] holds N <= n ) ) from NAT_1:sch_5(A69); A71: for n being Element of NAT holds (s * F) . n < x0 proof defpred S5[ Element of NAT ] means (s * F) . $1 < x0; A72: for k being Element of NAT st S5[k] holds S5[k + 1] proof let k be Element of NAT ; ::_thesis: ( S5[k] implies S5[k + 1] ) assume (s * F) . k < x0 ; ::_thesis: S5[k + 1] S3[F . k,F . (k + 1)] by A43; then s . (F . (k + 1)) < x0 ; hence S5[k + 1] by FUNCT_2:15; ::_thesis: verum end; A73: S5[ 0 ] by A36, A43, FUNCT_2:15; thus for k being Element of NAT holds S5[k] from NAT_1:sch_1(A73, A72); ::_thesis: verum end; A74: rng (s * F) c= (dom f) /\ (left_open_halfline x0) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (s * F) or x in (dom f) /\ (left_open_halfline x0) ) assume A75: x in rng (s * F) ; ::_thesis: x in (dom f) /\ (left_open_halfline x0) then consider n being Element of NAT such that A76: (s * F) . n = x by FUNCT_2:113; (s * F) . n < x0 by A71; then x in { g1 where g1 is Real : g1 < x0 } by A76; then A77: x in left_open_halfline x0 by XXREAL_1:229; x in dom f by A50, A75, XBOOLE_0:def_5; hence x in (dom f) /\ (left_open_halfline x0) by A77, XBOOLE_0:def_4; ::_thesis: verum end; defpred S5[ set , set ] means for n, m being Element of NAT st $1 = n & $2 = m holds ( n < m & s . m > x0 & ( for k being Element of NAT st n < k & s . k > x0 holds m <= k ) ); defpred S6[ Element of NAT , set , set ] means S5[$2,$3]; A78: s * F is convergent by A15, A49, SEQ_4:16; reconsider N9 = N as Element of NAT by ORDINAL1:def_12; A79: now__::_thesis:_for_n_being_Element_of_NAT_ex_m_being_Element_of_NAT_st_ (_n_<_m_&_s_._m_>_x0_) let n be Element of NAT ; ::_thesis: ex m being Element of NAT st ( n < m & s . m > x0 ) consider m being Element of NAT such that A80: n + 1 <= m and A81: s . m > x0 by A66; take m = m; ::_thesis: ( n < m & s . m > x0 ) thus ( n < m & s . m > x0 ) by A80, A81, NAT_1:13; ::_thesis: verum end; A82: for n, x being Element of NAT ex y being Element of NAT st S6[n,x,y] proof let n, x be Element of NAT ; ::_thesis: ex y being Element of NAT st S6[n,x,y] defpred S7[ Nat] means ( x < $1 & s . $1 > x0 ); ex m being Element of NAT st S7[m] by A79; then A83: ex m being Nat st S7[m] ; consider l being Nat such that A84: ( S7[l] & ( for k being Nat st S7[k] holds l <= k ) ) from NAT_1:sch_5(A83); reconsider l = l as Element of NAT by ORDINAL1:def_12; take l ; ::_thesis: S6[n,x,l] thus S6[n,x,l] by A84; ::_thesis: verum end; consider G being Function of NAT,NAT such that A85: ( G . 0 = N9 & ( for n being Element of NAT holds S6[n,G . n,G . (n + 1)] ) ) from RECDEF_1:sch_2(A82); A86: rng G c= NAT by RELAT_1:def_19; then A87: rng G c= REAL by XBOOLE_1:1; A88: dom G = NAT by FUNCT_2:def_1; then reconsider G = G as Real_Sequence by A87, RELSET_1:4; A89: now__::_thesis:_for_n_being_Element_of_NAT_holds_G_._n_is_Element_of_NAT let n be Element of NAT ; ::_thesis: G . n is Element of NAT G . n in rng G by A88, FUNCT_1:def_3; hence G . n is Element of NAT by A86; ::_thesis: verum end; now__::_thesis:_for_n_being_Element_of_NAT_holds_G_._n_<_G_._(n_+_1) let n be Element of NAT ; ::_thesis: G . n < G . (n + 1) A90: G . (n + 1) is Element of NAT by A89; G . n is Element of NAT by A89; hence G . n < G . (n + 1) by A85, A90; ::_thesis: verum end; then reconsider G = G as V37() sequence of NAT by SEQM_3:def_6; A91: s * G is subsequence of s by VALUED_0:def_17; then rng (s * G) c= rng s by VALUED_0:21; then A92: rng (s * G) c= (dom f) \ {x0} by A17, XBOOLE_1:1; defpred S7[ Nat] means ( s . $1 > x0 & ( for m being Element of NAT holds G . m <> $1 ) ); A93: for n being Element of NAT st s . n > x0 holds ex m being Element of NAT st G . m = n proof assume ex n being Element of NAT st S7[n] ; ::_thesis: contradiction then A94: ex n being Nat st S7[n] ; consider N1 being Nat such that A95: ( S7[N1] & ( for n being Nat st S7[n] holds N1 <= n ) ) from NAT_1:sch_5(A94); defpred S8[ Nat] means ( $1 < N1 & s . $1 > x0 & ex m being Element of NAT st G . m = $1 ); A96: ex n being Nat st S8[n] proof take N ; ::_thesis: S8[N] A97: N <> N1 by A85, A95; N <= N1 by A70, A95; hence N < N1 by A97, XXREAL_0:1; ::_thesis: ( s . N > x0 & ex m being Element of NAT st G . m = N ) thus s . N > x0 by A70; ::_thesis: ex m being Element of NAT st G . m = N take 0 ; ::_thesis: G . 0 = N thus G . 0 = N by A85; ::_thesis: verum end; A98: for n being Nat st S8[n] holds n <= N1 ; consider NX being Nat such that A99: ( S8[NX] & ( for n being Nat st S8[n] holds n <= NX ) ) from NAT_1:sch_6(A98, A96); A100: for k being Element of NAT st NX < k & k < N1 holds s . k <= x0 proof given k being Element of NAT such that A101: NX < k and A102: k < N1 and A103: s . k > x0 ; ::_thesis: contradiction now__::_thesis:_contradiction percases ( ex m being Element of NAT st G . m = k or for m being Element of NAT holds G . m <> k ) ; suppose ex m being Element of NAT st G . m = k ; ::_thesis: contradiction hence contradiction by A99, A101, A102, A103; ::_thesis: verum end; suppose for m being Element of NAT holds G . m <> k ; ::_thesis: contradiction hence contradiction by A95, A102, A103; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; consider m being Element of NAT such that A104: G . m = NX by A99; N1 in NAT by ORDINAL1:def_12; then A105: G . (m + 1) <= N1 by A85, A95, A99, A104; A106: s . (G . (m + 1)) > x0 by A85, A104; A107: NX < G . (m + 1) by A85, A104; now__::_thesis:_not_G_._(m_+_1)_<>_N1 assume G . (m + 1) <> N1 ; ::_thesis: contradiction then G . (m + 1) < N1 by A105, XXREAL_0:1; hence contradiction by A100, A107, A106; ::_thesis: verum end; hence contradiction by A95; ::_thesis: verum end; A108: for n being Element of NAT holds (s * G) . n > x0 proof defpred S8[ Element of NAT ] means (s * G) . $1 > x0; A109: for k being Element of NAT st S8[k] holds S8[k + 1] proof let k be Element of NAT ; ::_thesis: ( S8[k] implies S8[k + 1] ) assume (s * G) . k > x0 ; ::_thesis: S8[k + 1] S5[G . k,G . (k + 1)] by A85; then s . (G . (k + 1)) > x0 ; hence S8[k + 1] by FUNCT_2:15; ::_thesis: verum end; A110: S8[ 0 ] by A70, A85, FUNCT_2:15; thus for k being Element of NAT holds S8[k] from NAT_1:sch_1(A110, A109); ::_thesis: verum end; A111: rng (s * G) c= (dom f) /\ (right_open_halfline x0) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (s * G) or x in (dom f) /\ (right_open_halfline x0) ) assume A112: x in rng (s * G) ; ::_thesis: x in (dom f) /\ (right_open_halfline x0) then consider n being Element of NAT such that A113: (s * G) . n = x by FUNCT_2:113; (s * G) . n > x0 by A108; then x in { g1 where g1 is Real : x0 < g1 } by A113; then A114: x in right_open_halfline x0 by XXREAL_1:230; x in dom f by A92, A112, XBOOLE_0:def_5; hence x in (dom f) /\ (right_open_halfline x0) by A114, XBOOLE_0:def_4; ::_thesis: verum end; A115: s * G is convergent by A15, A91, SEQ_4:16; lim (s * G) = x0 by A15, A16, A91, SEQ_4:17; then A116: f /* (s * G) is divergent_to+infty by A12, A115, A111, LIMFUNC2:def_5; lim (s * F) = x0 by A15, A16, A49, SEQ_4:17; then A117: f /* (s * F) is divergent_to+infty by A11, A78, A74, LIMFUNC2:def_2; now__::_thesis:_for_r_being_Real_ex_n_being_Element_of_NAT_st_ for_k_being_Element_of_NAT_st_n_<=_k_holds_ r_<_(f_/*_s)_._k let r be Real; ::_thesis: ex n being Element of NAT st for k being Element of NAT st n <= k holds r < (f /* s) . k consider n1 being Element of NAT such that A118: for k being Element of NAT st n1 <= k holds r < (f /* (s * F)) . k by A117, LIMFUNC1:def_4; consider n2 being Element of NAT such that A119: for k being Element of NAT st n2 <= k holds r < (f /* (s * G)) . k by A116, LIMFUNC1:def_4; take n = max ((F . n1),(G . n2)); ::_thesis: for k being Element of NAT st n <= k holds r < (f /* s) . k let k be Element of NAT ; ::_thesis: ( n <= k implies r < (f /* s) . k ) assume A120: n <= k ; ::_thesis: r < (f /* s) . k s . k in rng s by VALUED_0:28; then not s . k in {x0} by A17, XBOOLE_0:def_5; then A121: s . k <> x0 by TARSKI:def_1; now__::_thesis:_r_<_(f_/*_s)_._k percases ( s . k < x0 or s . k > x0 ) by A121, XXREAL_0:1; suppose s . k < x0 ; ::_thesis: r < (f /* s) . k then consider l being Element of NAT such that A122: k = F . l by A51; F . n1 <= n by XXREAL_0:25; then F . n1 <= k by A120, XXREAL_0:2; then l >= n1 by A122, SEQM_3:1; then r < (f /* (s * F)) . l by A118; then r < f . ((s * F) . l) by A50, FUNCT_2:108, XBOOLE_1:1; then r < f . (s . k) by A122, FUNCT_2:15; hence r < (f /* s) . k by A17, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; suppose s . k > x0 ; ::_thesis: r < (f /* s) . k then consider l being Element of NAT such that A123: k = G . l by A93; G . n2 <= n by XXREAL_0:25; then G . n2 <= k by A120, XXREAL_0:2; then l >= n2 by A123, SEQM_3:1; then r < (f /* (s * G)) . l by A119; then r < f . ((s * G) . l) by A92, FUNCT_2:108, XBOOLE_1:1; then r < f . (s . k) by A123, FUNCT_2:15; hence r < (f /* s) . k by A17, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; end; end; hence r < (f /* s) . k ; ::_thesis: verum end; hence f /* s is divergent_to+infty by LIMFUNC1:def_4; ::_thesis: verum end; end; end; hence f /* s is divergent_to+infty ; ::_thesis: verum end; end; end; hence f /* s is divergent_to+infty ; ::_thesis: verum end; for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) by A11, LIMFUNC2:def_2; then for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) by A13, Th8; hence f is_divergent_to+infty_in x0 by A14, Def2; ::_thesis: verum end; theorem Th13: :: LIMFUNC3:13 for x0 being Real for f being PartFunc of REAL,REAL holds ( f is_divergent_to-infty_in x0 iff ( f is_left_divergent_to-infty_in x0 & f is_right_divergent_to-infty_in x0 ) ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL holds ( f is_divergent_to-infty_in x0 iff ( f is_left_divergent_to-infty_in x0 & f is_right_divergent_to-infty_in x0 ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_divergent_to-infty_in x0 iff ( f is_left_divergent_to-infty_in x0 & f is_right_divergent_to-infty_in x0 ) ) thus ( f is_divergent_to-infty_in x0 implies ( f is_left_divergent_to-infty_in x0 & f is_right_divergent_to-infty_in x0 ) ) ::_thesis: ( f is_left_divergent_to-infty_in x0 & f is_right_divergent_to-infty_in x0 implies f is_divergent_to-infty_in x0 ) proof assume A1: f is_divergent_to-infty_in x0 ; ::_thesis: ( f is_left_divergent_to-infty_in x0 & f is_right_divergent_to-infty_in x0 ) A2: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_f)_/\_(left_open_halfline_x0)_holds_ f_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom f) /\ (left_open_halfline x0) implies f /* s is divergent_to-infty ) assume that A3: s is convergent and A4: lim s = x0 and A5: rng s c= (dom f) /\ (left_open_halfline x0) ; ::_thesis: f /* s is divergent_to-infty rng s c= (dom f) \ {x0} by A5, Th1; hence f /* s is divergent_to-infty by A1, A3, A4, Def3; ::_thesis: verum end; A6: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_f)_/\_(right_open_halfline_x0)_holds_ f_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom f) /\ (right_open_halfline x0) implies f /* s is divergent_to-infty ) assume that A7: s is convergent and A8: lim s = x0 and A9: rng s c= (dom f) /\ (right_open_halfline x0) ; ::_thesis: f /* s is divergent_to-infty rng s c= (dom f) \ {x0} by A9, Th1; hence f /* s is divergent_to-infty by A1, A7, A8, Def3; ::_thesis: verum end; A10: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) by A1, Def3; then for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) by Th8; hence f is_left_divergent_to-infty_in x0 by A2, LIMFUNC2:def_3; ::_thesis: f is_right_divergent_to-infty_in x0 for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) by A10, Th8; hence f is_right_divergent_to-infty_in x0 by A6, LIMFUNC2:def_6; ::_thesis: verum end; assume that A11: f is_left_divergent_to-infty_in x0 and A12: f is_right_divergent_to-infty_in x0 ; ::_thesis: f is_divergent_to-infty_in x0 A13: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_f)_\_{x0}_holds_ f_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom f) \ {x0} implies f /* s is divergent_to-infty ) assume that A14: s is convergent and A15: lim s = x0 and A16: rng s c= (dom f) \ {x0} ; ::_thesis: f /* s is divergent_to-infty now__::_thesis:_f_/*_s_is_divergent_to-infty percases ( ex k being Element of NAT st for n being Element of NAT st k <= n holds s . n < x0 or for k being Element of NAT ex n being Element of NAT st ( k <= n & s . n >= x0 ) ) ; suppose ex k being Element of NAT st for n being Element of NAT st k <= n holds s . n < x0 ; ::_thesis: f /* s is divergent_to-infty then consider k being Element of NAT such that A17: for n being Element of NAT st k <= n holds s . n < x0 ; A18: rng s c= dom f by A16, XBOOLE_1:1; A19: rng (s ^\ k) c= (dom f) /\ (left_open_halfline x0) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (s ^\ k) or x in (dom f) /\ (left_open_halfline x0) ) assume x in rng (s ^\ k) ; ::_thesis: x in (dom f) /\ (left_open_halfline x0) then consider n being Element of NAT such that A20: (s ^\ k) . n = x by FUNCT_2:113; s . (n + k) < x0 by A17, NAT_1:12; then s . (n + k) in { g1 where g1 is Real : g1 < x0 } ; then s . (n + k) in left_open_halfline x0 by XXREAL_1:229; then A21: x in left_open_halfline x0 by A20, NAT_1:def_3; s . (n + k) in rng s by VALUED_0:28; then x in rng s by A20, NAT_1:def_3; hence x in (dom f) /\ (left_open_halfline x0) by A18, A21, XBOOLE_0:def_4; ::_thesis: verum end; A22: f /* (s ^\ k) = (f /* s) ^\ k by A16, VALUED_0:27, XBOOLE_1:1; lim (s ^\ k) = x0 by A14, A15, SEQ_4:20; then f /* (s ^\ k) is divergent_to-infty by A11, A14, A19, LIMFUNC2:def_3; hence f /* s is divergent_to-infty by A22, LIMFUNC1:7; ::_thesis: verum end; supposeA23: for k being Element of NAT ex n being Element of NAT st ( k <= n & s . n >= x0 ) ; ::_thesis: f /* s is divergent_to-infty now__::_thesis:_f_/*_s_is_divergent_to-infty percases ( ex k being Element of NAT st for n being Element of NAT st k <= n holds x0 < s . n or for k being Element of NAT ex n being Element of NAT st ( k <= n & x0 >= s . n ) ) ; suppose ex k being Element of NAT st for n being Element of NAT st k <= n holds x0 < s . n ; ::_thesis: f /* s is divergent_to-infty then consider k being Element of NAT such that A24: for n being Element of NAT st k <= n holds s . n > x0 ; A25: rng s c= dom f by A16, XBOOLE_1:1; A26: rng (s ^\ k) c= (dom f) /\ (right_open_halfline x0) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (s ^\ k) or x in (dom f) /\ (right_open_halfline x0) ) assume x in rng (s ^\ k) ; ::_thesis: x in (dom f) /\ (right_open_halfline x0) then consider n being Element of NAT such that A27: (s ^\ k) . n = x by FUNCT_2:113; x0 < s . (n + k) by A24, NAT_1:12; then s . (n + k) in { g1 where g1 is Real : x0 < g1 } ; then s . (n + k) in right_open_halfline x0 by XXREAL_1:230; then A28: x in right_open_halfline x0 by A27, NAT_1:def_3; s . (n + k) in rng s by VALUED_0:28; then x in rng s by A27, NAT_1:def_3; hence x in (dom f) /\ (right_open_halfline x0) by A25, A28, XBOOLE_0:def_4; ::_thesis: verum end; A29: f /* (s ^\ k) = (f /* s) ^\ k by A16, VALUED_0:27, XBOOLE_1:1; lim (s ^\ k) = x0 by A14, A15, SEQ_4:20; then f /* (s ^\ k) is divergent_to-infty by A12, A14, A26, LIMFUNC2:def_6; hence f /* s is divergent_to-infty by A29, LIMFUNC1:7; ::_thesis: verum end; supposeA30: for k being Element of NAT ex n being Element of NAT st ( k <= n & x0 >= s . n ) ; ::_thesis: f /* s is divergent_to-infty defpred S1[ set , set ] means for n, m being Element of NAT st $1 = n & $2 = m holds ( n < m & s . m < x0 & ( for k being Element of NAT st n < k & s . k < x0 holds m <= k ) ); defpred S2[ Element of NAT , set , set ] means S1[$2,$3]; defpred S3[ Nat] means s . $1 < x0; A31: now__::_thesis:_for_k_being_Element_of_NAT_ex_n_being_Element_of_NAT_st_ (_k_<=_n_&_s_._n_<_x0_) let k be Element of NAT ; ::_thesis: ex n being Element of NAT st ( k <= n & s . n < x0 ) consider n being Element of NAT such that A32: k <= n and A33: s . n <= x0 by A30; take n = n; ::_thesis: ( k <= n & s . n < x0 ) thus k <= n by A32; ::_thesis: s . n < x0 s . n in rng s by VALUED_0:28; then not s . n in {x0} by A16, XBOOLE_0:def_5; then s . n <> x0 by TARSKI:def_1; hence s . n < x0 by A33, XXREAL_0:1; ::_thesis: verum end; then ex m1 being Element of NAT st ( 0 <= m1 & s . m1 < x0 ) ; then A34: ex m being Nat st S3[m] ; consider M being Nat such that A35: ( S3[M] & ( for n being Nat st S3[n] holds M <= n ) ) from NAT_1:sch_5(A34); reconsider M9 = M as Element of NAT by ORDINAL1:def_12; A36: now__::_thesis:_for_n_being_Element_of_NAT_ex_m_being_Element_of_NAT_st_ (_n_<_m_&_s_._m_<_x0_) let n be Element of NAT ; ::_thesis: ex m being Element of NAT st ( n < m & s . m < x0 ) consider m being Element of NAT such that A37: n + 1 <= m and A38: s . m < x0 by A31; take m = m; ::_thesis: ( n < m & s . m < x0 ) thus ( n < m & s . m < x0 ) by A37, A38, NAT_1:13; ::_thesis: verum end; A39: for n, x being Element of NAT ex y being Element of NAT st S2[n,x,y] proof let n, x be Element of NAT ; ::_thesis: ex y being Element of NAT st S2[n,x,y] defpred S4[ Nat] means ( x < $1 & s . $1 < x0 ); ex m being Element of NAT st S4[m] by A36; then A40: ex m being Nat st S4[m] ; consider l being Nat such that A41: ( S4[l] & ( for k being Nat st S4[k] holds l <= k ) ) from NAT_1:sch_5(A40); take l ; ::_thesis: ( l is Element of REAL & l is Element of NAT & S2[n,x,l] ) l in NAT by ORDINAL1:def_12; hence ( l is Element of REAL & l is Element of NAT & S2[n,x,l] ) by A41; ::_thesis: verum end; consider F being Function of NAT,NAT such that A42: ( F . 0 = M9 & ( for n being Element of NAT holds S2[n,F . n,F . (n + 1)] ) ) from RECDEF_1:sch_2(A39); A43: rng F c= NAT by RELAT_1:def_19; then A44: rng F c= REAL by XBOOLE_1:1; A45: dom F = NAT by FUNCT_2:def_1; then reconsider F = F as Real_Sequence by A44, RELSET_1:4; A46: now__::_thesis:_for_n_being_Element_of_NAT_holds_F_._n_is_Element_of_NAT let n be Element of NAT ; ::_thesis: F . n is Element of NAT F . n in rng F by A45, FUNCT_1:def_3; hence F . n is Element of NAT by A43; ::_thesis: verum end; now__::_thesis:_for_n_being_Element_of_NAT_holds_F_._n_<_F_._(n_+_1) let n be Element of NAT ; ::_thesis: F . n < F . (n + 1) A47: F . (n + 1) is Element of NAT by A46; F . n is Element of NAT by A46; hence F . n < F . (n + 1) by A42, A47; ::_thesis: verum end; then reconsider F = F as V37() sequence of NAT by SEQM_3:def_6; A48: s * F is subsequence of s by VALUED_0:def_17; then rng (s * F) c= rng s by VALUED_0:21; then A49: rng (s * F) c= (dom f) \ {x0} by A16, XBOOLE_1:1; defpred S4[ Nat] means ( s . $1 < x0 & ( for m being Element of NAT holds F . m <> $1 ) ); A50: for n being Element of NAT st s . n < x0 holds ex m being Element of NAT st F . m = n proof assume ex n being Element of NAT st S4[n] ; ::_thesis: contradiction then A51: ex n being Nat st S4[n] ; consider M1 being Nat such that A52: ( S4[M1] & ( for n being Nat st S4[n] holds M1 <= n ) ) from NAT_1:sch_5(A51); defpred S5[ Nat] means ( $1 < M1 & s . $1 < x0 & ex m being Element of NAT st F . m = $1 ); A53: ex n being Nat st S5[n] proof take M ; ::_thesis: S5[M] A54: M <> M1 by A42, A52; M <= M1 by A35, A52; hence M < M1 by A54, XXREAL_0:1; ::_thesis: ( s . M < x0 & ex m being Element of NAT st F . m = M ) thus s . M < x0 by A35; ::_thesis: ex m being Element of NAT st F . m = M take 0 ; ::_thesis: F . 0 = M thus F . 0 = M by A42; ::_thesis: verum end; A55: for n being Nat st S5[n] holds n <= M1 ; consider MX being Nat such that A56: ( S5[MX] & ( for n being Nat st S5[n] holds n <= MX ) ) from NAT_1:sch_6(A55, A53); A57: for k being Element of NAT st MX < k & k < M1 holds s . k >= x0 proof given k being Element of NAT such that A58: MX < k and A59: k < M1 and A60: s . k < x0 ; ::_thesis: contradiction now__::_thesis:_contradiction percases ( ex m being Element of NAT st F . m = k or for m being Element of NAT holds F . m <> k ) ; suppose ex m being Element of NAT st F . m = k ; ::_thesis: contradiction hence contradiction by A56, A58, A59, A60; ::_thesis: verum end; suppose for m being Element of NAT holds F . m <> k ; ::_thesis: contradiction hence contradiction by A52, A59, A60; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; consider m being Element of NAT such that A61: F . m = MX by A56; M1 in NAT by ORDINAL1:def_12; then A62: F . (m + 1) <= M1 by A42, A52, A56, A61; A63: s . (F . (m + 1)) < x0 by A42, A61; A64: MX < F . (m + 1) by A42, A61; now__::_thesis:_not_F_._(m_+_1)_<>_M1 assume F . (m + 1) <> M1 ; ::_thesis: contradiction then F . (m + 1) < M1 by A62, XXREAL_0:1; hence contradiction by A57, A64, A63; ::_thesis: verum end; hence contradiction by A52; ::_thesis: verum end; defpred S5[ Nat] means s . $1 > x0; A65: now__::_thesis:_for_k_being_Element_of_NAT_ex_n_being_Element_of_NAT_st_ (_k_<=_n_&_s_._n_>_x0_) let k be Element of NAT ; ::_thesis: ex n being Element of NAT st ( k <= n & s . n > x0 ) consider n being Element of NAT such that A66: k <= n and A67: s . n >= x0 by A23; take n = n; ::_thesis: ( k <= n & s . n > x0 ) thus k <= n by A66; ::_thesis: s . n > x0 s . n in rng s by VALUED_0:28; then not s . n in {x0} by A16, XBOOLE_0:def_5; then s . n <> x0 by TARSKI:def_1; hence s . n > x0 by A67, XXREAL_0:1; ::_thesis: verum end; then ex mn being Element of NAT st ( 0 <= mn & s . mn > x0 ) ; then A68: ex m being Nat st S5[m] ; consider N being Nat such that A69: ( S5[N] & ( for n being Nat st S5[n] holds N <= n ) ) from NAT_1:sch_5(A68); A70: for n being Element of NAT holds (s * F) . n < x0 proof defpred S6[ Element of NAT ] means (s * F) . $1 < x0; A71: for k being Element of NAT st S6[k] holds S6[k + 1] proof let k be Element of NAT ; ::_thesis: ( S6[k] implies S6[k + 1] ) assume (s * F) . k < x0 ; ::_thesis: S6[k + 1] S1[F . k,F . (k + 1)] by A42; then s . (F . (k + 1)) < x0 ; hence S6[k + 1] by FUNCT_2:15; ::_thesis: verum end; A72: S6[ 0 ] by A35, A42, FUNCT_2:15; thus for k being Element of NAT holds S6[k] from NAT_1:sch_1(A72, A71); ::_thesis: verum end; A73: rng (s * F) c= (dom f) /\ (left_open_halfline x0) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (s * F) or x in (dom f) /\ (left_open_halfline x0) ) assume A74: x in rng (s * F) ; ::_thesis: x in (dom f) /\ (left_open_halfline x0) then consider n being Element of NAT such that A75: (s * F) . n = x by FUNCT_2:113; (s * F) . n < x0 by A70; then x in { g1 where g1 is Real : g1 < x0 } by A75; then A76: x in left_open_halfline x0 by XXREAL_1:229; x in dom f by A49, A74, XBOOLE_0:def_5; hence x in (dom f) /\ (left_open_halfline x0) by A76, XBOOLE_0:def_4; ::_thesis: verum end; defpred S6[ set , set ] means for n, m being Element of NAT st $1 = n & $2 = m holds ( n < m & s . m > x0 & ( for k being Element of NAT st n < k & s . k > x0 holds m <= k ) ); defpred S7[ Element of NAT , set , set ] means S6[$2,$3]; A77: s * F is convergent by A14, A48, SEQ_4:16; lim (s * F) = x0 by A14, A15, A48, SEQ_4:17; then A78: f /* (s * F) is divergent_to-infty by A11, A77, A73, LIMFUNC2:def_3; reconsider N9 = N as Element of NAT by ORDINAL1:def_12; A79: now__::_thesis:_for_n_being_Element_of_NAT_ex_m_being_Element_of_NAT_st_ (_n_<_m_&_s_._m_>_x0_) let n be Element of NAT ; ::_thesis: ex m being Element of NAT st ( n < m & s . m > x0 ) consider m being Element of NAT such that A80: n + 1 <= m and A81: s . m > x0 by A65; take m = m; ::_thesis: ( n < m & s . m > x0 ) thus ( n < m & s . m > x0 ) by A80, A81, NAT_1:13; ::_thesis: verum end; A82: for n, x being Element of NAT ex y being Element of NAT st S7[n,x,y] proof let n, x be Element of NAT ; ::_thesis: ex y being Element of NAT st S7[n,x,y] defpred S8[ Nat] means ( x < $1 & s . $1 > x0 ); ex m being Element of NAT st S8[m] by A79; then A83: ex m being Nat st S8[m] ; consider l being Nat such that A84: ( S8[l] & ( for k being Nat st S8[k] holds l <= k ) ) from NAT_1:sch_5(A83); take l ; ::_thesis: ( l is Element of REAL & l is Element of NAT & S7[n,x,l] ) l in NAT by ORDINAL1:def_12; hence ( l is Element of REAL & l is Element of NAT & S7[n,x,l] ) by A84; ::_thesis: verum end; consider G being Function of NAT,NAT such that A85: ( G . 0 = N9 & ( for n being Element of NAT holds S7[n,G . n,G . (n + 1)] ) ) from RECDEF_1:sch_2(A82); A86: rng G c= NAT by RELAT_1:def_19; then A87: rng G c= REAL by XBOOLE_1:1; A88: dom G = NAT by FUNCT_2:def_1; then reconsider G = G as Real_Sequence by A87, RELSET_1:4; A89: now__::_thesis:_for_n_being_Element_of_NAT_holds_G_._n_is_Element_of_NAT let n be Element of NAT ; ::_thesis: G . n is Element of NAT G . n in rng G by A88, FUNCT_1:def_3; hence G . n is Element of NAT by A86; ::_thesis: verum end; now__::_thesis:_for_n_being_Element_of_NAT_holds_G_._n_<_G_._(n_+_1) let n be Element of NAT ; ::_thesis: G . n < G . (n + 1) A90: G . (n + 1) is Element of NAT by A89; G . n is Element of NAT by A89; hence G . n < G . (n + 1) by A85, A90; ::_thesis: verum end; then reconsider G = G as V37() sequence of NAT by SEQM_3:def_6; A91: s * G is subsequence of s by VALUED_0:def_17; then rng (s * G) c= rng s by VALUED_0:21; then A92: rng (s * G) c= (dom f) \ {x0} by A16, XBOOLE_1:1; defpred S8[ Nat] means ( s . $1 > x0 & ( for m being Element of NAT holds G . m <> $1 ) ); A93: for n being Element of NAT st s . n > x0 holds ex m being Element of NAT st G . m = n proof assume ex n being Element of NAT st S8[n] ; ::_thesis: contradiction then A94: ex n being Nat st S8[n] ; consider N1 being Nat such that A95: ( S8[N1] & ( for n being Nat st S8[n] holds N1 <= n ) ) from NAT_1:sch_5(A94); defpred S9[ Nat] means ( $1 < N1 & s . $1 > x0 & ex m being Element of NAT st G . m = $1 ); A96: ex n being Nat st S9[n] proof take N ; ::_thesis: S9[N] A97: N <> N1 by A85, A95; N <= N1 by A69, A95; hence N < N1 by A97, XXREAL_0:1; ::_thesis: ( s . N > x0 & ex m being Element of NAT st G . m = N ) thus s . N > x0 by A69; ::_thesis: ex m being Element of NAT st G . m = N take 0 ; ::_thesis: G . 0 = N thus G . 0 = N by A85; ::_thesis: verum end; A98: for n being Nat st S9[n] holds n <= N1 ; consider NX being Nat such that A99: ( S9[NX] & ( for n being Nat st S9[n] holds n <= NX ) ) from NAT_1:sch_6(A98, A96); A100: for k being Element of NAT st NX < k & k < N1 holds s . k <= x0 proof given k being Element of NAT such that A101: NX < k and A102: k < N1 and A103: s . k > x0 ; ::_thesis: contradiction now__::_thesis:_contradiction percases ( ex m being Element of NAT st G . m = k or for m being Element of NAT holds G . m <> k ) ; suppose ex m being Element of NAT st G . m = k ; ::_thesis: contradiction hence contradiction by A99, A101, A102, A103; ::_thesis: verum end; suppose for m being Element of NAT holds G . m <> k ; ::_thesis: contradiction hence contradiction by A95, A102, A103; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; consider m being Element of NAT such that A104: G . m = NX by A99; N1 in NAT by ORDINAL1:def_12; then A105: G . (m + 1) <= N1 by A85, A95, A99, A104; A106: s . (G . (m + 1)) > x0 by A85, A104; A107: NX < G . (m + 1) by A85, A104; now__::_thesis:_not_G_._(m_+_1)_<>_N1 assume G . (m + 1) <> N1 ; ::_thesis: contradiction then G . (m + 1) < N1 by A105, XXREAL_0:1; hence contradiction by A100, A107, A106; ::_thesis: verum end; hence contradiction by A95; ::_thesis: verum end; A108: for n being Element of NAT holds (s * G) . n > x0 proof defpred S9[ Element of NAT ] means (s * G) . $1 > x0; A109: for k being Element of NAT st S9[k] holds S9[k + 1] proof let k be Element of NAT ; ::_thesis: ( S9[k] implies S9[k + 1] ) assume (s * G) . k > x0 ; ::_thesis: S9[k + 1] S6[G . k,G . (k + 1)] by A85; then s . (G . (k + 1)) > x0 ; hence S9[k + 1] by FUNCT_2:15; ::_thesis: verum end; A110: S9[ 0 ] by A69, A85, FUNCT_2:15; thus for k being Element of NAT holds S9[k] from NAT_1:sch_1(A110, A109); ::_thesis: verum end; A111: rng (s * G) c= (dom f) /\ (right_open_halfline x0) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (s * G) or x in (dom f) /\ (right_open_halfline x0) ) assume A112: x in rng (s * G) ; ::_thesis: x in (dom f) /\ (right_open_halfline x0) then consider n being Element of NAT such that A113: (s * G) . n = x by FUNCT_2:113; (s * G) . n > x0 by A108; then x in { g1 where g1 is Real : x0 < g1 } by A113; then A114: x in right_open_halfline x0 by XXREAL_1:230; x in dom f by A92, A112, XBOOLE_0:def_5; hence x in (dom f) /\ (right_open_halfline x0) by A114, XBOOLE_0:def_4; ::_thesis: verum end; A115: s * G is convergent by A14, A91, SEQ_4:16; lim (s * G) = x0 by A14, A15, A91, SEQ_4:17; then A116: f /* (s * G) is divergent_to-infty by A12, A115, A111, LIMFUNC2:def_6; now__::_thesis:_for_r_being_Real_ex_n_being_Element_of_NAT_st_ for_k_being_Element_of_NAT_st_n_<=_k_holds_ (f_/*_s)_._k_<_r let r be Real; ::_thesis: ex n being Element of NAT st for k being Element of NAT st n <= k holds (f /* s) . k < r consider n1 being Element of NAT such that A117: for k being Element of NAT st n1 <= k holds (f /* (s * F)) . k < r by A78, LIMFUNC1:def_5; consider n2 being Element of NAT such that A118: for k being Element of NAT st n2 <= k holds (f /* (s * G)) . k < r by A116, LIMFUNC1:def_5; take n = max ((F . n1),(G . n2)); ::_thesis: for k being Element of NAT st n <= k holds (f /* s) . k < r let k be Element of NAT ; ::_thesis: ( n <= k implies (f /* s) . k < r ) assume A119: n <= k ; ::_thesis: (f /* s) . k < r s . k in rng s by VALUED_0:28; then not s . k in {x0} by A16, XBOOLE_0:def_5; then A120: s . k <> x0 by TARSKI:def_1; now__::_thesis:_(f_/*_s)_._k_<_r percases ( s . k < x0 or s . k > x0 ) by A120, XXREAL_0:1; suppose s . k < x0 ; ::_thesis: (f /* s) . k < r then consider l being Element of NAT such that A121: k = F . l by A50; F . n1 <= n by XXREAL_0:25; then F . n1 <= k by A119, XXREAL_0:2; then l >= n1 by A121, SEQM_3:1; then (f /* (s * F)) . l < r by A117; then f . ((s * F) . l) < r by A49, FUNCT_2:108, XBOOLE_1:1; then f . (s . k) < r by A121, FUNCT_2:15; hence (f /* s) . k < r by A16, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; suppose s . k > x0 ; ::_thesis: (f /* s) . k < r then consider l being Element of NAT such that A122: k = G . l by A93; G . n2 <= n by XXREAL_0:25; then G . n2 <= k by A119, XXREAL_0:2; then l >= n2 by A122, SEQM_3:1; then (f /* (s * G)) . l < r by A118; then f . ((s * G) . l) < r by A92, FUNCT_2:108, XBOOLE_1:1; then f . (s . k) < r by A122, FUNCT_2:15; hence (f /* s) . k < r by A16, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; end; end; hence (f /* s) . k < r ; ::_thesis: verum end; hence f /* s is divergent_to-infty by LIMFUNC1:def_5; ::_thesis: verum end; end; end; hence f /* s is divergent_to-infty ; ::_thesis: verum end; end; end; hence f /* s is divergent_to-infty ; ::_thesis: verum end; now__::_thesis:_for_r1,_r2_being_Real_st_r1_<_x0_&_x0_<_r2_holds_ ex_g1,_g2_being_Real_st_ (_r1_<_g1_&_g1_<_x0_&_g1_in_dom_f_&_g2_<_r2_&_x0_<_g2_&_g2_in_dom_f_) let r1, r2 be Real; ::_thesis: ( r1 < x0 & x0 < r2 implies ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) assume that A123: r1 < x0 and A124: x0 < r2 ; ::_thesis: ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) consider g1 being Real such that A125: r1 < g1 and A126: g1 < x0 and A127: g1 in dom f by A11, A123, LIMFUNC2:def_3; consider g2 being Real such that A128: g2 < r2 and A129: x0 < g2 and A130: g2 in dom f by A12, A124, LIMFUNC2:def_6; take g1 = g1; ::_thesis: ex g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) take g2 = g2; ::_thesis: ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) thus ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) by A125, A126, A127, A128, A129, A130; ::_thesis: verum end; hence f is_divergent_to-infty_in x0 by A13, Def3; ::_thesis: verum end; theorem :: LIMFUNC3:14 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_divergent_to+infty_in x0 & f2 is_divergent_to+infty_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in (dom f1) /\ (dom f2) & g2 < r2 & x0 < g2 & g2 in (dom f1) /\ (dom f2) ) ) holds ( f1 + f2 is_divergent_to+infty_in x0 & f1 (#) f2 is_divergent_to+infty_in x0 ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_divergent_to+infty_in x0 & f2 is_divergent_to+infty_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in (dom f1) /\ (dom f2) & g2 < r2 & x0 < g2 & g2 in (dom f1) /\ (dom f2) ) ) holds ( f1 + f2 is_divergent_to+infty_in x0 & f1 (#) f2 is_divergent_to+infty_in x0 ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_divergent_to+infty_in x0 & f2 is_divergent_to+infty_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in (dom f1) /\ (dom f2) & g2 < r2 & x0 < g2 & g2 in (dom f1) /\ (dom f2) ) ) implies ( f1 + f2 is_divergent_to+infty_in x0 & f1 (#) f2 is_divergent_to+infty_in x0 ) ) assume that A1: f1 is_divergent_to+infty_in x0 and A2: f2 is_divergent_to+infty_in x0 and A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in (dom f1) /\ (dom f2) & g2 < r2 & x0 < g2 & g2 in (dom f1) /\ (dom f2) ) ; ::_thesis: ( f1 + f2 is_divergent_to+infty_in x0 & f1 (#) f2 is_divergent_to+infty_in x0 ) A4: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f1_+_f2))_\_{x0}_holds_ (f1_+_f2)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f1 + f2)) \ {x0} implies (f1 + f2) /* s is divergent_to+infty ) assume that A5: s is convergent and A6: lim s = x0 and A7: rng s c= (dom (f1 + f2)) \ {x0} ; ::_thesis: (f1 + f2) /* s is divergent_to+infty rng s c= (dom f2) \ {x0} by A7, Lm4; then A8: f2 /* s is divergent_to+infty by A2, A5, A6, Def2; rng s c= (dom f1) \ {x0} by A7, Lm4; then f1 /* s is divergent_to+infty by A1, A5, A6, Def2; then A9: (f1 /* s) + (f2 /* s) is divergent_to+infty by A8, LIMFUNC1:8; A10: dom (f1 + f2) = (dom f1) /\ (dom f2) by A7, Lm4; rng s c= dom (f1 + f2) by A7, Lm4; hence (f1 + f2) /* s is divergent_to+infty by A10, A9, RFUNCT_2:8; ::_thesis: verum end; A11: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f1_(#)_f2))_\_{x0}_holds_ (f1_(#)_f2)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f1 (#) f2)) \ {x0} implies (f1 (#) f2) /* s is divergent_to+infty ) assume that A12: s is convergent and A13: lim s = x0 and A14: rng s c= (dom (f1 (#) f2)) \ {x0} ; ::_thesis: (f1 (#) f2) /* s is divergent_to+infty rng s c= (dom f2) \ {x0} by A14, Lm2; then A15: f2 /* s is divergent_to+infty by A2, A12, A13, Def2; rng s c= (dom f1) \ {x0} by A14, Lm2; then f1 /* s is divergent_to+infty by A1, A12, A13, Def2; then A16: (f1 /* s) (#) (f2 /* s) is divergent_to+infty by A15, LIMFUNC1:10; A17: dom (f1 (#) f2) = (dom f1) /\ (dom f2) by A14, Lm2; rng s c= dom (f1 (#) f2) by A14, Lm2; hence (f1 (#) f2) /* s is divergent_to+infty by A17, A16, RFUNCT_2:8; ::_thesis: verum end; now__::_thesis:_for_r1,_r2_being_Real_st_r1_<_x0_&_x0_<_r2_holds_ ex_g1,_g2_being_Real_st_ (_r1_<_g1_&_g1_<_x0_&_g1_in_dom_(f1_+_f2)_&_g2_<_r2_&_x0_<_g2_&_g2_in_dom_(f1_+_f2)_) let r1, r2 be Real; ::_thesis: ( r1 < x0 & x0 < r2 implies ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 + f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 + f2) ) ) assume that A18: r1 < x0 and A19: x0 < r2 ; ::_thesis: ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 + f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 + f2) ) consider g1, g2 being Real such that A20: r1 < g1 and A21: g1 < x0 and A22: g1 in (dom f1) /\ (dom f2) and A23: g2 < r2 and A24: x0 < g2 and A25: g2 in (dom f1) /\ (dom f2) by A3, A18, A19; take g1 = g1; ::_thesis: ex g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 + f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 + f2) ) take g2 = g2; ::_thesis: ( r1 < g1 & g1 < x0 & g1 in dom (f1 + f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 + f2) ) thus ( r1 < g1 & g1 < x0 & g1 in dom (f1 + f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 + f2) ) by A20, A21, A22, A23, A24, A25, VALUED_1:def_1; ::_thesis: verum end; hence f1 + f2 is_divergent_to+infty_in x0 by A4, Def2; ::_thesis: f1 (#) f2 is_divergent_to+infty_in x0 now__::_thesis:_for_r1,_r2_being_Real_st_r1_<_x0_&_x0_<_r2_holds_ ex_g1,_g2_being_Real_st_ (_r1_<_g1_&_g1_<_x0_&_g1_in_dom_(f1_(#)_f2)_&_g2_<_r2_&_x0_<_g2_&_g2_in_dom_(f1_(#)_f2)_) let r1, r2 be Real; ::_thesis: ( r1 < x0 & x0 < r2 implies ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 (#) f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 (#) f2) ) ) assume that A26: r1 < x0 and A27: x0 < r2 ; ::_thesis: ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 (#) f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 (#) f2) ) consider g1, g2 being Real such that A28: r1 < g1 and A29: g1 < x0 and A30: g1 in (dom f1) /\ (dom f2) and A31: g2 < r2 and A32: x0 < g2 and A33: g2 in (dom f1) /\ (dom f2) by A3, A26, A27; take g1 = g1; ::_thesis: ex g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 (#) f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 (#) f2) ) take g2 = g2; ::_thesis: ( r1 < g1 & g1 < x0 & g1 in dom (f1 (#) f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 (#) f2) ) thus ( r1 < g1 & g1 < x0 & g1 in dom (f1 (#) f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 (#) f2) ) by A28, A29, A30, A31, A32, A33, VALUED_1:def_4; ::_thesis: verum end; hence f1 (#) f2 is_divergent_to+infty_in x0 by A11, Def2; ::_thesis: verum end; theorem :: LIMFUNC3:15 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_divergent_to-infty_in x0 & f2 is_divergent_to-infty_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in (dom f1) /\ (dom f2) & g2 < r2 & x0 < g2 & g2 in (dom f1) /\ (dom f2) ) ) holds ( f1 + f2 is_divergent_to-infty_in x0 & f1 (#) f2 is_divergent_to+infty_in x0 ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_divergent_to-infty_in x0 & f2 is_divergent_to-infty_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in (dom f1) /\ (dom f2) & g2 < r2 & x0 < g2 & g2 in (dom f1) /\ (dom f2) ) ) holds ( f1 + f2 is_divergent_to-infty_in x0 & f1 (#) f2 is_divergent_to+infty_in x0 ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_divergent_to-infty_in x0 & f2 is_divergent_to-infty_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in (dom f1) /\ (dom f2) & g2 < r2 & x0 < g2 & g2 in (dom f1) /\ (dom f2) ) ) implies ( f1 + f2 is_divergent_to-infty_in x0 & f1 (#) f2 is_divergent_to+infty_in x0 ) ) assume that A1: f1 is_divergent_to-infty_in x0 and A2: f2 is_divergent_to-infty_in x0 and A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in (dom f1) /\ (dom f2) & g2 < r2 & x0 < g2 & g2 in (dom f1) /\ (dom f2) ) ; ::_thesis: ( f1 + f2 is_divergent_to-infty_in x0 & f1 (#) f2 is_divergent_to+infty_in x0 ) A4: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f1_+_f2))_\_{x0}_holds_ (f1_+_f2)_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f1 + f2)) \ {x0} implies (f1 + f2) /* s is divergent_to-infty ) assume that A5: s is convergent and A6: lim s = x0 and A7: rng s c= (dom (f1 + f2)) \ {x0} ; ::_thesis: (f1 + f2) /* s is divergent_to-infty rng s c= (dom f2) \ {x0} by A7, Lm4; then A8: f2 /* s is divergent_to-infty by A2, A5, A6, Def3; rng s c= (dom f1) \ {x0} by A7, Lm4; then f1 /* s is divergent_to-infty by A1, A5, A6, Def3; then A9: (f1 /* s) + (f2 /* s) is divergent_to-infty by A8, LIMFUNC1:11; A10: dom (f1 + f2) = (dom f1) /\ (dom f2) by A7, Lm4; rng s c= dom (f1 + f2) by A7, Lm4; hence (f1 + f2) /* s is divergent_to-infty by A10, A9, RFUNCT_2:8; ::_thesis: verum end; A11: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f1_(#)_f2))_\_{x0}_holds_ (f1_(#)_f2)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f1 (#) f2)) \ {x0} implies (f1 (#) f2) /* s is divergent_to+infty ) assume that A12: s is convergent and A13: lim s = x0 and A14: rng s c= (dom (f1 (#) f2)) \ {x0} ; ::_thesis: (f1 (#) f2) /* s is divergent_to+infty rng s c= (dom f2) \ {x0} by A14, Lm2; then A15: f2 /* s is divergent_to-infty by A2, A12, A13, Def3; rng s c= (dom f1) \ {x0} by A14, Lm2; then f1 /* s is divergent_to-infty by A1, A12, A13, Def3; then A16: (f1 /* s) (#) (f2 /* s) is divergent_to+infty by A15, LIMFUNC1:24; A17: dom (f1 (#) f2) = (dom f1) /\ (dom f2) by A14, Lm2; rng s c= dom (f1 (#) f2) by A14, Lm2; hence (f1 (#) f2) /* s is divergent_to+infty by A17, A16, RFUNCT_2:8; ::_thesis: verum end; now__::_thesis:_for_r1,_r2_being_Real_st_r1_<_x0_&_x0_<_r2_holds_ ex_g1,_g2_being_Real_st_ (_r1_<_g1_&_g1_<_x0_&_g1_in_dom_(f1_+_f2)_&_g2_<_r2_&_x0_<_g2_&_g2_in_dom_(f1_+_f2)_) let r1, r2 be Real; ::_thesis: ( r1 < x0 & x0 < r2 implies ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 + f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 + f2) ) ) assume that A18: r1 < x0 and A19: x0 < r2 ; ::_thesis: ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 + f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 + f2) ) consider g1, g2 being Real such that A20: r1 < g1 and A21: g1 < x0 and A22: g1 in (dom f1) /\ (dom f2) and A23: g2 < r2 and A24: x0 < g2 and A25: g2 in (dom f1) /\ (dom f2) by A3, A18, A19; take g1 = g1; ::_thesis: ex g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 + f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 + f2) ) take g2 = g2; ::_thesis: ( r1 < g1 & g1 < x0 & g1 in dom (f1 + f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 + f2) ) thus ( r1 < g1 & g1 < x0 & g1 in dom (f1 + f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 + f2) ) by A20, A21, A22, A23, A24, A25, VALUED_1:def_1; ::_thesis: verum end; hence f1 + f2 is_divergent_to-infty_in x0 by A4, Def3; ::_thesis: f1 (#) f2 is_divergent_to+infty_in x0 now__::_thesis:_for_r1,_r2_being_Real_st_r1_<_x0_&_x0_<_r2_holds_ ex_g1,_g2_being_Real_st_ (_r1_<_g1_&_g1_<_x0_&_g1_in_dom_(f1_(#)_f2)_&_g2_<_r2_&_x0_<_g2_&_g2_in_dom_(f1_(#)_f2)_) let r1, r2 be Real; ::_thesis: ( r1 < x0 & x0 < r2 implies ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 (#) f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 (#) f2) ) ) assume that A26: r1 < x0 and A27: x0 < r2 ; ::_thesis: ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 (#) f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 (#) f2) ) consider g1, g2 being Real such that A28: r1 < g1 and A29: g1 < x0 and A30: g1 in (dom f1) /\ (dom f2) and A31: g2 < r2 and A32: x0 < g2 and A33: g2 in (dom f1) /\ (dom f2) by A3, A26, A27; take g1 = g1; ::_thesis: ex g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 (#) f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 (#) f2) ) take g2 = g2; ::_thesis: ( r1 < g1 & g1 < x0 & g1 in dom (f1 (#) f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 (#) f2) ) thus ( r1 < g1 & g1 < x0 & g1 in dom (f1 (#) f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 (#) f2) ) by A28, A29, A30, A31, A32, A33, VALUED_1:def_4; ::_thesis: verum end; hence f1 (#) f2 is_divergent_to+infty_in x0 by A11, Def2; ::_thesis: verum end; theorem :: LIMFUNC3:16 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_divergent_to+infty_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 + f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 + f2) ) ) & ex r being Real st ( 0 < r & f2 | (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) is bounded_below ) holds f1 + f2 is_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_divergent_to+infty_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 + f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 + f2) ) ) & ex r being Real st ( 0 < r & f2 | (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) is bounded_below ) holds f1 + f2 is_divergent_to+infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_divergent_to+infty_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 + f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 + f2) ) ) & ex r being Real st ( 0 < r & f2 | (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) is bounded_below ) implies f1 + f2 is_divergent_to+infty_in x0 ) assume that A1: f1 is_divergent_to+infty_in x0 and A2: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 + f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 + f2) ) ; ::_thesis: ( for r being Real holds ( not 0 < r or not f2 | (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) is bounded_below ) or f1 + f2 is_divergent_to+infty_in x0 ) given r being Real such that A3: 0 < r and A4: f2 | (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) is bounded_below ; ::_thesis: f1 + f2 is_divergent_to+infty_in x0 now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f1_+_f2))_\_{x0}_holds_ (f1_+_f2)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f1 + f2)) \ {x0} implies (f1 + f2) /* s is divergent_to+infty ) assume that A5: s is convergent and A6: lim s = x0 and A7: rng s c= (dom (f1 + f2)) \ {x0} ; ::_thesis: (f1 + f2) /* s is divergent_to+infty consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds ( x0 - r < s . n & s . n < x0 + r ) by A3, A5, A6, Th7; rng (s ^\ k) c= rng s by VALUED_0:21; then A9: rng (s ^\ k) c= (dom (f1 + f2)) \ {x0} by A7, XBOOLE_1:1; then A10: rng (s ^\ k) c= (dom f1) \ {x0} by Lm4; A11: rng (s ^\ k) c= dom f2 by A9, Lm4; now__::_thesis:_ex_r2_being_Element_of_REAL_st_ for_n_being_Element_of_NAT_holds_r2_<_(f2_/*_(s_^\_k))_._n consider r1 being real number such that A12: for g being set st g in (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) /\ (dom f2) holds r1 <= f2 . g by A4, RFUNCT_1:71; take r2 = r1 - 1; ::_thesis: for n being Element of NAT holds r2 < (f2 /* (s ^\ k)) . n let n be Element of NAT ; ::_thesis: r2 < (f2 /* (s ^\ k)) . n A13: k <= n + k by NAT_1:12; then s . (n + k) < x0 + r by A8; then A14: (s ^\ k) . n < x0 + r by NAT_1:def_3; x0 - r < s . (n + k) by A8, A13; then x0 - r < (s ^\ k) . n by NAT_1:def_3; then (s ^\ k) . n in { g2 where g2 is Real : ( x0 - r < g2 & g2 < x0 + r ) } by A14; then A15: (s ^\ k) . n in ].(x0 - r),(x0 + r).[ by RCOMP_1:def_2; A16: (s ^\ k) . n in rng (s ^\ k) by VALUED_0:28; then not (s ^\ k) . n in {x0} by A9, XBOOLE_0:def_5; then (s ^\ k) . n in ].(x0 - r),(x0 + r).[ \ {x0} by A15, XBOOLE_0:def_5; then (s ^\ k) . n in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ by A3, Th4; then (s ^\ k) . n in (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) /\ (dom f2) by A11, A16, XBOOLE_0:def_4; then r1 - 1 < (f2 . ((s ^\ k) . n)) - 0 by A12, XREAL_1:15; hence r2 < (f2 /* (s ^\ k)) . n by A11, FUNCT_2:108; ::_thesis: verum end; then A17: f2 /* (s ^\ k) is bounded_below by SEQ_2:def_4; lim (s ^\ k) = x0 by A5, A6, SEQ_4:20; then f1 /* (s ^\ k) is divergent_to+infty by A1, A5, A10, Def2; then A18: (f1 /* (s ^\ k)) + (f2 /* (s ^\ k)) is divergent_to+infty by A17, LIMFUNC1:9; A19: rng s c= dom (f1 + f2) by A7, Lm4; rng (s ^\ k) c= dom (f1 + f2) by A9, Lm4; then rng (s ^\ k) c= (dom f1) /\ (dom f2) by VALUED_1:def_1; then (f1 /* (s ^\ k)) + (f2 /* (s ^\ k)) = (f1 + f2) /* (s ^\ k) by RFUNCT_2:8 .= ((f1 + f2) /* s) ^\ k by A19, VALUED_0:27 ; hence (f1 + f2) /* s is divergent_to+infty by A18, LIMFUNC1:7; ::_thesis: verum end; hence f1 + f2 is_divergent_to+infty_in x0 by A2, Def2; ::_thesis: verum end; theorem :: LIMFUNC3:17 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_divergent_to+infty_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 (#) f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 (#) f2) ) ) & ex r, r1 being Real st ( 0 < r & 0 < r1 & ( for g being Real st g in (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds r1 <= f2 . g ) ) holds f1 (#) f2 is_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_divergent_to+infty_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 (#) f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 (#) f2) ) ) & ex r, r1 being Real st ( 0 < r & 0 < r1 & ( for g being Real st g in (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds r1 <= f2 . g ) ) holds f1 (#) f2 is_divergent_to+infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_divergent_to+infty_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 (#) f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 (#) f2) ) ) & ex r, r1 being Real st ( 0 < r & 0 < r1 & ( for g being Real st g in (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds r1 <= f2 . g ) ) implies f1 (#) f2 is_divergent_to+infty_in x0 ) assume that A1: f1 is_divergent_to+infty_in x0 and A2: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 (#) f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 (#) f2) ) ; ::_thesis: ( for r, r1 being Real holds ( not 0 < r or not 0 < r1 or ex g being Real st ( g in (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & not r1 <= f2 . g ) ) or f1 (#) f2 is_divergent_to+infty_in x0 ) given r, t being Real such that A3: 0 < r and A4: 0 < t and A5: for g being Real st g in (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds t <= f2 . g ; ::_thesis: f1 (#) f2 is_divergent_to+infty_in x0 now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f1_(#)_f2))_\_{x0}_holds_ (f1_(#)_f2)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f1 (#) f2)) \ {x0} implies (f1 (#) f2) /* s is divergent_to+infty ) assume that A6: s is convergent and A7: lim s = x0 and A8: rng s c= (dom (f1 (#) f2)) \ {x0} ; ::_thesis: (f1 (#) f2) /* s is divergent_to+infty consider k being Element of NAT such that A9: for n being Element of NAT st k <= n holds ( x0 - r < s . n & s . n < x0 + r ) by A3, A6, A7, Th7; A10: rng s c= dom (f1 (#) f2) by A8, Lm2; A11: dom (f1 (#) f2) = (dom f1) /\ (dom f2) by A8, Lm2; rng (s ^\ k) c= rng s by VALUED_0:21; then A12: rng (s ^\ k) c= (dom (f1 (#) f2)) \ {x0} by A8, XBOOLE_1:1; then A13: rng (s ^\ k) c= (dom f1) \ {x0} by Lm2; A14: rng (s ^\ k) c= dom f2 by A12, Lm2; A15: now__::_thesis:_(_0_<_t_&_(_for_n_being_Element_of_NAT_holds_t_<=_(f2_/*_(s_^\_k))_._n_)_) thus 0 < t by A4; ::_thesis: for n being Element of NAT holds t <= (f2 /* (s ^\ k)) . n let n be Element of NAT ; ::_thesis: t <= (f2 /* (s ^\ k)) . n A16: k <= n + k by NAT_1:12; then s . (n + k) < x0 + r by A9; then A17: (s ^\ k) . n < x0 + r by NAT_1:def_3; x0 - r < s . (n + k) by A9, A16; then x0 - r < (s ^\ k) . n by NAT_1:def_3; then (s ^\ k) . n in { g2 where g2 is Real : ( x0 - r < g2 & g2 < x0 + r ) } by A17; then A18: (s ^\ k) . n in ].(x0 - r),(x0 + r).[ by RCOMP_1:def_2; A19: (s ^\ k) . n in rng (s ^\ k) by VALUED_0:28; then not (s ^\ k) . n in {x0} by A12, XBOOLE_0:def_5; then (s ^\ k) . n in ].(x0 - r),(x0 + r).[ \ {x0} by A18, XBOOLE_0:def_5; then (s ^\ k) . n in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ by A3, Th4; then (s ^\ k) . n in (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) by A14, A19, XBOOLE_0:def_4; then t <= f2 . ((s ^\ k) . n) by A5; hence t <= (f2 /* (s ^\ k)) . n by A14, FUNCT_2:108; ::_thesis: verum end; lim (s ^\ k) = x0 by A6, A7, SEQ_4:20; then f1 /* (s ^\ k) is divergent_to+infty by A1, A6, A13, Def2; then A20: (f1 /* (s ^\ k)) (#) (f2 /* (s ^\ k)) is divergent_to+infty by A15, LIMFUNC1:22; rng (s ^\ k) c= dom (f1 (#) f2) by A12, Lm2; then (f1 /* (s ^\ k)) (#) (f2 /* (s ^\ k)) = (f1 (#) f2) /* (s ^\ k) by A11, RFUNCT_2:8 .= ((f1 (#) f2) /* s) ^\ k by A10, VALUED_0:27 ; hence (f1 (#) f2) /* s is divergent_to+infty by A20, LIMFUNC1:7; ::_thesis: verum end; hence f1 (#) f2 is_divergent_to+infty_in x0 by A2, Def2; ::_thesis: verum end; theorem :: LIMFUNC3:18 for x0, r being Real for f being PartFunc of REAL,REAL holds ( ( f is_divergent_to+infty_in x0 & r > 0 implies r (#) f is_divergent_to+infty_in x0 ) & ( f is_divergent_to+infty_in x0 & r < 0 implies r (#) f is_divergent_to-infty_in x0 ) & ( f is_divergent_to-infty_in x0 & r > 0 implies r (#) f is_divergent_to-infty_in x0 ) & ( f is_divergent_to-infty_in x0 & r < 0 implies r (#) f is_divergent_to+infty_in x0 ) ) proof let x0, r be Real; ::_thesis: for f being PartFunc of REAL,REAL holds ( ( f is_divergent_to+infty_in x0 & r > 0 implies r (#) f is_divergent_to+infty_in x0 ) & ( f is_divergent_to+infty_in x0 & r < 0 implies r (#) f is_divergent_to-infty_in x0 ) & ( f is_divergent_to-infty_in x0 & r > 0 implies r (#) f is_divergent_to-infty_in x0 ) & ( f is_divergent_to-infty_in x0 & r < 0 implies r (#) f is_divergent_to+infty_in x0 ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( ( f is_divergent_to+infty_in x0 & r > 0 implies r (#) f is_divergent_to+infty_in x0 ) & ( f is_divergent_to+infty_in x0 & r < 0 implies r (#) f is_divergent_to-infty_in x0 ) & ( f is_divergent_to-infty_in x0 & r > 0 implies r (#) f is_divergent_to-infty_in x0 ) & ( f is_divergent_to-infty_in x0 & r < 0 implies r (#) f is_divergent_to+infty_in x0 ) ) thus ( f is_divergent_to+infty_in x0 & r > 0 implies r (#) f is_divergent_to+infty_in x0 ) ::_thesis: ( ( f is_divergent_to+infty_in x0 & r < 0 implies r (#) f is_divergent_to-infty_in x0 ) & ( f is_divergent_to-infty_in x0 & r > 0 implies r (#) f is_divergent_to-infty_in x0 ) & ( f is_divergent_to-infty_in x0 & r < 0 implies r (#) f is_divergent_to+infty_in x0 ) ) proof assume that A1: f is_divergent_to+infty_in x0 and A2: r > 0 ; ::_thesis: r (#) f is_divergent_to+infty_in x0 thus for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) :: according to LIMFUNC3:def_2 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom (r (#) f)) \ {x0} holds (r (#) f) /* seq is divergent_to+infty proof let r1, r2 be Real; ::_thesis: ( r1 < x0 & x0 < r2 implies ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) ) assume that A3: r1 < x0 and A4: x0 < r2 ; ::_thesis: ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) consider g1, g2 being Real such that A5: r1 < g1 and A6: g1 < x0 and A7: g1 in dom f and A8: g2 < r2 and A9: x0 < g2 and A10: g2 in dom f by A1, A3, A4, Def2; take g1 ; ::_thesis: ex g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) take g2 ; ::_thesis: ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) thus ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) by A5, A6, A7, A8, A9, A10, VALUED_1:def_5; ::_thesis: verum end; let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (r (#) f)) \ {x0} implies (r (#) f) /* seq is divergent_to+infty ) assume that A11: seq is convergent and A12: lim seq = x0 and A13: rng seq c= (dom (r (#) f)) \ {x0} ; ::_thesis: (r (#) f) /* seq is divergent_to+infty A14: rng seq c= (dom f) \ {x0} by A13, VALUED_1:def_5; then f /* seq is divergent_to+infty by A1, A11, A12, Def2; then r (#) (f /* seq) is divergent_to+infty by A2, LIMFUNC1:13; hence (r (#) f) /* seq is divergent_to+infty by A14, RFUNCT_2:9, XBOOLE_1:1; ::_thesis: verum end; thus ( f is_divergent_to+infty_in x0 & r < 0 implies r (#) f is_divergent_to-infty_in x0 ) ::_thesis: ( ( f is_divergent_to-infty_in x0 & r > 0 implies r (#) f is_divergent_to-infty_in x0 ) & ( f is_divergent_to-infty_in x0 & r < 0 implies r (#) f is_divergent_to+infty_in x0 ) ) proof assume that A15: f is_divergent_to+infty_in x0 and A16: r < 0 ; ::_thesis: r (#) f is_divergent_to-infty_in x0 thus for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) :: according to LIMFUNC3:def_3 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom (r (#) f)) \ {x0} holds (r (#) f) /* seq is divergent_to-infty proof let r1, r2 be Real; ::_thesis: ( r1 < x0 & x0 < r2 implies ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) ) assume that A17: r1 < x0 and A18: x0 < r2 ; ::_thesis: ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) consider g1, g2 being Real such that A19: r1 < g1 and A20: g1 < x0 and A21: g1 in dom f and A22: g2 < r2 and A23: x0 < g2 and A24: g2 in dom f by A15, A17, A18, Def2; take g1 ; ::_thesis: ex g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) take g2 ; ::_thesis: ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) thus ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) by A19, A20, A21, A22, A23, A24, VALUED_1:def_5; ::_thesis: verum end; let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (r (#) f)) \ {x0} implies (r (#) f) /* seq is divergent_to-infty ) assume that A25: seq is convergent and A26: lim seq = x0 and A27: rng seq c= (dom (r (#) f)) \ {x0} ; ::_thesis: (r (#) f) /* seq is divergent_to-infty A28: rng seq c= (dom f) \ {x0} by A27, VALUED_1:def_5; then f /* seq is divergent_to+infty by A15, A25, A26, Def2; then r (#) (f /* seq) is divergent_to-infty by A16, LIMFUNC1:13; hence (r (#) f) /* seq is divergent_to-infty by A28, RFUNCT_2:9, XBOOLE_1:1; ::_thesis: verum end; thus ( f is_divergent_to-infty_in x0 & r > 0 implies r (#) f is_divergent_to-infty_in x0 ) ::_thesis: ( f is_divergent_to-infty_in x0 & r < 0 implies r (#) f is_divergent_to+infty_in x0 ) proof assume that A29: f is_divergent_to-infty_in x0 and A30: r > 0 ; ::_thesis: r (#) f is_divergent_to-infty_in x0 thus for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) :: according to LIMFUNC3:def_3 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom (r (#) f)) \ {x0} holds (r (#) f) /* seq is divergent_to-infty proof let r1, r2 be Real; ::_thesis: ( r1 < x0 & x0 < r2 implies ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) ) assume that A31: r1 < x0 and A32: x0 < r2 ; ::_thesis: ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) consider g1, g2 being Real such that A33: r1 < g1 and A34: g1 < x0 and A35: g1 in dom f and A36: g2 < r2 and A37: x0 < g2 and A38: g2 in dom f by A29, A31, A32, Def3; take g1 ; ::_thesis: ex g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) take g2 ; ::_thesis: ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) thus ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) by A33, A34, A35, A36, A37, A38, VALUED_1:def_5; ::_thesis: verum end; let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (r (#) f)) \ {x0} implies (r (#) f) /* seq is divergent_to-infty ) assume that A39: seq is convergent and A40: lim seq = x0 and A41: rng seq c= (dom (r (#) f)) \ {x0} ; ::_thesis: (r (#) f) /* seq is divergent_to-infty A42: rng seq c= (dom f) \ {x0} by A41, VALUED_1:def_5; then f /* seq is divergent_to-infty by A29, A39, A40, Def3; then r (#) (f /* seq) is divergent_to-infty by A30, LIMFUNC1:14; hence (r (#) f) /* seq is divergent_to-infty by A42, RFUNCT_2:9, XBOOLE_1:1; ::_thesis: verum end; assume that A43: f is_divergent_to-infty_in x0 and A44: r < 0 ; ::_thesis: r (#) f is_divergent_to+infty_in x0 thus for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) :: according to LIMFUNC3:def_2 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom (r (#) f)) \ {x0} holds (r (#) f) /* seq is divergent_to+infty proof let r1, r2 be Real; ::_thesis: ( r1 < x0 & x0 < r2 implies ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) ) assume that A45: r1 < x0 and A46: x0 < r2 ; ::_thesis: ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) consider g1, g2 being Real such that A47: r1 < g1 and A48: g1 < x0 and A49: g1 in dom f and A50: g2 < r2 and A51: x0 < g2 and A52: g2 in dom f by A43, A45, A46, Def3; take g1 ; ::_thesis: ex g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) take g2 ; ::_thesis: ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) thus ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) by A47, A48, A49, A50, A51, A52, VALUED_1:def_5; ::_thesis: verum end; let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (r (#) f)) \ {x0} implies (r (#) f) /* seq is divergent_to+infty ) assume that A53: seq is convergent and A54: lim seq = x0 and A55: rng seq c= (dom (r (#) f)) \ {x0} ; ::_thesis: (r (#) f) /* seq is divergent_to+infty A56: rng seq c= (dom f) \ {x0} by A55, VALUED_1:def_5; then f /* seq is divergent_to-infty by A43, A53, A54, Def3; then r (#) (f /* seq) is divergent_to+infty by A44, LIMFUNC1:14; hence (r (#) f) /* seq is divergent_to+infty by A56, RFUNCT_2:9, XBOOLE_1:1; ::_thesis: verum end; theorem :: LIMFUNC3:19 for x0 being Real for f being PartFunc of REAL,REAL st ( f is_divergent_to+infty_in x0 or f is_divergent_to-infty_in x0 ) holds abs f is_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st ( f is_divergent_to+infty_in x0 or f is_divergent_to-infty_in x0 ) holds abs f is_divergent_to+infty_in x0 let f be PartFunc of REAL,REAL; ::_thesis: ( ( f is_divergent_to+infty_in x0 or f is_divergent_to-infty_in x0 ) implies abs f is_divergent_to+infty_in x0 ) assume A1: ( f is_divergent_to+infty_in x0 or f is_divergent_to-infty_in x0 ) ; ::_thesis: abs f is_divergent_to+infty_in x0 now__::_thesis:_abs_f_is_divergent_to+infty_in_x0 percases ( f is_divergent_to+infty_in x0 or f is_divergent_to-infty_in x0 ) by A1; supposeA2: f is_divergent_to+infty_in x0 ; ::_thesis: abs f is_divergent_to+infty_in x0 A3: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(abs_f))_\_{x0}_holds_ (abs_f)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (abs f)) \ {x0} implies (abs f) /* seq is divergent_to+infty ) assume that A4: seq is convergent and A5: lim seq = x0 and A6: rng seq c= (dom (abs f)) \ {x0} ; ::_thesis: (abs f) /* seq is divergent_to+infty A7: rng seq c= (dom f) \ {x0} by A6, VALUED_1:def_11; then f /* seq is divergent_to+infty by A2, A4, A5, Def2; then A8: abs (f /* seq) is divergent_to+infty by LIMFUNC1:25; rng seq c= dom f by A7, XBOOLE_1:1; hence (abs f) /* seq is divergent_to+infty by A8, RFUNCT_2:10; ::_thesis: verum end; now__::_thesis:_for_r1,_r2_being_Real_st_r1_<_x0_&_x0_<_r2_holds_ ex_g1,_g2_being_Real_st_ (_r1_<_g1_&_g1_<_x0_&_g1_in_dom_(abs_f)_&_g2_<_r2_&_x0_<_g2_&_g2_in_dom_(abs_f)_) let r1, r2 be Real; ::_thesis: ( r1 < x0 & x0 < r2 implies ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (abs f) & g2 < r2 & x0 < g2 & g2 in dom (abs f) ) ) assume that A9: r1 < x0 and A10: x0 < r2 ; ::_thesis: ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (abs f) & g2 < r2 & x0 < g2 & g2 in dom (abs f) ) consider g1, g2 being Real such that A11: r1 < g1 and A12: g1 < x0 and A13: g1 in dom f and A14: g2 < r2 and A15: x0 < g2 and A16: g2 in dom f by A2, A9, A10, Def2; take g1 = g1; ::_thesis: ex g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (abs f) & g2 < r2 & x0 < g2 & g2 in dom (abs f) ) take g2 = g2; ::_thesis: ( r1 < g1 & g1 < x0 & g1 in dom (abs f) & g2 < r2 & x0 < g2 & g2 in dom (abs f) ) thus ( r1 < g1 & g1 < x0 & g1 in dom (abs f) & g2 < r2 & x0 < g2 & g2 in dom (abs f) ) by A11, A12, A13, A14, A15, A16, VALUED_1:def_11; ::_thesis: verum end; hence abs f is_divergent_to+infty_in x0 by A3, Def2; ::_thesis: verum end; supposeA17: f is_divergent_to-infty_in x0 ; ::_thesis: abs f is_divergent_to+infty_in x0 A18: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(abs_f))_\_{x0}_holds_ (abs_f)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (abs f)) \ {x0} implies (abs f) /* seq is divergent_to+infty ) assume that A19: seq is convergent and A20: lim seq = x0 and A21: rng seq c= (dom (abs f)) \ {x0} ; ::_thesis: (abs f) /* seq is divergent_to+infty A22: rng seq c= (dom f) \ {x0} by A21, VALUED_1:def_11; then f /* seq is divergent_to-infty by A17, A19, A20, Def3; then A23: abs (f /* seq) is divergent_to+infty by LIMFUNC1:25; rng seq c= dom f by A22, XBOOLE_1:1; hence (abs f) /* seq is divergent_to+infty by A23, RFUNCT_2:10; ::_thesis: verum end; now__::_thesis:_for_r1,_r2_being_Real_st_r1_<_x0_&_x0_<_r2_holds_ ex_g1,_g2_being_Real_st_ (_r1_<_g1_&_g1_<_x0_&_g1_in_dom_(abs_f)_&_g2_<_r2_&_x0_<_g2_&_g2_in_dom_(abs_f)_) let r1, r2 be Real; ::_thesis: ( r1 < x0 & x0 < r2 implies ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (abs f) & g2 < r2 & x0 < g2 & g2 in dom (abs f) ) ) assume that A24: r1 < x0 and A25: x0 < r2 ; ::_thesis: ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (abs f) & g2 < r2 & x0 < g2 & g2 in dom (abs f) ) consider g1, g2 being Real such that A26: r1 < g1 and A27: g1 < x0 and A28: g1 in dom f and A29: g2 < r2 and A30: x0 < g2 and A31: g2 in dom f by A17, A24, A25, Def3; take g1 = g1; ::_thesis: ex g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (abs f) & g2 < r2 & x0 < g2 & g2 in dom (abs f) ) take g2 = g2; ::_thesis: ( r1 < g1 & g1 < x0 & g1 in dom (abs f) & g2 < r2 & x0 < g2 & g2 in dom (abs f) ) thus ( r1 < g1 & g1 < x0 & g1 in dom (abs f) & g2 < r2 & x0 < g2 & g2 in dom (abs f) ) by A26, A27, A28, A29, A30, A31, VALUED_1:def_11; ::_thesis: verum end; hence abs f is_divergent_to+infty_in x0 by A18, Def2; ::_thesis: verum end; end; end; hence abs f is_divergent_to+infty_in x0 ; ::_thesis: verum end; theorem Th20: :: LIMFUNC3:20 for x0 being Real for f being PartFunc of REAL,REAL st ex r being Real st ( f | ].(x0 - r),x0.[ is non-decreasing & f | ].x0,(x0 + r).[ is non-increasing & not f | ].(x0 - r),x0.[ is bounded_above & not f | ].x0,(x0 + r).[ is bounded_above ) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) holds f is_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st ex r being Real st ( f | ].(x0 - r),x0.[ is non-decreasing & f | ].x0,(x0 + r).[ is non-increasing & not f | ].(x0 - r),x0.[ is bounded_above & not f | ].x0,(x0 + r).[ is bounded_above ) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) holds f is_divergent_to+infty_in x0 let f be PartFunc of REAL,REAL; ::_thesis: ( ex r being Real st ( f | ].(x0 - r),x0.[ is non-decreasing & f | ].x0,(x0 + r).[ is non-increasing & not f | ].(x0 - r),x0.[ is bounded_above & not f | ].x0,(x0 + r).[ is bounded_above ) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) implies f is_divergent_to+infty_in x0 ) given r being Real such that A1: f | ].(x0 - r),x0.[ is non-decreasing and A2: f | ].x0,(x0 + r).[ is non-increasing and A3: not f | ].(x0 - r),x0.[ is bounded_above and A4: not f | ].x0,(x0 + r).[ is bounded_above ; ::_thesis: ( ex r1, r2 being Real st ( r1 < x0 & x0 < r2 & ( for g1, g2 being Real holds ( not r1 < g1 or not g1 < x0 or not g1 in dom f or not g2 < r2 or not x0 < g2 or not g2 in dom f ) ) ) or f is_divergent_to+infty_in x0 ) assume A5: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ; ::_thesis: f is_divergent_to+infty_in x0 then for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) by Th8; then A6: f is_right_divergent_to+infty_in x0 by A2, A4, LIMFUNC2:29; for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) by A5, Th8; then f is_left_divergent_to+infty_in x0 by A1, A3, LIMFUNC2:25; hence f is_divergent_to+infty_in x0 by A6, Th12; ::_thesis: verum end; theorem :: LIMFUNC3:21 for x0 being Real for f being PartFunc of REAL,REAL st ex r being Real st ( 0 < r & f | ].(x0 - r),x0.[ is increasing & f | ].x0,(x0 + r).[ is decreasing & not f | ].(x0 - r),x0.[ is bounded_above & not f | ].x0,(x0 + r).[ is bounded_above ) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) holds f is_divergent_to+infty_in x0 by Th20; theorem Th22: :: LIMFUNC3:22 for x0 being Real for f being PartFunc of REAL,REAL st ex r being Real st ( f | ].(x0 - r),x0.[ is non-increasing & f | ].x0,(x0 + r).[ is non-decreasing & not f | ].(x0 - r),x0.[ is bounded_below & not f | ].x0,(x0 + r).[ is bounded_below ) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) holds f is_divergent_to-infty_in x0 proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st ex r being Real st ( f | ].(x0 - r),x0.[ is non-increasing & f | ].x0,(x0 + r).[ is non-decreasing & not f | ].(x0 - r),x0.[ is bounded_below & not f | ].x0,(x0 + r).[ is bounded_below ) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) holds f is_divergent_to-infty_in x0 let f be PartFunc of REAL,REAL; ::_thesis: ( ex r being Real st ( f | ].(x0 - r),x0.[ is non-increasing & f | ].x0,(x0 + r).[ is non-decreasing & not f | ].(x0 - r),x0.[ is bounded_below & not f | ].x0,(x0 + r).[ is bounded_below ) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) implies f is_divergent_to-infty_in x0 ) given r being Real such that A1: f | ].(x0 - r),x0.[ is non-increasing and A2: f | ].x0,(x0 + r).[ is non-decreasing and A3: not f | ].(x0 - r),x0.[ is bounded_below and A4: not f | ].x0,(x0 + r).[ is bounded_below ; ::_thesis: ( ex r1, r2 being Real st ( r1 < x0 & x0 < r2 & ( for g1, g2 being Real holds ( not r1 < g1 or not g1 < x0 or not g1 in dom f or not g2 < r2 or not x0 < g2 or not g2 in dom f ) ) ) or f is_divergent_to-infty_in x0 ) assume A5: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ; ::_thesis: f is_divergent_to-infty_in x0 then for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) by Th8; then A6: f is_right_divergent_to-infty_in x0 by A2, A4, LIMFUNC2:31; for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) by A5, Th8; then f is_left_divergent_to-infty_in x0 by A1, A3, LIMFUNC2:27; hence f is_divergent_to-infty_in x0 by A6, Th13; ::_thesis: verum end; theorem :: LIMFUNC3:23 for x0 being Real for f being PartFunc of REAL,REAL st ex r being Real st ( 0 < r & f | ].(x0 - r),x0.[ is decreasing & f | ].x0,(x0 + r).[ is increasing & not f | ].(x0 - r),x0.[ is bounded_below & not f | ].x0,(x0 + r).[ is bounded_below ) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) holds f is_divergent_to-infty_in x0 by Th22; theorem Th24: :: LIMFUNC3:24 for x0 being Real for f1, f being PartFunc of REAL,REAL st f1 is_divergent_to+infty_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ex r being Real st ( 0 < r & (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f1 . g <= f . g ) ) holds f is_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f1, f being PartFunc of REAL,REAL st f1 is_divergent_to+infty_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ex r being Real st ( 0 < r & (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f1 . g <= f . g ) ) holds f is_divergent_to+infty_in x0 let f1, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is_divergent_to+infty_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ex r being Real st ( 0 < r & (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f1 . g <= f . g ) ) implies f is_divergent_to+infty_in x0 ) assume that A1: f1 is_divergent_to+infty_in x0 and A2: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ; ::_thesis: ( for r being Real holds ( not 0 < r or not (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) or ex g being Real st ( g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & not f1 . g <= f . g ) ) or f is_divergent_to+infty_in x0 ) given r being Real such that A3: 0 < r and A4: (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) and A5: for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f1 . g <= f . g ; ::_thesis: f is_divergent_to+infty_in x0 thus for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) by A2; :: according to LIMFUNC3:def_2 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} holds f /* seq is divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom f) \ {x0} implies f /* s is divergent_to+infty ) assume that A6: s is convergent and A7: lim s = x0 and A8: rng s c= (dom f) \ {x0} ; ::_thesis: f /* s is divergent_to+infty consider k being Element of NAT such that A9: for n being Element of NAT st k <= n holds ( x0 - r < s . n & s . n < x0 + r ) by A3, A6, A7, Th7; A10: rng (s ^\ k) c= rng s by VALUED_0:21; then A11: rng (s ^\ k) c= (dom f) \ {x0} by A8, XBOOLE_1:1; now__::_thesis:_for_x_being_set_st_x_in_rng_(s_^\_k)_holds_ x_in_].(x0_-_r),x0.[_\/_].x0,(x0_+_r).[ let x be set ; ::_thesis: ( x in rng (s ^\ k) implies x in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ ) assume x in rng (s ^\ k) ; ::_thesis: x in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ then consider n being Element of NAT such that A12: (s ^\ k) . n = x by FUNCT_2:113; A13: k <= n + k by NAT_1:12; then s . (n + k) < x0 + r by A9; then A14: (s ^\ k) . n < x0 + r by NAT_1:def_3; x0 - r < s . (n + k) by A9, A13; then x0 - r < (s ^\ k) . n by NAT_1:def_3; then (s ^\ k) . n in { g1 where g1 is Real : ( x0 - r < g1 & g1 < x0 + r ) } by A14; then A15: (s ^\ k) . n in ].(x0 - r),(x0 + r).[ by RCOMP_1:def_2; (s ^\ k) . n in rng (s ^\ k) by VALUED_0:28; then not (s ^\ k) . n in {x0} by A11, XBOOLE_0:def_5; then (s ^\ k) . n in ].(x0 - r),(x0 + r).[ \ {x0} by A15, XBOOLE_0:def_5; hence x in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ by A3, A12, Th4; ::_thesis: verum end; then A16: rng (s ^\ k) c= ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ by TARSKI:def_3; A17: rng s c= dom f by A8, XBOOLE_1:1; then rng (s ^\ k) c= dom f by A10, XBOOLE_1:1; then A18: rng (s ^\ k) c= (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) by A16, XBOOLE_1:19; then A19: rng (s ^\ k) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) by A4, XBOOLE_1:1; A20: now__::_thesis:_for_n_being_Element_of_NAT_holds_(f1_/*_(s_^\_k))_._n_<=_(f_/*_(s_^\_k))_._n let n be Element of NAT ; ::_thesis: (f1 /* (s ^\ k)) . n <= (f /* (s ^\ k)) . n (s ^\ k) . n in rng (s ^\ k) by VALUED_0:28; then f1 . ((s ^\ k) . n) <= f . ((s ^\ k) . n) by A5, A18; then (f1 /* (s ^\ k)) . n <= f . ((s ^\ k) . n) by A19, FUNCT_2:108, XBOOLE_1:18; hence (f1 /* (s ^\ k)) . n <= (f /* (s ^\ k)) . n by A17, A10, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; A21: rng (s ^\ k) c= dom f1 by A19, XBOOLE_1:18; now__::_thesis:_for_x_being_set_st_x_in_rng_(s_^\_k)_holds_ x_in_(dom_f1)_\_{x0} let x be set ; ::_thesis: ( x in rng (s ^\ k) implies x in (dom f1) \ {x0} ) assume A22: x in rng (s ^\ k) ; ::_thesis: x in (dom f1) \ {x0} then not x in {x0} by A11, XBOOLE_0:def_5; hence x in (dom f1) \ {x0} by A21, A22, XBOOLE_0:def_5; ::_thesis: verum end; then A23: rng (s ^\ k) c= (dom f1) \ {x0} by TARSKI:def_3; lim (s ^\ k) = x0 by A6, A7, SEQ_4:20; then f1 /* (s ^\ k) is divergent_to+infty by A1, A6, A23, Def2; then f /* (s ^\ k) is divergent_to+infty by A20, LIMFUNC1:42; then (f /* s) ^\ k is divergent_to+infty by A8, VALUED_0:27, XBOOLE_1:1; hence f /* s is divergent_to+infty by LIMFUNC1:7; ::_thesis: verum end; theorem Th25: :: LIMFUNC3:25 for x0 being Real for f1, f being PartFunc of REAL,REAL st f1 is_divergent_to-infty_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ex r being Real st ( 0 < r & (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f . g <= f1 . g ) ) holds f is_divergent_to-infty_in x0 proof let x0 be Real; ::_thesis: for f1, f being PartFunc of REAL,REAL st f1 is_divergent_to-infty_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ex r being Real st ( 0 < r & (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f . g <= f1 . g ) ) holds f is_divergent_to-infty_in x0 let f1, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is_divergent_to-infty_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ex r being Real st ( 0 < r & (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f . g <= f1 . g ) ) implies f is_divergent_to-infty_in x0 ) assume that A1: f1 is_divergent_to-infty_in x0 and A2: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ; ::_thesis: ( for r being Real holds ( not 0 < r or not (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) or ex g being Real st ( g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & not f . g <= f1 . g ) ) or f is_divergent_to-infty_in x0 ) given r being Real such that A3: 0 < r and A4: (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) and A5: for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f . g <= f1 . g ; ::_thesis: f is_divergent_to-infty_in x0 thus for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) by A2; :: according to LIMFUNC3:def_3 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} holds f /* seq is divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom f) \ {x0} implies f /* s is divergent_to-infty ) assume that A6: s is convergent and A7: lim s = x0 and A8: rng s c= (dom f) \ {x0} ; ::_thesis: f /* s is divergent_to-infty consider k being Element of NAT such that A9: for n being Element of NAT st k <= n holds ( x0 - r < s . n & s . n < x0 + r ) by A3, A6, A7, Th7; A10: rng (s ^\ k) c= rng s by VALUED_0:21; then A11: rng (s ^\ k) c= (dom f) \ {x0} by A8, XBOOLE_1:1; now__::_thesis:_for_x_being_set_st_x_in_rng_(s_^\_k)_holds_ x_in_].(x0_-_r),x0.[_\/_].x0,(x0_+_r).[ let x be set ; ::_thesis: ( x in rng (s ^\ k) implies x in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ ) assume x in rng (s ^\ k) ; ::_thesis: x in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ then consider n being Element of NAT such that A12: (s ^\ k) . n = x by FUNCT_2:113; A13: k <= n + k by NAT_1:12; then s . (n + k) < x0 + r by A9; then A14: (s ^\ k) . n < x0 + r by NAT_1:def_3; x0 - r < s . (n + k) by A9, A13; then x0 - r < (s ^\ k) . n by NAT_1:def_3; then (s ^\ k) . n in { g1 where g1 is Real : ( x0 - r < g1 & g1 < x0 + r ) } by A14; then A15: (s ^\ k) . n in ].(x0 - r),(x0 + r).[ by RCOMP_1:def_2; (s ^\ k) . n in rng (s ^\ k) by VALUED_0:28; then not (s ^\ k) . n in {x0} by A11, XBOOLE_0:def_5; then (s ^\ k) . n in ].(x0 - r),(x0 + r).[ \ {x0} by A15, XBOOLE_0:def_5; hence x in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ by A3, A12, Th4; ::_thesis: verum end; then A16: rng (s ^\ k) c= ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ by TARSKI:def_3; A17: rng s c= dom f by A8, XBOOLE_1:1; then rng (s ^\ k) c= dom f by A10, XBOOLE_1:1; then A18: rng (s ^\ k) c= (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) by A16, XBOOLE_1:19; then A19: rng (s ^\ k) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) by A4, XBOOLE_1:1; A20: now__::_thesis:_for_n_being_Element_of_NAT_holds_(f_/*_(s_^\_k))_._n_<=_(f1_/*_(s_^\_k))_._n let n be Element of NAT ; ::_thesis: (f /* (s ^\ k)) . n <= (f1 /* (s ^\ k)) . n (s ^\ k) . n in rng (s ^\ k) by VALUED_0:28; then f . ((s ^\ k) . n) <= f1 . ((s ^\ k) . n) by A5, A18; then (f /* (s ^\ k)) . n <= f1 . ((s ^\ k) . n) by A17, A10, FUNCT_2:108, XBOOLE_1:1; hence (f /* (s ^\ k)) . n <= (f1 /* (s ^\ k)) . n by A19, FUNCT_2:108, XBOOLE_1:18; ::_thesis: verum end; A21: rng (s ^\ k) c= dom f1 by A19, XBOOLE_1:18; now__::_thesis:_for_x_being_set_st_x_in_rng_(s_^\_k)_holds_ x_in_(dom_f1)_\_{x0} let x be set ; ::_thesis: ( x in rng (s ^\ k) implies x in (dom f1) \ {x0} ) assume A22: x in rng (s ^\ k) ; ::_thesis: x in (dom f1) \ {x0} then not x in {x0} by A11, XBOOLE_0:def_5; hence x in (dom f1) \ {x0} by A21, A22, XBOOLE_0:def_5; ::_thesis: verum end; then A23: rng (s ^\ k) c= (dom f1) \ {x0} by TARSKI:def_3; lim (s ^\ k) = x0 by A6, A7, SEQ_4:20; then f1 /* (s ^\ k) is divergent_to-infty by A1, A6, A23, Def3; then f /* (s ^\ k) is divergent_to-infty by A20, LIMFUNC1:43; then (f /* s) ^\ k is divergent_to-infty by A8, VALUED_0:27, XBOOLE_1:1; hence f /* s is divergent_to-infty by LIMFUNC1:7; ::_thesis: verum end; theorem :: LIMFUNC3:26 for x0 being Real for f1, f being PartFunc of REAL,REAL st f1 is_divergent_to+infty_in x0 & ex r being Real st ( 0 < r & ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ c= (dom f) /\ (dom f1) & ( for g being Real st g in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ holds f1 . g <= f . g ) ) holds f is_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f1, f being PartFunc of REAL,REAL st f1 is_divergent_to+infty_in x0 & ex r being Real st ( 0 < r & ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ c= (dom f) /\ (dom f1) & ( for g being Real st g in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ holds f1 . g <= f . g ) ) holds f is_divergent_to+infty_in x0 let f1, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is_divergent_to+infty_in x0 & ex r being Real st ( 0 < r & ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ c= (dom f) /\ (dom f1) & ( for g being Real st g in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ holds f1 . g <= f . g ) ) implies f is_divergent_to+infty_in x0 ) assume A1: f1 is_divergent_to+infty_in x0 ; ::_thesis: ( for r being Real holds ( not 0 < r or not ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ c= (dom f) /\ (dom f1) or ex g being Real st ( g in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ & not f1 . g <= f . g ) ) or f is_divergent_to+infty_in x0 ) given r being Real such that A2: 0 < r and A3: ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ c= (dom f) /\ (dom f1) and A4: for g being Real st g in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ holds f1 . g <= f . g ; ::_thesis: f is_divergent_to+infty_in x0 A5: ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ = (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) by A3, XBOOLE_1:18, XBOOLE_1:28; A6: ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ = (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) by A3, XBOOLE_1:18, XBOOLE_1:28; for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) by A2, A3, Th5, XBOOLE_1:18; hence f is_divergent_to+infty_in x0 by A1, A2, A4, A5, A6, Th24; ::_thesis: verum end; theorem :: LIMFUNC3:27 for x0 being Real for f1, f being PartFunc of REAL,REAL st f1 is_divergent_to-infty_in x0 & ex r being Real st ( 0 < r & ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ c= (dom f) /\ (dom f1) & ( for g being Real st g in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ holds f . g <= f1 . g ) ) holds f is_divergent_to-infty_in x0 proof let x0 be Real; ::_thesis: for f1, f being PartFunc of REAL,REAL st f1 is_divergent_to-infty_in x0 & ex r being Real st ( 0 < r & ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ c= (dom f) /\ (dom f1) & ( for g being Real st g in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ holds f . g <= f1 . g ) ) holds f is_divergent_to-infty_in x0 let f1, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is_divergent_to-infty_in x0 & ex r being Real st ( 0 < r & ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ c= (dom f) /\ (dom f1) & ( for g being Real st g in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ holds f . g <= f1 . g ) ) implies f is_divergent_to-infty_in x0 ) assume A1: f1 is_divergent_to-infty_in x0 ; ::_thesis: ( for r being Real holds ( not 0 < r or not ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ c= (dom f) /\ (dom f1) or ex g being Real st ( g in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ & not f . g <= f1 . g ) ) or f is_divergent_to-infty_in x0 ) given r being Real such that A2: 0 < r and A3: ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ c= (dom f) /\ (dom f1) and A4: for g being Real st g in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ holds f . g <= f1 . g ; ::_thesis: f is_divergent_to-infty_in x0 A5: ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ = (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) by A3, XBOOLE_1:18, XBOOLE_1:28; A6: ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ = (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) by A3, XBOOLE_1:18, XBOOLE_1:28; for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) by A2, A3, Th5, XBOOLE_1:18; hence f is_divergent_to-infty_in x0 by A1, A2, A4, A5, A6, Th25; ::_thesis: verum end; definition let f be PartFunc of REAL,REAL; let x0 be Real; assume A1: f is_convergent_in x0 ; func lim (f,x0) -> Real means :Def4: :: LIMFUNC3:def 4 for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} holds ( f /* seq is convergent & lim (f /* seq) = it ); existence ex b1 being Real st for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} holds ( f /* seq is convergent & lim (f /* seq) = b1 ) by A1, Def1; uniqueness for b1, b2 being Real st ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} holds ( f /* seq is convergent & lim (f /* seq) = b1 ) ) & ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} holds ( f /* seq is convergent & lim (f /* seq) = b2 ) ) holds b1 = b2 proof defpred S1[ Element of NAT , real number ] means ( x0 - (1 / ($1 + 1)) < $2 & $2 < x0 & $2 in dom f ); A2: now__::_thesis:_for_n_being_Element_of_NAT_ex_g1_being_Real_st_S1[n,g1] let n be Element of NAT ; ::_thesis: ex g1 being Real st S1[n,g1] A3: x0 + 0 < x0 + 1 by XREAL_1:8; x0 - (1 / (n + 1)) < x0 by Lm3; then consider g1, g2 being Real such that A4: x0 - (1 / (n + 1)) < g1 and A5: g1 < x0 and A6: g1 in dom f and g2 < x0 + 1 and x0 < g2 and g2 in dom f by A1, A3, Def1; take g1 = g1; ::_thesis: S1[n,g1] thus S1[n,g1] by A4, A5, A6; ::_thesis: verum end; consider s being Real_Sequence such that A7: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A2); A8: rng s c= (dom f) \ {x0} by A7, Th6; A9: lim s = x0 by A7, Th6; let g1, g2 be Real; ::_thesis: ( ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} holds ( f /* seq is convergent & lim (f /* seq) = g1 ) ) & ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} holds ( f /* seq is convergent & lim (f /* seq) = g2 ) ) implies g1 = g2 ) assume that A10: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} holds ( f /* seq is convergent & lim (f /* seq) = g1 ) and A11: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} holds ( f /* seq is convergent & lim (f /* seq) = g2 ) ; ::_thesis: g1 = g2 A12: s is convergent by A7, Th6; then lim (f /* s) = g1 by A9, A8, A10; hence g1 = g2 by A12, A9, A8, A11; ::_thesis: verum end; end; :: deftheorem Def4 defines lim LIMFUNC3:def_4_:_ for f being PartFunc of REAL,REAL for x0 being Real st f is_convergent_in x0 holds for b3 being Real holds ( b3 = lim (f,x0) iff for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} holds ( f /* seq is convergent & lim (f /* seq) = b3 ) ); theorem :: LIMFUNC3:28 for x0, g being Real for f being PartFunc of REAL,REAL st f is_convergent_in x0 holds ( lim (f,x0) = g iff for g1 being Real st 0 < g1 holds ex g2 being Real st ( 0 < g2 & ( for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ) proof let x0, g be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_convergent_in x0 holds ( lim (f,x0) = g iff for g1 being Real st 0 < g1 holds ex g2 being Real st ( 0 < g2 & ( for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_convergent_in x0 implies ( lim (f,x0) = g iff for g1 being Real st 0 < g1 holds ex g2 being Real st ( 0 < g2 & ( for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ) ) assume A1: f is_convergent_in x0 ; ::_thesis: ( lim (f,x0) = g iff for g1 being Real st 0 < g1 holds ex g2 being Real st ( 0 < g2 & ( for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ) thus ( lim (f,x0) = g implies for g1 being Real st 0 < g1 holds ex g2 being Real st ( 0 < g2 & ( for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ) ::_thesis: ( ( for g1 being Real st 0 < g1 holds ex g2 being Real st ( 0 < g2 & ( for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ) implies lim (f,x0) = g ) proof assume that A2: lim (f,x0) = g and A3: ex g1 being Real st ( 0 < g1 & ( for g2 being Real st 0 < g2 holds ex r1 being Real st ( 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f & g1 <= abs ((f . r1) - g) ) ) ) ; ::_thesis: contradiction consider g1 being Real such that A4: 0 < g1 and A5: for g2 being Real st 0 < g2 holds ex r1 being Real st ( 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f & g1 <= abs ((f . r1) - g) ) by A3; defpred S1[ Element of NAT , real number ] means ( 0 < abs (x0 - $2) & abs (x0 - $2) < 1 / ($1 + 1) & $2 in dom f & abs ((f . $2) - g) >= g1 ); A6: for n being Element of NAT ex r1 being Real st S1[n,r1] by A5, XREAL_1:139; consider s being Real_Sequence such that A7: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A6); A8: rng s c= (dom f) \ {x0} by A7, Th2; A9: lim s = x0 by A7, Th2; A10: s is convergent by A7, Th2; then A11: lim (f /* s) = g by A1, A2, A9, A8, Def4; f /* s is convergent by A1, A2, A10, A9, A8, Def4; then consider n being Element of NAT such that A12: for k being Element of NAT st n <= k holds abs (((f /* s) . k) - g) < g1 by A4, A11, SEQ_2:def_7; A13: abs (((f /* s) . n) - g) < g1 by A12; rng s c= dom f by A7, Th2; then abs ((f . (s . n)) - g) < g1 by A13, FUNCT_2:108; hence contradiction by A7; ::_thesis: verum end; assume A14: for g1 being Real st 0 < g1 holds ex g2 being Real st ( 0 < g2 & ( for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ; ::_thesis: lim (f,x0) = g now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_f)_\_{x0}_holds_ (_f_/*_s_is_convergent_&_lim_(f_/*_s)_=_g_) let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom f) \ {x0} implies ( f /* s is convergent & lim (f /* s) = g ) ) assume that A15: s is convergent and A16: lim s = x0 and A17: rng s c= (dom f) \ {x0} ; ::_thesis: ( f /* s is convergent & lim (f /* s) = g ) A18: now__::_thesis:_for_g1_being_real_number_st_0_<_g1_holds_ ex_n_being_Element_of_NAT_st_ for_k_being_Element_of_NAT_st_n_<=_k_holds_ abs_(((f_/*_s)_._k)_-_g)_<_g1 let g1 be real number ; ::_thesis: ( 0 < g1 implies ex n being Element of NAT st for k being Element of NAT st n <= k holds abs (((f /* s) . k) - g) < g1 ) assume A19: 0 < g1 ; ::_thesis: ex n being Element of NAT st for k being Element of NAT st n <= k holds abs (((f /* s) . k) - g) < g1 g1 is Real by XREAL_0:def_1; then consider g2 being Real such that A20: 0 < g2 and A21: for r1 being Real st 0 < abs (x0 - r1) & abs (x0 - r1) < g2 & r1 in dom f holds abs ((f . r1) - g) < g1 by A14, A19; consider n being Element of NAT such that A22: for k being Element of NAT st n <= k holds ( 0 < abs (x0 - (s . k)) & abs (x0 - (s . k)) < g2 & s . k in dom f ) by A15, A16, A17, A20, Th3; take n = n; ::_thesis: for k being Element of NAT st n <= k holds abs (((f /* s) . k) - g) < g1 let k be Element of NAT ; ::_thesis: ( n <= k implies abs (((f /* s) . k) - g) < g1 ) assume A23: n <= k ; ::_thesis: abs (((f /* s) . k) - g) < g1 then A24: abs (x0 - (s . k)) < g2 by A22; A25: s . k in dom f by A22, A23; 0 < abs (x0 - (s . k)) by A22, A23; then abs ((f . (s . k)) - g) < g1 by A21, A24, A25; hence abs (((f /* s) . k) - g) < g1 by A17, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; hence f /* s is convergent by SEQ_2:def_6; ::_thesis: lim (f /* s) = g hence lim (f /* s) = g by A18, SEQ_2:def_7; ::_thesis: verum end; hence lim (f,x0) = g by A1, Def4; ::_thesis: verum end; theorem Th29: :: LIMFUNC3:29 for x0 being Real for f being PartFunc of REAL,REAL st f is_convergent_in x0 holds ( f is_left_convergent_in x0 & f is_right_convergent_in x0 & lim_left (f,x0) = lim_right (f,x0) & lim (f,x0) = lim_left (f,x0) & lim (f,x0) = lim_right (f,x0) ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_convergent_in x0 holds ( f is_left_convergent_in x0 & f is_right_convergent_in x0 & lim_left (f,x0) = lim_right (f,x0) & lim (f,x0) = lim_left (f,x0) & lim (f,x0) = lim_right (f,x0) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_convergent_in x0 implies ( f is_left_convergent_in x0 & f is_right_convergent_in x0 & lim_left (f,x0) = lim_right (f,x0) & lim (f,x0) = lim_left (f,x0) & lim (f,x0) = lim_right (f,x0) ) ) assume A1: f is_convergent_in x0 ; ::_thesis: ( f is_left_convergent_in x0 & f is_right_convergent_in x0 & lim_left (f,x0) = lim_right (f,x0) & lim (f,x0) = lim_left (f,x0) & lim (f,x0) = lim_right (f,x0) ) A2: lim (f,x0) = lim (f,x0) ; A3: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_f)_/\_(right_open_halfline_x0)_holds_ (_f_/*_s_is_convergent_&_lim_(f_/*_s)_=_lim_(f,x0)_) let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom f) /\ (right_open_halfline x0) implies ( f /* s is convergent & lim (f /* s) = lim (f,x0) ) ) assume that A4: s is convergent and A5: lim s = x0 and A6: rng s c= (dom f) /\ (right_open_halfline x0) ; ::_thesis: ( f /* s is convergent & lim (f /* s) = lim (f,x0) ) rng s c= (dom f) \ {x0} by A6, Th1; hence ( f /* s is convergent & lim (f /* s) = lim (f,x0) ) by A1, A2, A4, A5, Def4; ::_thesis: verum end; A7: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_f)_/\_(left_open_halfline_x0)_holds_ (_f_/*_s_is_convergent_&_lim_(f_/*_s)_=_lim_(f,x0)_) let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom f) /\ (left_open_halfline x0) implies ( f /* s is convergent & lim (f /* s) = lim (f,x0) ) ) assume that A8: s is convergent and A9: lim s = x0 and A10: rng s c= (dom f) /\ (left_open_halfline x0) ; ::_thesis: ( f /* s is convergent & lim (f /* s) = lim (f,x0) ) rng s c= (dom f) \ {x0} by A10, Th1; hence ( f /* s is convergent & lim (f /* s) = lim (f,x0) ) by A1, A2, A8, A9, Def4; ::_thesis: verum end; A11: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) by A1, Def1; then for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) by Th8; hence f is_left_convergent_in x0 by A7, LIMFUNC2:def_1; ::_thesis: ( f is_right_convergent_in x0 & lim_left (f,x0) = lim_right (f,x0) & lim (f,x0) = lim_left (f,x0) & lim (f,x0) = lim_right (f,x0) ) then A12: lim_left (f,x0) = lim (f,x0) by A7, LIMFUNC2:def_7; for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) by A11, Th8; hence f is_right_convergent_in x0 by A3, LIMFUNC2:def_4; ::_thesis: ( lim_left (f,x0) = lim_right (f,x0) & lim (f,x0) = lim_left (f,x0) & lim (f,x0) = lim_right (f,x0) ) hence ( lim_left (f,x0) = lim_right (f,x0) & lim (f,x0) = lim_left (f,x0) & lim (f,x0) = lim_right (f,x0) ) by A12, A3, LIMFUNC2:def_8; ::_thesis: verum end; theorem :: LIMFUNC3:30 for x0 being Real for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 & f is_right_convergent_in x0 & lim_left (f,x0) = lim_right (f,x0) holds ( f is_convergent_in x0 & lim (f,x0) = lim_left (f,x0) & lim (f,x0) = lim_right (f,x0) ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 & f is_right_convergent_in x0 & lim_left (f,x0) = lim_right (f,x0) holds ( f is_convergent_in x0 & lim (f,x0) = lim_left (f,x0) & lim (f,x0) = lim_right (f,x0) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_left_convergent_in x0 & f is_right_convergent_in x0 & lim_left (f,x0) = lim_right (f,x0) implies ( f is_convergent_in x0 & lim (f,x0) = lim_left (f,x0) & lim (f,x0) = lim_right (f,x0) ) ) assume that A1: f is_left_convergent_in x0 and A2: f is_right_convergent_in x0 and A3: lim_left (f,x0) = lim_right (f,x0) ; ::_thesis: ( f is_convergent_in x0 & lim (f,x0) = lim_left (f,x0) & lim (f,x0) = lim_right (f,x0) ) A4: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_f)_\_{x0}_holds_ (_f_/*_s_is_convergent_&_lim_(f_/*_s)_=_lim_left_(f,x0)_) let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom f) \ {x0} implies ( f /* s is convergent & lim (f /* s) = lim_left (f,x0) ) ) assume that A5: s is convergent and A6: lim s = x0 and A7: rng s c= (dom f) \ {x0} ; ::_thesis: ( f /* s is convergent & lim (f /* s) = lim_left (f,x0) ) now__::_thesis:_(_f_/*_s_is_convergent_&_lim_(f_/*_s)_=_lim_left_(f,x0)_) percases ( ex k being Element of NAT st for n being Element of NAT st k <= n holds s . n < x0 or for k being Element of NAT ex n being Element of NAT st ( k <= n & s . n >= x0 ) ) ; suppose ex k being Element of NAT st for n being Element of NAT st k <= n holds s . n < x0 ; ::_thesis: ( f /* s is convergent & lim (f /* s) = lim_left (f,x0) ) then consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds s . n < x0 ; A9: rng s c= dom f by A7, XBOOLE_1:1; A10: rng (s ^\ k) c= (dom f) /\ (left_open_halfline x0) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (s ^\ k) or x in (dom f) /\ (left_open_halfline x0) ) assume x in rng (s ^\ k) ; ::_thesis: x in (dom f) /\ (left_open_halfline x0) then consider n being Element of NAT such that A11: (s ^\ k) . n = x by FUNCT_2:113; s . (n + k) < x0 by A8, NAT_1:12; then s . (n + k) in { g1 where g1 is Real : g1 < x0 } ; then s . (n + k) in left_open_halfline x0 by XXREAL_1:229; then A12: x in left_open_halfline x0 by A11, NAT_1:def_3; s . (n + k) in rng s by VALUED_0:28; then x in rng s by A11, NAT_1:def_3; hence x in (dom f) /\ (left_open_halfline x0) by A9, A12, XBOOLE_0:def_4; ::_thesis: verum end; A13: f /* (s ^\ k) = (f /* s) ^\ k by A7, VALUED_0:27, XBOOLE_1:1; A14: lim (s ^\ k) = x0 by A5, A6, SEQ_4:20; then A15: f /* (s ^\ k) is convergent by A1, A3, A5, A10, LIMFUNC2:def_7; hence f /* s is convergent by A13, SEQ_4:21; ::_thesis: lim (f /* s) = lim_left (f,x0) lim (f /* (s ^\ k)) = lim_left (f,x0) by A1, A5, A14, A10, LIMFUNC2:def_7; hence lim (f /* s) = lim_left (f,x0) by A15, A13, SEQ_4:22; ::_thesis: verum end; supposeA16: for k being Element of NAT ex n being Element of NAT st ( k <= n & s . n >= x0 ) ; ::_thesis: ( f /* s is convergent & lim (f /* s) = lim_left (f,x0) ) now__::_thesis:_(_f_/*_s_is_convergent_&_lim_(f_/*_s)_=_lim_left_(f,x0)_) percases ( ex k being Element of NAT st for n being Element of NAT st k <= n holds x0 < s . n or for k being Element of NAT ex n being Element of NAT st ( k <= n & x0 >= s . n ) ) ; suppose ex k being Element of NAT st for n being Element of NAT st k <= n holds x0 < s . n ; ::_thesis: ( f /* s is convergent & lim (f /* s) = lim_left (f,x0) ) then consider k being Element of NAT such that A17: for n being Element of NAT st k <= n holds s . n > x0 ; A18: rng s c= dom f by A7, XBOOLE_1:1; A19: rng (s ^\ k) c= (dom f) /\ (right_open_halfline x0) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (s ^\ k) or x in (dom f) /\ (right_open_halfline x0) ) assume x in rng (s ^\ k) ; ::_thesis: x in (dom f) /\ (right_open_halfline x0) then consider n being Element of NAT such that A20: (s ^\ k) . n = x by FUNCT_2:113; x0 < s . (n + k) by A17, NAT_1:12; then s . (n + k) in { g1 where g1 is Real : x0 < g1 } ; then s . (n + k) in right_open_halfline x0 by XXREAL_1:230; then A21: x in right_open_halfline x0 by A20, NAT_1:def_3; s . (n + k) in rng s by VALUED_0:28; then x in rng s by A20, NAT_1:def_3; hence x in (dom f) /\ (right_open_halfline x0) by A18, A21, XBOOLE_0:def_4; ::_thesis: verum end; A22: f /* (s ^\ k) = (f /* s) ^\ k by A7, VALUED_0:27, XBOOLE_1:1; A23: lim (s ^\ k) = x0 by A5, A6, SEQ_4:20; then A24: f /* (s ^\ k) is convergent by A2, A3, A5, A19, LIMFUNC2:def_8; hence f /* s is convergent by A22, SEQ_4:21; ::_thesis: lim (f /* s) = lim_left (f,x0) lim (f /* (s ^\ k)) = lim_left (f,x0) by A2, A3, A5, A23, A19, LIMFUNC2:def_8; hence lim (f /* s) = lim_left (f,x0) by A24, A22, SEQ_4:22; ::_thesis: verum end; supposeA25: for k being Element of NAT ex n being Element of NAT st ( k <= n & x0 >= s . n ) ; ::_thesis: ( f /* s is convergent & lim (f /* s) = lim_left (f,x0) ) set GR = lim_left (f,x0); defpred S1[ set , set ] means for n, m being Element of NAT st $1 = n & $2 = m holds ( n < m & s . m < x0 & ( for k being Element of NAT st n < k & s . k < x0 holds m <= k ) ); defpred S2[ Element of NAT , set , set ] means S1[$2,$3]; defpred S3[ Nat] means s . $1 < x0; A26: now__::_thesis:_for_k_being_Element_of_NAT_ex_n_being_Element_of_NAT_st_ (_k_<=_n_&_s_._n_<_x0_) let k be Element of NAT ; ::_thesis: ex n being Element of NAT st ( k <= n & s . n < x0 ) consider n being Element of NAT such that A27: k <= n and A28: s . n <= x0 by A25; take n = n; ::_thesis: ( k <= n & s . n < x0 ) thus k <= n by A27; ::_thesis: s . n < x0 s . n in rng s by VALUED_0:28; then not s . n in {x0} by A7, XBOOLE_0:def_5; then s . n <> x0 by TARSKI:def_1; hence s . n < x0 by A28, XXREAL_0:1; ::_thesis: verum end; then ex m1 being Element of NAT st ( 0 <= m1 & s . m1 < x0 ) ; then A29: ex m being Nat st S3[m] ; consider M being Nat such that A30: ( S3[M] & ( for n being Nat st S3[n] holds M <= n ) ) from NAT_1:sch_5(A29); reconsider M9 = M as Element of NAT by ORDINAL1:def_12; A31: now__::_thesis:_for_n_being_Element_of_NAT_ex_m_being_Element_of_NAT_st_ (_n_<_m_&_s_._m_<_x0_) let n be Element of NAT ; ::_thesis: ex m being Element of NAT st ( n < m & s . m < x0 ) consider m being Element of NAT such that A32: n + 1 <= m and A33: s . m < x0 by A26; take m = m; ::_thesis: ( n < m & s . m < x0 ) thus ( n < m & s . m < x0 ) by A32, A33, NAT_1:13; ::_thesis: verum end; A34: for n, x being Element of NAT ex y being Element of NAT st S2[n,x,y] proof let n, x be Element of NAT ; ::_thesis: ex y being Element of NAT st S2[n,x,y] defpred S4[ Nat] means ( x < $1 & s . $1 < x0 ); ex m being Element of NAT st S4[m] by A31; then A35: ex m being Nat st S4[m] ; consider l being Nat such that A36: ( S4[l] & ( for k being Nat st S4[k] holds l <= k ) ) from NAT_1:sch_5(A35); take l ; ::_thesis: ( l is Element of REAL & l is Element of NAT & S2[n,x,l] ) l in NAT by ORDINAL1:def_12; hence ( l is Element of REAL & l is Element of NAT & S2[n,x,l] ) by A36; ::_thesis: verum end; consider F being Function of NAT,NAT such that A37: ( F . 0 = M9 & ( for n being Element of NAT holds S2[n,F . n,F . (n + 1)] ) ) from RECDEF_1:sch_2(A34); A38: rng F c= NAT by RELAT_1:def_19; then A39: rng F c= REAL by XBOOLE_1:1; A40: dom F = NAT by FUNCT_2:def_1; then reconsider F = F as Real_Sequence by A39, RELSET_1:4; A41: now__::_thesis:_for_n_being_Element_of_NAT_holds_F_._n_is_Element_of_NAT let n be Element of NAT ; ::_thesis: F . n is Element of NAT F . n in rng F by A40, FUNCT_1:def_3; hence F . n is Element of NAT by A38; ::_thesis: verum end; now__::_thesis:_for_n_being_Element_of_NAT_holds_F_._n_<_F_._(n_+_1) let n be Element of NAT ; ::_thesis: F . n < F . (n + 1) A42: F . (n + 1) is Element of NAT by A41; F . n is Element of NAT by A41; hence F . n < F . (n + 1) by A37, A42; ::_thesis: verum end; then reconsider F = F as V37() sequence of NAT by SEQM_3:def_6; A43: s * F is subsequence of s by VALUED_0:def_17; then rng (s * F) c= rng s by VALUED_0:21; then A44: rng (s * F) c= (dom f) \ {x0} by A7, XBOOLE_1:1; defpred S4[ Nat] means ( s . $1 < x0 & ( for m being Element of NAT holds F . m <> $1 ) ); A45: for n being Element of NAT st s . n < x0 holds ex m being Element of NAT st F . m = n proof assume ex n being Element of NAT st S4[n] ; ::_thesis: contradiction then A46: ex n being Nat st S4[n] ; consider M1 being Nat such that A47: ( S4[M1] & ( for n being Nat st S4[n] holds M1 <= n ) ) from NAT_1:sch_5(A46); defpred S5[ Nat] means ( $1 < M1 & s . $1 < x0 & ex m being Element of NAT st F . m = $1 ); A48: ex n being Nat st S5[n] proof take M ; ::_thesis: S5[M] A49: M <> M1 by A37, A47; M <= M1 by A30, A47; hence M < M1 by A49, XXREAL_0:1; ::_thesis: ( s . M < x0 & ex m being Element of NAT st F . m = M ) thus s . M < x0 by A30; ::_thesis: ex m being Element of NAT st F . m = M take 0 ; ::_thesis: F . 0 = M thus F . 0 = M by A37; ::_thesis: verum end; A50: for n being Nat st S5[n] holds n <= M1 ; consider MX being Nat such that A51: ( S5[MX] & ( for n being Nat st S5[n] holds n <= MX ) ) from NAT_1:sch_6(A50, A48); A52: for k being Element of NAT st MX < k & k < M1 holds s . k >= x0 proof given k being Element of NAT such that A53: MX < k and A54: k < M1 and A55: s . k < x0 ; ::_thesis: contradiction now__::_thesis:_contradiction percases ( ex m being Element of NAT st F . m = k or for m being Element of NAT holds F . m <> k ) ; suppose ex m being Element of NAT st F . m = k ; ::_thesis: contradiction hence contradiction by A51, A53, A54, A55; ::_thesis: verum end; suppose for m being Element of NAT holds F . m <> k ; ::_thesis: contradiction hence contradiction by A47, A54, A55; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; consider m being Element of NAT such that A56: F . m = MX by A51; M1 in NAT by ORDINAL1:def_12; then A57: F . (m + 1) <= M1 by A37, A47, A51, A56; A58: s . (F . (m + 1)) < x0 by A37, A56; A59: MX < F . (m + 1) by A37, A56; now__::_thesis:_not_F_._(m_+_1)_<>_M1 assume F . (m + 1) <> M1 ; ::_thesis: contradiction then F . (m + 1) < M1 by A57, XXREAL_0:1; hence contradiction by A52, A59, A58; ::_thesis: verum end; hence contradiction by A47; ::_thesis: verum end; defpred S5[ Nat] means s . $1 > x0; A60: now__::_thesis:_for_k_being_Element_of_NAT_ex_n_being_Element_of_NAT_st_ (_k_<=_n_&_s_._n_>_x0_) let k be Element of NAT ; ::_thesis: ex n being Element of NAT st ( k <= n & s . n > x0 ) consider n being Element of NAT such that A61: k <= n and A62: s . n >= x0 by A16; take n = n; ::_thesis: ( k <= n & s . n > x0 ) thus k <= n by A61; ::_thesis: s . n > x0 s . n in rng s by VALUED_0:28; then not s . n in {x0} by A7, XBOOLE_0:def_5; then s . n <> x0 by TARSKI:def_1; hence s . n > x0 by A62, XXREAL_0:1; ::_thesis: verum end; then ex mn being Element of NAT st ( 0 <= mn & s . mn > x0 ) ; then A63: ex m being Nat st S5[m] ; consider N being Nat such that A64: ( S5[N] & ( for n being Nat st S5[n] holds N <= n ) ) from NAT_1:sch_5(A63); A65: for n being Element of NAT holds (s * F) . n < x0 proof defpred S6[ Element of NAT ] means (s * F) . $1 < x0; A66: for k being Element of NAT st S6[k] holds S6[k + 1] proof let k be Element of NAT ; ::_thesis: ( S6[k] implies S6[k + 1] ) assume (s * F) . k < x0 ; ::_thesis: S6[k + 1] S1[F . k,F . (k + 1)] by A37; then s . (F . (k + 1)) < x0 ; hence S6[k + 1] by FUNCT_2:15; ::_thesis: verum end; A67: S6[ 0 ] by A30, A37, FUNCT_2:15; thus for k being Element of NAT holds S6[k] from NAT_1:sch_1(A67, A66); ::_thesis: verum end; A68: rng (s * F) c= (dom f) /\ (left_open_halfline x0) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (s * F) or x in (dom f) /\ (left_open_halfline x0) ) assume A69: x in rng (s * F) ; ::_thesis: x in (dom f) /\ (left_open_halfline x0) then consider n being Element of NAT such that A70: (s * F) . n = x by FUNCT_2:113; (s * F) . n < x0 by A65; then x in { g1 where g1 is Real : g1 < x0 } by A70; then A71: x in left_open_halfline x0 by XXREAL_1:229; x in dom f by A44, A69, XBOOLE_0:def_5; hence x in (dom f) /\ (left_open_halfline x0) by A71, XBOOLE_0:def_4; ::_thesis: verum end; defpred S6[ set , set ] means for n, m being Element of NAT st $1 = n & $2 = m holds ( n < m & s . m > x0 & ( for k being Element of NAT st n < k & s . k > x0 holds m <= k ) ); defpred S7[ Element of NAT , set , set ] means S6[$2,$3]; A72: s * F is convergent by A5, A43, SEQ_4:16; reconsider N9 = N as Element of NAT by ORDINAL1:def_12; A73: now__::_thesis:_for_n_being_Element_of_NAT_ex_m_being_Element_of_NAT_st_ (_n_<_m_&_s_._m_>_x0_) let n be Element of NAT ; ::_thesis: ex m being Element of NAT st ( n < m & s . m > x0 ) consider m being Element of NAT such that A74: n + 1 <= m and A75: s . m > x0 by A60; take m = m; ::_thesis: ( n < m & s . m > x0 ) thus ( n < m & s . m > x0 ) by A74, A75, NAT_1:13; ::_thesis: verum end; A76: for n, x being Element of NAT ex y being Element of NAT st S7[n,x,y] proof let n, x be Element of NAT ; ::_thesis: ex y being Element of NAT st S7[n,x,y] defpred S8[ Nat] means ( x < $1 & s . $1 > x0 ); ex m being Element of NAT st S8[m] by A73; then A77: ex m being Nat st S8[m] ; consider l being Nat such that A78: ( S8[l] & ( for k being Nat st S8[k] holds l <= k ) ) from NAT_1:sch_5(A77); take l ; ::_thesis: ( l is Element of REAL & l is Element of NAT & S7[n,x,l] ) l in NAT by ORDINAL1:def_12; hence ( l is Element of REAL & l is Element of NAT & S7[n,x,l] ) by A78; ::_thesis: verum end; consider G being Function of NAT,NAT such that A79: ( G . 0 = N9 & ( for n being Element of NAT holds S7[n,G . n,G . (n + 1)] ) ) from RECDEF_1:sch_2(A76); A80: rng G c= NAT by RELAT_1:def_19; then A81: rng G c= REAL by XBOOLE_1:1; A82: dom G = NAT by FUNCT_2:def_1; then reconsider G = G as Real_Sequence by A81, RELSET_1:4; A83: now__::_thesis:_for_n_being_Element_of_NAT_holds_G_._n_is_Element_of_NAT let n be Element of NAT ; ::_thesis: G . n is Element of NAT G . n in rng G by A82, FUNCT_1:def_3; hence G . n is Element of NAT by A80; ::_thesis: verum end; now__::_thesis:_for_n_being_Element_of_NAT_holds_G_._n_<_G_._(n_+_1) let n be Element of NAT ; ::_thesis: G . n < G . (n + 1) A84: G . (n + 1) is Element of NAT by A83; G . n is Element of NAT by A83; hence G . n < G . (n + 1) by A79, A84; ::_thesis: verum end; then reconsider G = G as V37() sequence of NAT by SEQM_3:def_6; A85: s * G is subsequence of s by VALUED_0:def_17; then rng (s * G) c= rng s by VALUED_0:21; then A86: rng (s * G) c= (dom f) \ {x0} by A7, XBOOLE_1:1; A87: lim (s * F) = x0 by A5, A6, A43, SEQ_4:17; then A88: lim (f /* (s * F)) = lim_left (f,x0) by A1, A72, A68, LIMFUNC2:def_7; A89: f /* (s * F) is convergent by A1, A3, A72, A87, A68, LIMFUNC2:def_7; A90: s * G is convergent by A5, A85, SEQ_4:16; defpred S8[ Nat] means ( s . $1 > x0 & ( for m being Element of NAT holds G . m <> $1 ) ); A91: for n being Element of NAT st s . n > x0 holds ex m being Element of NAT st G . m = n proof assume ex n being Element of NAT st S8[n] ; ::_thesis: contradiction then A92: ex n being Nat st S8[n] ; consider N1 being Nat such that A93: ( S8[N1] & ( for n being Nat st S8[n] holds N1 <= n ) ) from NAT_1:sch_5(A92); defpred S9[ Nat] means ( $1 < N1 & s . $1 > x0 & ex m being Element of NAT st G . m = $1 ); A94: ex n being Nat st S9[n] proof take N ; ::_thesis: S9[N] A95: N <> N1 by A79, A93; N <= N1 by A64, A93; hence N < N1 by A95, XXREAL_0:1; ::_thesis: ( s . N > x0 & ex m being Element of NAT st G . m = N ) thus s . N > x0 by A64; ::_thesis: ex m being Element of NAT st G . m = N take 0 ; ::_thesis: G . 0 = N thus G . 0 = N by A79; ::_thesis: verum end; A96: for n being Nat st S9[n] holds n <= N1 ; consider NX being Nat such that A97: ( S9[NX] & ( for n being Nat st S9[n] holds n <= NX ) ) from NAT_1:sch_6(A96, A94); A98: for k being Element of NAT st NX < k & k < N1 holds s . k <= x0 proof given k being Element of NAT such that A99: NX < k and A100: k < N1 and A101: s . k > x0 ; ::_thesis: contradiction now__::_thesis:_contradiction percases ( ex m being Element of NAT st G . m = k or for m being Element of NAT holds G . m <> k ) ; suppose ex m being Element of NAT st G . m = k ; ::_thesis: contradiction hence contradiction by A97, A99, A100, A101; ::_thesis: verum end; suppose for m being Element of NAT holds G . m <> k ; ::_thesis: contradiction hence contradiction by A93, A100, A101; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; consider m being Element of NAT such that A102: G . m = NX by A97; N1 in NAT by ORDINAL1:def_12; then A103: G . (m + 1) <= N1 by A79, A93, A97, A102; A104: s . (G . (m + 1)) > x0 by A79, A102; A105: NX < G . (m + 1) by A79, A102; now__::_thesis:_not_G_._(m_+_1)_<>_N1 assume G . (m + 1) <> N1 ; ::_thesis: contradiction then G . (m + 1) < N1 by A103, XXREAL_0:1; hence contradiction by A98, A105, A104; ::_thesis: verum end; hence contradiction by A93; ::_thesis: verum end; A106: for n being Element of NAT holds (s * G) . n > x0 proof defpred S9[ Element of NAT ] means (s * G) . $1 > x0; A107: for k being Element of NAT st S9[k] holds S9[k + 1] proof let k be Element of NAT ; ::_thesis: ( S9[k] implies S9[k + 1] ) assume (s * G) . k > x0 ; ::_thesis: S9[k + 1] S6[G . k,G . (k + 1)] by A79; then s . (G . (k + 1)) > x0 ; hence S9[k + 1] by FUNCT_2:15; ::_thesis: verum end; A108: S9[ 0 ] by A64, A79, FUNCT_2:15; thus for k being Element of NAT holds S9[k] from NAT_1:sch_1(A108, A107); ::_thesis: verum end; A109: rng (s * G) c= (dom f) /\ (right_open_halfline x0) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (s * G) or x in (dom f) /\ (right_open_halfline x0) ) assume A110: x in rng (s * G) ; ::_thesis: x in (dom f) /\ (right_open_halfline x0) then consider n being Element of NAT such that A111: (s * G) . n = x by FUNCT_2:113; (s * G) . n > x0 by A106; then x in { g1 where g1 is Real : x0 < g1 } by A111; then A112: x in right_open_halfline x0 by XXREAL_1:230; x in dom f by A86, A110, XBOOLE_0:def_5; hence x in (dom f) /\ (right_open_halfline x0) by A112, XBOOLE_0:def_4; ::_thesis: verum end; A113: lim (s * G) = x0 by A5, A6, A85, SEQ_4:17; then A114: lim (f /* (s * G)) = lim_left (f,x0) by A2, A3, A90, A109, LIMFUNC2:def_8; A115: f /* (s * G) is convergent by A2, A3, A90, A113, A109, LIMFUNC2:def_8; A116: now__::_thesis:_for_r_being_real_number_st_0_<_r_holds_ ex_n_being_Element_of_NAT_st_ for_k_being_Element_of_NAT_st_n_<=_k_holds_ abs_(((f_/*_s)_._k)_-_(lim_left_(f,x0)))_<_r let r be real number ; ::_thesis: ( 0 < r implies ex n being Element of NAT st for k being Element of NAT st n <= k holds abs (((f /* s) . k) - (lim_left (f,x0))) < r ) assume A117: 0 < r ; ::_thesis: ex n being Element of NAT st for k being Element of NAT st n <= k holds abs (((f /* s) . k) - (lim_left (f,x0))) < r then consider n1 being Element of NAT such that A118: for k being Element of NAT st n1 <= k holds abs (((f /* (s * F)) . k) - (lim_left (f,x0))) < r by A89, A88, SEQ_2:def_7; consider n2 being Element of NAT such that A119: for k being Element of NAT st n2 <= k holds abs (((f /* (s * G)) . k) - (lim_left (f,x0))) < r by A115, A114, A117, SEQ_2:def_7; take n = max ((F . n1),(G . n2)); ::_thesis: for k being Element of NAT st n <= k holds abs (((f /* s) . k) - (lim_left (f,x0))) < r let k be Element of NAT ; ::_thesis: ( n <= k implies abs (((f /* s) . k) - (lim_left (f,x0))) < r ) assume A120: n <= k ; ::_thesis: abs (((f /* s) . k) - (lim_left (f,x0))) < r s . k in rng s by VALUED_0:28; then not s . k in {x0} by A7, XBOOLE_0:def_5; then A121: s . k <> x0 by TARSKI:def_1; now__::_thesis:_abs_(((f_/*_s)_._k)_-_(lim_left_(f,x0)))_<_r percases ( s . k < x0 or s . k > x0 ) by A121, XXREAL_0:1; suppose s . k < x0 ; ::_thesis: abs (((f /* s) . k) - (lim_left (f,x0))) < r then consider l being Element of NAT such that A122: k = F . l by A45; F . n1 <= n by XXREAL_0:25; then F . n1 <= k by A120, XXREAL_0:2; then l >= n1 by A122, SEQM_3:1; then abs (((f /* (s * F)) . l) - (lim_left (f,x0))) < r by A118; then abs ((f . ((s * F) . l)) - (lim_left (f,x0))) < r by A44, FUNCT_2:108, XBOOLE_1:1; then abs ((f . (s . k)) - (lim_left (f,x0))) < r by A122, FUNCT_2:15; hence abs (((f /* s) . k) - (lim_left (f,x0))) < r by A7, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; suppose s . k > x0 ; ::_thesis: abs (((f /* s) . k) - (lim_left (f,x0))) < r then consider l being Element of NAT such that A123: k = G . l by A91; G . n2 <= n by XXREAL_0:25; then G . n2 <= k by A120, XXREAL_0:2; then l >= n2 by A123, SEQM_3:1; then abs (((f /* (s * G)) . l) - (lim_left (f,x0))) < r by A119; then abs ((f . ((s * G) . l)) - (lim_left (f,x0))) < r by A86, FUNCT_2:108, XBOOLE_1:1; then abs ((f . (s . k)) - (lim_left (f,x0))) < r by A123, FUNCT_2:15; hence abs (((f /* s) . k) - (lim_left (f,x0))) < r by A7, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; end; end; hence abs (((f /* s) . k) - (lim_left (f,x0))) < r ; ::_thesis: verum end; hence f /* s is convergent by SEQ_2:def_6; ::_thesis: lim (f /* s) = lim_left (f,x0) hence lim (f /* s) = lim_left (f,x0) by A116, SEQ_2:def_7; ::_thesis: verum end; end; end; hence ( f /* s is convergent & lim (f /* s) = lim_left (f,x0) ) ; ::_thesis: verum end; end; end; hence ( f /* s is convergent & lim (f /* s) = lim_left (f,x0) ) ; ::_thesis: verum end; now__::_thesis:_for_r1,_r2_being_Real_st_r1_<_x0_&_x0_<_r2_holds_ ex_g1,_g2_being_Real_st_ (_r1_<_g1_&_g1_<_x0_&_g1_in_dom_f_&_g2_<_r2_&_x0_<_g2_&_g2_in_dom_f_) let r1, r2 be Real; ::_thesis: ( r1 < x0 & x0 < r2 implies ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) assume that A124: r1 < x0 and A125: x0 < r2 ; ::_thesis: ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) consider g1 being Real such that A126: r1 < g1 and A127: g1 < x0 and A128: g1 in dom f by A1, A124, LIMFUNC2:def_1; consider g2 being Real such that A129: g2 < r2 and A130: x0 < g2 and A131: g2 in dom f by A2, A125, LIMFUNC2:def_4; take g1 = g1; ::_thesis: ex g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) take g2 = g2; ::_thesis: ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) thus ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) by A126, A127, A128, A129, A130, A131; ::_thesis: verum end; hence f is_convergent_in x0 by A4, Def1; ::_thesis: ( lim (f,x0) = lim_left (f,x0) & lim (f,x0) = lim_right (f,x0) ) hence ( lim (f,x0) = lim_left (f,x0) & lim (f,x0) = lim_right (f,x0) ) by A3, A4, Def4; ::_thesis: verum end; theorem Th31: :: LIMFUNC3:31 for x0, r being Real for f being PartFunc of REAL,REAL st f is_convergent_in x0 holds ( r (#) f is_convergent_in x0 & lim ((r (#) f),x0) = r * (lim (f,x0)) ) proof let x0, r be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_convergent_in x0 holds ( r (#) f is_convergent_in x0 & lim ((r (#) f),x0) = r * (lim (f,x0)) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_convergent_in x0 implies ( r (#) f is_convergent_in x0 & lim ((r (#) f),x0) = r * (lim (f,x0)) ) ) assume A1: f is_convergent_in x0 ; ::_thesis: ( r (#) f is_convergent_in x0 & lim ((r (#) f),x0) = r * (lim (f,x0)) ) A2: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(r_(#)_f))_\_{x0}_holds_ (_(r_(#)_f)_/*_seq_is_convergent_&_lim_((r_(#)_f)_/*_seq)_=_r_*_(lim_(f,x0))_) let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (r (#) f)) \ {x0} implies ( (r (#) f) /* seq is convergent & lim ((r (#) f) /* seq) = r * (lim (f,x0)) ) ) assume that A3: seq is convergent and A4: lim seq = x0 and A5: rng seq c= (dom (r (#) f)) \ {x0} ; ::_thesis: ( (r (#) f) /* seq is convergent & lim ((r (#) f) /* seq) = r * (lim (f,x0)) ) A6: rng seq c= (dom f) \ {x0} by A5, VALUED_1:def_5; then A7: r (#) (f /* seq) = (r (#) f) /* seq by RFUNCT_2:9, XBOOLE_1:1; lim (f,x0) = lim (f,x0) ; then A8: f /* seq is convergent by A1, A3, A4, A6, Def4; then r (#) (f /* seq) is convergent by SEQ_2:7; hence (r (#) f) /* seq is convergent by A6, RFUNCT_2:9, XBOOLE_1:1; ::_thesis: lim ((r (#) f) /* seq) = r * (lim (f,x0)) lim (f /* seq) = lim (f,x0) by A1, A3, A4, A6, Def4; hence lim ((r (#) f) /* seq) = r * (lim (f,x0)) by A8, A7, SEQ_2:8; ::_thesis: verum end; now__::_thesis:_for_r1,_r2_being_Real_st_r1_<_x0_&_x0_<_r2_holds_ ex_g1,_g2_being_Real_st_ (_r1_<_g1_&_g1_<_x0_&_g1_in_dom_(r_(#)_f)_&_g2_<_r2_&_x0_<_g2_&_g2_in_dom_(r_(#)_f)_) let r1, r2 be Real; ::_thesis: ( r1 < x0 & x0 < r2 implies ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) ) assume that A9: r1 < x0 and A10: x0 < r2 ; ::_thesis: ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) consider g1, g2 being Real such that A11: r1 < g1 and A12: g1 < x0 and A13: g1 in dom f and A14: g2 < r2 and A15: x0 < g2 and A16: g2 in dom f by A1, A9, A10, Def1; take g1 = g1; ::_thesis: ex g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) take g2 = g2; ::_thesis: ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) thus ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) by A11, A12, A13, A14, A15, A16, VALUED_1:def_5; ::_thesis: verum end; hence r (#) f is_convergent_in x0 by A2, Def1; ::_thesis: lim ((r (#) f),x0) = r * (lim (f,x0)) hence lim ((r (#) f),x0) = r * (lim (f,x0)) by A2, Def4; ::_thesis: verum end; theorem Th32: :: LIMFUNC3:32 for x0 being Real for f being PartFunc of REAL,REAL st f is_convergent_in x0 holds ( - f is_convergent_in x0 & lim ((- f),x0) = - (lim (f,x0)) ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_convergent_in x0 holds ( - f is_convergent_in x0 & lim ((- f),x0) = - (lim (f,x0)) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_convergent_in x0 implies ( - f is_convergent_in x0 & lim ((- f),x0) = - (lim (f,x0)) ) ) assume A1: f is_convergent_in x0 ; ::_thesis: ( - f is_convergent_in x0 & lim ((- f),x0) = - (lim (f,x0)) ) (- 1) (#) f = - f ; hence - f is_convergent_in x0 by A1, Th31; ::_thesis: lim ((- f),x0) = - (lim (f,x0)) thus lim ((- f),x0) = (- 1) * (lim (f,x0)) by A1, Th31 .= - (lim (f,x0)) ; ::_thesis: verum end; theorem Th33: :: LIMFUNC3:33 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_convergent_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 + f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 + f2) ) ) holds ( f1 + f2 is_convergent_in x0 & lim ((f1 + f2),x0) = (lim (f1,x0)) + (lim (f2,x0)) ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_convergent_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 + f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 + f2) ) ) holds ( f1 + f2 is_convergent_in x0 & lim ((f1 + f2),x0) = (lim (f1,x0)) + (lim (f2,x0)) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_convergent_in x0 & f2 is_convergent_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 + f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 + f2) ) ) implies ( f1 + f2 is_convergent_in x0 & lim ((f1 + f2),x0) = (lim (f1,x0)) + (lim (f2,x0)) ) ) assume that A1: f1 is_convergent_in x0 and A2: f2 is_convergent_in x0 and A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 + f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 + f2) ) ; ::_thesis: ( f1 + f2 is_convergent_in x0 & lim ((f1 + f2),x0) = (lim (f1,x0)) + (lim (f2,x0)) ) A4: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(f1_+_f2))_\_{x0}_holds_ (_(f1_+_f2)_/*_seq_is_convergent_&_lim_((f1_+_f2)_/*_seq)_=_(lim_(f1,x0))_+_(lim_(f2,x0))_) let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f1 + f2)) \ {x0} implies ( (f1 + f2) /* seq is convergent & lim ((f1 + f2) /* seq) = (lim (f1,x0)) + (lim (f2,x0)) ) ) assume that A5: seq is convergent and A6: lim seq = x0 and A7: rng seq c= (dom (f1 + f2)) \ {x0} ; ::_thesis: ( (f1 + f2) /* seq is convergent & lim ((f1 + f2) /* seq) = (lim (f1,x0)) + (lim (f2,x0)) ) A8: dom (f1 + f2) = (dom f1) /\ (dom f2) by A7, Lm4; A9: rng seq c= (dom f1) \ {x0} by A7, Lm4; A10: rng seq c= (dom f2) \ {x0} by A7, Lm4; then A11: lim (f2 /* seq) = lim (f2,x0) by A2, A5, A6, Def4; lim (f2,x0) = lim (f2,x0) ; then A12: f2 /* seq is convergent by A2, A5, A6, A10, Def4; rng seq c= dom (f1 + f2) by A7, Lm4; then A13: (f1 /* seq) + (f2 /* seq) = (f1 + f2) /* seq by A8, RFUNCT_2:8; lim (f1,x0) = lim (f1,x0) ; then A14: f1 /* seq is convergent by A1, A5, A6, A9, Def4; hence (f1 + f2) /* seq is convergent by A12, A13, SEQ_2:5; ::_thesis: lim ((f1 + f2) /* seq) = (lim (f1,x0)) + (lim (f2,x0)) lim (f1 /* seq) = lim (f1,x0) by A1, A5, A6, A9, Def4; hence lim ((f1 + f2) /* seq) = (lim (f1,x0)) + (lim (f2,x0)) by A14, A12, A11, A13, SEQ_2:6; ::_thesis: verum end; hence f1 + f2 is_convergent_in x0 by A3, Def1; ::_thesis: lim ((f1 + f2),x0) = (lim (f1,x0)) + (lim (f2,x0)) hence lim ((f1 + f2),x0) = (lim (f1,x0)) + (lim (f2,x0)) by A4, Def4; ::_thesis: verum end; theorem :: LIMFUNC3:34 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_convergent_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 - f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 - f2) ) ) holds ( f1 - f2 is_convergent_in x0 & lim ((f1 - f2),x0) = (lim (f1,x0)) - (lim (f2,x0)) ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_convergent_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 - f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 - f2) ) ) holds ( f1 - f2 is_convergent_in x0 & lim ((f1 - f2),x0) = (lim (f1,x0)) - (lim (f2,x0)) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_convergent_in x0 & f2 is_convergent_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 - f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 - f2) ) ) implies ( f1 - f2 is_convergent_in x0 & lim ((f1 - f2),x0) = (lim (f1,x0)) - (lim (f2,x0)) ) ) assume that A1: f1 is_convergent_in x0 and A2: f2 is_convergent_in x0 and A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 - f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 - f2) ) ; ::_thesis: ( f1 - f2 is_convergent_in x0 & lim ((f1 - f2),x0) = (lim (f1,x0)) - (lim (f2,x0)) ) A4: - f2 is_convergent_in x0 by A2, Th32; hence f1 - f2 is_convergent_in x0 by A1, A3, Th33; ::_thesis: lim ((f1 - f2),x0) = (lim (f1,x0)) - (lim (f2,x0)) thus lim ((f1 - f2),x0) = (lim (f1,x0)) + (lim ((- f2),x0)) by A1, A3, A4, Th33 .= (lim (f1,x0)) + (- (lim (f2,x0))) by A2, Th32 .= (lim (f1,x0)) - (lim (f2,x0)) ; ::_thesis: verum end; theorem :: LIMFUNC3:35 for x0 being Real for f being PartFunc of REAL,REAL st f is_convergent_in x0 & f " {0} = {} & lim (f,x0) <> 0 holds ( f ^ is_convergent_in x0 & lim ((f ^),x0) = (lim (f,x0)) " ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_convergent_in x0 & f " {0} = {} & lim (f,x0) <> 0 holds ( f ^ is_convergent_in x0 & lim ((f ^),x0) = (lim (f,x0)) " ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_convergent_in x0 & f " {0} = {} & lim (f,x0) <> 0 implies ( f ^ is_convergent_in x0 & lim ((f ^),x0) = (lim (f,x0)) " ) ) assume that A1: f is_convergent_in x0 and A2: f " {0} = {} and A3: lim (f,x0) <> 0 ; ::_thesis: ( f ^ is_convergent_in x0 & lim ((f ^),x0) = (lim (f,x0)) " ) A4: dom f = (dom f) \ (f " {0}) by A2 .= dom (f ^) by RFUNCT_1:def_2 ; A5: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(f_^))_\_{x0}_holds_ (_(f_^)_/*_seq_is_convergent_&_lim_((f_^)_/*_seq)_=_(lim_(f,x0))_"_) let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f ^)) \ {x0} implies ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = (lim (f,x0)) " ) ) assume that A6: seq is convergent and A7: lim seq = x0 and A8: rng seq c= (dom (f ^)) \ {x0} ; ::_thesis: ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = (lim (f,x0)) " ) A9: lim (f /* seq) = lim (f,x0) by A1, A4, A6, A7, A8, Def4; A10: (f /* seq) " = (f ^) /* seq by A8, RFUNCT_2:12, XBOOLE_1:1; A11: rng seq c= dom f by A4, A8, XBOOLE_1:1; A12: f /* seq is convergent by A1, A3, A4, A6, A7, A8, Def4; hence (f ^) /* seq is convergent by A3, A4, A9, A11, A10, RFUNCT_2:11, SEQ_2:21; ::_thesis: lim ((f ^) /* seq) = (lim (f,x0)) " thus lim ((f ^) /* seq) = (lim (f,x0)) " by A3, A4, A12, A9, A11, A10, RFUNCT_2:11, SEQ_2:22; ::_thesis: verum end; for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) & g2 < r2 & x0 < g2 & g2 in dom (f ^) ) by A1, A4, Def1; hence f ^ is_convergent_in x0 by A5, Def1; ::_thesis: lim ((f ^),x0) = (lim (f,x0)) " hence lim ((f ^),x0) = (lim (f,x0)) " by A5, Def4; ::_thesis: verum end; theorem :: LIMFUNC3:36 for x0 being Real for f being PartFunc of REAL,REAL st f is_convergent_in x0 holds ( abs f is_convergent_in x0 & lim ((abs f),x0) = abs (lim (f,x0)) ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_convergent_in x0 holds ( abs f is_convergent_in x0 & lim ((abs f),x0) = abs (lim (f,x0)) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_convergent_in x0 implies ( abs f is_convergent_in x0 & lim ((abs f),x0) = abs (lim (f,x0)) ) ) assume A1: f is_convergent_in x0 ; ::_thesis: ( abs f is_convergent_in x0 & lim ((abs f),x0) = abs (lim (f,x0)) ) A2: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(abs_f))_\_{x0}_holds_ (_(abs_f)_/*_seq_is_convergent_&_lim_((abs_f)_/*_seq)_=_abs_(lim_(f,x0))_) let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (abs f)) \ {x0} implies ( (abs f) /* seq is convergent & lim ((abs f) /* seq) = abs (lim (f,x0)) ) ) assume that A3: seq is convergent and A4: lim seq = x0 and A5: rng seq c= (dom (abs f)) \ {x0} ; ::_thesis: ( (abs f) /* seq is convergent & lim ((abs f) /* seq) = abs (lim (f,x0)) ) A6: rng seq c= (dom f) \ {x0} by A5, VALUED_1:def_11; then rng seq c= dom f by XBOOLE_1:1; then A7: abs (f /* seq) = (abs f) /* seq by RFUNCT_2:10; lim (f,x0) = lim (f,x0) ; then A8: f /* seq is convergent by A1, A3, A4, A6, Def4; hence (abs f) /* seq is convergent by A7; ::_thesis: lim ((abs f) /* seq) = abs (lim (f,x0)) lim (f /* seq) = lim (f,x0) by A1, A3, A4, A6, Def4; hence lim ((abs f) /* seq) = abs (lim (f,x0)) by A8, A7, SEQ_4:14; ::_thesis: verum end; now__::_thesis:_for_r1,_r2_being_Real_st_r1_<_x0_&_x0_<_r2_holds_ ex_g1,_g2_being_Real_st_ (_r1_<_g1_&_g1_<_x0_&_g1_in_dom_(abs_f)_&_g2_<_r2_&_x0_<_g2_&_g2_in_dom_(abs_f)_) let r1, r2 be Real; ::_thesis: ( r1 < x0 & x0 < r2 implies ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (abs f) & g2 < r2 & x0 < g2 & g2 in dom (abs f) ) ) assume that A9: r1 < x0 and A10: x0 < r2 ; ::_thesis: ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (abs f) & g2 < r2 & x0 < g2 & g2 in dom (abs f) ) consider g1, g2 being Real such that A11: r1 < g1 and A12: g1 < x0 and A13: g1 in dom f and A14: g2 < r2 and A15: x0 < g2 and A16: g2 in dom f by A1, A9, A10, Def1; take g1 = g1; ::_thesis: ex g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (abs f) & g2 < r2 & x0 < g2 & g2 in dom (abs f) ) take g2 = g2; ::_thesis: ( r1 < g1 & g1 < x0 & g1 in dom (abs f) & g2 < r2 & x0 < g2 & g2 in dom (abs f) ) thus ( r1 < g1 & g1 < x0 & g1 in dom (abs f) & g2 < r2 & x0 < g2 & g2 in dom (abs f) ) by A11, A12, A13, A14, A15, A16, VALUED_1:def_11; ::_thesis: verum end; hence abs f is_convergent_in x0 by A2, Def1; ::_thesis: lim ((abs f),x0) = abs (lim (f,x0)) hence lim ((abs f),x0) = abs (lim (f,x0)) by A2, Def4; ::_thesis: verum end; theorem Th37: :: LIMFUNC3:37 for x0 being Real for f being PartFunc of REAL,REAL st f is_convergent_in x0 & lim (f,x0) <> 0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f & f . g1 <> 0 & f . g2 <> 0 ) ) holds ( f ^ is_convergent_in x0 & lim ((f ^),x0) = (lim (f,x0)) " ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_convergent_in x0 & lim (f,x0) <> 0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f & f . g1 <> 0 & f . g2 <> 0 ) ) holds ( f ^ is_convergent_in x0 & lim ((f ^),x0) = (lim (f,x0)) " ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_convergent_in x0 & lim (f,x0) <> 0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f & f . g1 <> 0 & f . g2 <> 0 ) ) implies ( f ^ is_convergent_in x0 & lim ((f ^),x0) = (lim (f,x0)) " ) ) assume that A1: f is_convergent_in x0 and A2: lim (f,x0) <> 0 and A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f & f . g1 <> 0 & f . g2 <> 0 ) ; ::_thesis: ( f ^ is_convergent_in x0 & lim ((f ^),x0) = (lim (f,x0)) " ) A4: (dom f) \ (f " {0}) = dom (f ^) by RFUNCT_1:def_2; A5: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(f_^))_\_{x0}_holds_ (_(f_^)_/*_seq_is_convergent_&_lim_((f_^)_/*_seq)_=_(lim_(f,x0))_"_) let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f ^)) \ {x0} implies ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = (lim (f,x0)) " ) ) assume that A6: seq is convergent and A7: lim seq = x0 and A8: rng seq c= (dom (f ^)) \ {x0} ; ::_thesis: ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = (lim (f,x0)) " ) A9: f /* seq is non-zero by A8, RFUNCT_2:11, XBOOLE_1:1; rng seq c= dom (f ^) by A8, XBOOLE_1:1; then A10: rng seq c= dom f by A4, XBOOLE_1:1; now__::_thesis:_for_x_being_set_st_x_in_rng_seq_holds_ x_in_(dom_f)_\_{x0} let x be set ; ::_thesis: ( x in rng seq implies x in (dom f) \ {x0} ) assume A11: x in rng seq ; ::_thesis: x in (dom f) \ {x0} then not x in {x0} by A8, XBOOLE_0:def_5; hence x in (dom f) \ {x0} by A10, A11, XBOOLE_0:def_5; ::_thesis: verum end; then A12: rng seq c= (dom f) \ {x0} by TARSKI:def_3; then A13: lim (f /* seq) = lim (f,x0) by A1, A6, A7, Def4; A14: (f /* seq) " = (f ^) /* seq by A8, RFUNCT_2:12, XBOOLE_1:1; A15: f /* seq is convergent by A1, A2, A6, A7, A12, Def4; hence (f ^) /* seq is convergent by A2, A13, A9, A14, SEQ_2:21; ::_thesis: lim ((f ^) /* seq) = (lim (f,x0)) " thus lim ((f ^) /* seq) = (lim (f,x0)) " by A2, A15, A13, A9, A14, SEQ_2:22; ::_thesis: verum end; now__::_thesis:_for_r1,_r2_being_Real_st_r1_<_x0_&_x0_<_r2_holds_ ex_g1,_g2_being_Real_st_ (_r1_<_g1_&_g1_<_x0_&_g1_in_dom_(f_^)_&_g2_<_r2_&_x0_<_g2_&_g2_in_dom_(f_^)_) let r1, r2 be Real; ::_thesis: ( r1 < x0 & x0 < r2 implies ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) & g2 < r2 & x0 < g2 & g2 in dom (f ^) ) ) assume that A16: r1 < x0 and A17: x0 < r2 ; ::_thesis: ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) & g2 < r2 & x0 < g2 & g2 in dom (f ^) ) consider g1, g2 being Real such that A18: r1 < g1 and A19: g1 < x0 and A20: g1 in dom f and A21: g2 < r2 and A22: x0 < g2 and A23: g2 in dom f and A24: f . g1 <> 0 and A25: f . g2 <> 0 by A3, A16, A17; take g1 = g1; ::_thesis: ex g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) & g2 < r2 & x0 < g2 & g2 in dom (f ^) ) take g2 = g2; ::_thesis: ( r1 < g1 & g1 < x0 & g1 in dom (f ^) & g2 < r2 & x0 < g2 & g2 in dom (f ^) ) not f . g2 in {0} by A25, TARSKI:def_1; then A26: not g2 in f " {0} by FUNCT_1:def_7; not f . g1 in {0} by A24, TARSKI:def_1; then not g1 in f " {0} by FUNCT_1:def_7; hence ( r1 < g1 & g1 < x0 & g1 in dom (f ^) & g2 < r2 & x0 < g2 & g2 in dom (f ^) ) by A4, A18, A19, A20, A21, A22, A23, A26, XBOOLE_0:def_5; ::_thesis: verum end; hence f ^ is_convergent_in x0 by A5, Def1; ::_thesis: lim ((f ^),x0) = (lim (f,x0)) " hence lim ((f ^),x0) = (lim (f,x0)) " by A5, Def4; ::_thesis: verum end; theorem Th38: :: LIMFUNC3:38 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_convergent_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 (#) f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 (#) f2) ) ) holds ( f1 (#) f2 is_convergent_in x0 & lim ((f1 (#) f2),x0) = (lim (f1,x0)) * (lim (f2,x0)) ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_convergent_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 (#) f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 (#) f2) ) ) holds ( f1 (#) f2 is_convergent_in x0 & lim ((f1 (#) f2),x0) = (lim (f1,x0)) * (lim (f2,x0)) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_convergent_in x0 & f2 is_convergent_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 (#) f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 (#) f2) ) ) implies ( f1 (#) f2 is_convergent_in x0 & lim ((f1 (#) f2),x0) = (lim (f1,x0)) * (lim (f2,x0)) ) ) assume that A1: f1 is_convergent_in x0 and A2: f2 is_convergent_in x0 and A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 (#) f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 (#) f2) ) ; ::_thesis: ( f1 (#) f2 is_convergent_in x0 & lim ((f1 (#) f2),x0) = (lim (f1,x0)) * (lim (f2,x0)) ) A4: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(f1_(#)_f2))_\_{x0}_holds_ (_(f1_(#)_f2)_/*_seq_is_convergent_&_lim_((f1_(#)_f2)_/*_seq)_=_(lim_(f1,x0))_*_(lim_(f2,x0))_) let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f1 (#) f2)) \ {x0} implies ( (f1 (#) f2) /* seq is convergent & lim ((f1 (#) f2) /* seq) = (lim (f1,x0)) * (lim (f2,x0)) ) ) assume that A5: seq is convergent and A6: lim seq = x0 and A7: rng seq c= (dom (f1 (#) f2)) \ {x0} ; ::_thesis: ( (f1 (#) f2) /* seq is convergent & lim ((f1 (#) f2) /* seq) = (lim (f1,x0)) * (lim (f2,x0)) ) A8: dom (f1 (#) f2) = (dom f1) /\ (dom f2) by A7, Lm2; A9: rng seq c= (dom f1) \ {x0} by A7, Lm2; A10: rng seq c= (dom f2) \ {x0} by A7, Lm2; then A11: lim (f2 /* seq) = lim (f2,x0) by A2, A5, A6, Def4; lim (f2,x0) = lim (f2,x0) ; then A12: f2 /* seq is convergent by A2, A5, A6, A10, Def4; rng seq c= dom (f1 (#) f2) by A7, Lm2; then A13: (f1 /* seq) (#) (f2 /* seq) = (f1 (#) f2) /* seq by A8, RFUNCT_2:8; lim (f1,x0) = lim (f1,x0) ; then A14: f1 /* seq is convergent by A1, A5, A6, A9, Def4; hence (f1 (#) f2) /* seq is convergent by A12, A13, SEQ_2:14; ::_thesis: lim ((f1 (#) f2) /* seq) = (lim (f1,x0)) * (lim (f2,x0)) lim (f1 /* seq) = lim (f1,x0) by A1, A5, A6, A9, Def4; hence lim ((f1 (#) f2) /* seq) = (lim (f1,x0)) * (lim (f2,x0)) by A14, A12, A11, A13, SEQ_2:15; ::_thesis: verum end; hence f1 (#) f2 is_convergent_in x0 by A3, Def1; ::_thesis: lim ((f1 (#) f2),x0) = (lim (f1,x0)) * (lim (f2,x0)) hence lim ((f1 (#) f2),x0) = (lim (f1,x0)) * (lim (f2,x0)) by A4, Def4; ::_thesis: verum end; theorem :: LIMFUNC3:39 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_convergent_in x0 & lim (f2,x0) <> 0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 / f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 / f2) ) ) holds ( f1 / f2 is_convergent_in x0 & lim ((f1 / f2),x0) = (lim (f1,x0)) / (lim (f2,x0)) ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_convergent_in x0 & lim (f2,x0) <> 0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 / f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 / f2) ) ) holds ( f1 / f2 is_convergent_in x0 & lim ((f1 / f2),x0) = (lim (f1,x0)) / (lim (f2,x0)) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_convergent_in x0 & f2 is_convergent_in x0 & lim (f2,x0) <> 0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 / f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 / f2) ) ) implies ( f1 / f2 is_convergent_in x0 & lim ((f1 / f2),x0) = (lim (f1,x0)) / (lim (f2,x0)) ) ) assume that A1: f1 is_convergent_in x0 and A2: f2 is_convergent_in x0 and A3: lim (f2,x0) <> 0 and A4: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 / f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 / f2) ) ; ::_thesis: ( f1 / f2 is_convergent_in x0 & lim ((f1 / f2),x0) = (lim (f1,x0)) / (lim (f2,x0)) ) A5: now__::_thesis:_for_r1,_r2_being_Real_st_r1_<_x0_&_x0_<_r2_holds_ ex_g1,_g2_being_Real_st_ (_r1_<_g1_&_g1_<_x0_&_g1_in_dom_f2_&_g2_<_r2_&_x0_<_g2_&_g2_in_dom_f2_&_f2_._g1_<>_0_&_f2_._g2_<>_0_) let r1, r2 be Real; ::_thesis: ( r1 < x0 & x0 < r2 implies ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f2 & g2 < r2 & x0 < g2 & g2 in dom f2 & f2 . g1 <> 0 & f2 . g2 <> 0 ) ) assume that A6: r1 < x0 and A7: x0 < r2 ; ::_thesis: ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f2 & g2 < r2 & x0 < g2 & g2 in dom f2 & f2 . g1 <> 0 & f2 . g2 <> 0 ) consider g1, g2 being Real such that A8: r1 < g1 and A9: g1 < x0 and A10: g1 in dom (f1 / f2) and A11: g2 < r2 and A12: x0 < g2 and A13: g2 in dom (f1 / f2) by A4, A6, A7; take g1 = g1; ::_thesis: ex g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f2 & g2 < r2 & x0 < g2 & g2 in dom f2 & f2 . g1 <> 0 & f2 . g2 <> 0 ) take g2 = g2; ::_thesis: ( r1 < g1 & g1 < x0 & g1 in dom f2 & g2 < r2 & x0 < g2 & g2 in dom f2 & f2 . g1 <> 0 & f2 . g2 <> 0 ) thus ( r1 < g1 & g1 < x0 ) by A8, A9; ::_thesis: ( g1 in dom f2 & g2 < r2 & x0 < g2 & g2 in dom f2 & f2 . g1 <> 0 & f2 . g2 <> 0 ) A14: dom (f1 / f2) = (dom f1) /\ ((dom f2) \ (f2 " {0})) by RFUNCT_1:def_1; then g2 in (dom f2) \ (f2 " {0}) by A13, XBOOLE_0:def_4; then not g2 in f2 " {0} by XBOOLE_0:def_5; then A15: not f2 . g2 in {0} by A13, A14, FUNCT_1:def_7; g1 in (dom f2) \ (f2 " {0}) by A10, A14, XBOOLE_0:def_4; then not g1 in f2 " {0} by XBOOLE_0:def_5; then not f2 . g1 in {0} by A10, A14, FUNCT_1:def_7; hence ( g1 in dom f2 & g2 < r2 & x0 < g2 & g2 in dom f2 & f2 . g1 <> 0 & f2 . g2 <> 0 ) by A10, A11, A12, A13, A14, A15, TARSKI:def_1; ::_thesis: verum end; then A16: f2 ^ is_convergent_in x0 by A2, A3, Th37; A17: f1 / f2 = f1 (#) (f2 ^) by RFUNCT_1:31; hence f1 / f2 is_convergent_in x0 by A1, A4, A16, Th38; ::_thesis: lim ((f1 / f2),x0) = (lim (f1,x0)) / (lim (f2,x0)) lim ((f2 ^),x0) = (lim (f2,x0)) " by A2, A3, A5, Th37; hence lim ((f1 / f2),x0) = (lim (f1,x0)) * ((lim (f2,x0)) ") by A1, A4, A17, A16, Th38 .= (lim (f1,x0)) / (lim (f2,x0)) by XCMPLX_0:def_9 ; ::_thesis: verum end; theorem :: LIMFUNC3:40 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & lim (f1,x0) = 0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 (#) f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 (#) f2) ) ) & ex r being Real st ( 0 < r & f2 | (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) is bounded ) holds ( f1 (#) f2 is_convergent_in x0 & lim ((f1 (#) f2),x0) = 0 ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & lim (f1,x0) = 0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 (#) f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 (#) f2) ) ) & ex r being Real st ( 0 < r & f2 | (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) is bounded ) holds ( f1 (#) f2 is_convergent_in x0 & lim ((f1 (#) f2),x0) = 0 ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_convergent_in x0 & lim (f1,x0) = 0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 (#) f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 (#) f2) ) ) & ex r being Real st ( 0 < r & f2 | (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) is bounded ) implies ( f1 (#) f2 is_convergent_in x0 & lim ((f1 (#) f2),x0) = 0 ) ) assume that A1: f1 is_convergent_in x0 and A2: lim (f1,x0) = 0 and A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f1 (#) f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 (#) f2) ) ; ::_thesis: ( for r being Real holds ( not 0 < r or not f2 | (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) is bounded ) or ( f1 (#) f2 is_convergent_in x0 & lim ((f1 (#) f2),x0) = 0 ) ) given r being Real such that A4: 0 < r and A5: f2 | (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) is bounded ; ::_thesis: ( f1 (#) f2 is_convergent_in x0 & lim ((f1 (#) f2),x0) = 0 ) consider g being real number such that A6: for r1 being set st r1 in (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) /\ (dom f2) holds abs (f2 . r1) <= g by A5, RFUNCT_1:73; A7: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f1_(#)_f2))_\_{x0}_holds_ (_(f1_(#)_f2)_/*_s_is_convergent_&_lim_((f1_(#)_f2)_/*_s)_=_0_) let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f1 (#) f2)) \ {x0} implies ( (f1 (#) f2) /* s is convergent & lim ((f1 (#) f2) /* s) = 0 ) ) assume that A8: s is convergent and A9: lim s = x0 and A10: rng s c= (dom (f1 (#) f2)) \ {x0} ; ::_thesis: ( (f1 (#) f2) /* s is convergent & lim ((f1 (#) f2) /* s) = 0 ) consider k being Element of NAT such that A11: for n being Element of NAT st k <= n holds ( x0 - r < s . n & s . n < x0 + r ) by A4, A8, A9, Th7; A12: rng (s ^\ k) c= rng s by VALUED_0:21; rng s c= (dom f1) \ {x0} by A10, Lm2; then A13: rng (s ^\ k) c= (dom f1) \ {x0} by A12, XBOOLE_1:1; A14: lim (s ^\ k) = x0 by A8, A9, SEQ_4:20; then A15: f1 /* (s ^\ k) is convergent by A1, A2, A8, A13, Def4; A16: rng s c= dom f2 by A10, Lm2; then A17: rng (s ^\ k) c= dom f2 by A12, XBOOLE_1:1; now__::_thesis:_(_0_<_(abs_g)_+_1_&_(_for_n_being_Element_of_NAT_holds_abs_((f2_/*_(s_^\_k))_._n)_<_(abs_g)_+_1_)_) set t = (abs g) + 1; 0 <= abs g by COMPLEX1:46; hence 0 < (abs g) + 1 ; ::_thesis: for n being Element of NAT holds abs ((f2 /* (s ^\ k)) . n) < (abs g) + 1 let n be Element of NAT ; ::_thesis: abs ((f2 /* (s ^\ k)) . n) < (abs g) + 1 A18: k <= n + k by NAT_1:12; then s . (n + k) < x0 + r by A11; then A19: (s ^\ k) . n < x0 + r by NAT_1:def_3; x0 - r < s . (n + k) by A11, A18; then x0 - r < (s ^\ k) . n by NAT_1:def_3; then (s ^\ k) . n in { g1 where g1 is Real : ( x0 - r < g1 & g1 < x0 + r ) } by A19; then A20: (s ^\ k) . n in ].(x0 - r),(x0 + r).[ by RCOMP_1:def_2; A21: (s ^\ k) . n in rng (s ^\ k) by VALUED_0:28; then not (s ^\ k) . n in {x0} by A13, XBOOLE_0:def_5; then (s ^\ k) . n in ].(x0 - r),(x0 + r).[ \ {x0} by A20, XBOOLE_0:def_5; then (s ^\ k) . n in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ by A4, Th4; then (s ^\ k) . n in (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) /\ (dom f2) by A17, A21, XBOOLE_0:def_4; then abs (f2 . ((s ^\ k) . n)) <= g by A6; then A22: abs ((f2 /* (s ^\ k)) . n) <= g by A16, A12, FUNCT_2:108, XBOOLE_1:1; g <= abs g by ABSVALUE:4; then g < (abs g) + 1 by Lm1; hence abs ((f2 /* (s ^\ k)) . n) < (abs g) + 1 by A22, XXREAL_0:2; ::_thesis: verum end; then A23: f2 /* (s ^\ k) is bounded by SEQ_2:3; A24: rng s c= dom (f1 (#) f2) by A10, Lm2; dom (f1 (#) f2) = (dom f1) /\ (dom f2) by A10, Lm2; then rng (s ^\ k) c= (dom f1) /\ (dom f2) by A24, A12, XBOOLE_1:1; then A25: (f1 /* (s ^\ k)) (#) (f2 /* (s ^\ k)) = (f1 (#) f2) /* (s ^\ k) by RFUNCT_2:8 .= ((f1 (#) f2) /* s) ^\ k by A24, VALUED_0:27 ; A26: lim (f1 /* (s ^\ k)) = 0 by A1, A2, A8, A14, A13, Def4; then A27: (f1 /* (s ^\ k)) (#) (f2 /* (s ^\ k)) is convergent by A15, A23, SEQ_2:25; hence (f1 (#) f2) /* s is convergent by A25, SEQ_4:21; ::_thesis: lim ((f1 (#) f2) /* s) = 0 lim ((f1 /* (s ^\ k)) (#) (f2 /* (s ^\ k))) = 0 by A15, A26, A23, SEQ_2:26; hence lim ((f1 (#) f2) /* s) = 0 by A27, A25, SEQ_4:22; ::_thesis: verum end; hence f1 (#) f2 is_convergent_in x0 by A3, Def1; ::_thesis: lim ((f1 (#) f2),x0) = 0 hence lim ((f1 (#) f2),x0) = 0 by A7, Def4; ::_thesis: verum end; theorem Th41: :: LIMFUNC3:41 for x0 being Real for f1, f2, f being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_convergent_in x0 & lim (f1,x0) = lim (f2,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds ( f1 . g <= f . g & f . g <= f2 . g ) ) & ( ( (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) ) or ( (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) ) ) ) holds ( f is_convergent_in x0 & lim (f,x0) = lim (f1,x0) ) proof let x0 be Real; ::_thesis: for f1, f2, f being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_convergent_in x0 & lim (f1,x0) = lim (f2,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds ( f1 . g <= f . g & f . g <= f2 . g ) ) & ( ( (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) ) or ( (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) ) ) ) holds ( f is_convergent_in x0 & lim (f,x0) = lim (f1,x0) ) let f1, f2, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is_convergent_in x0 & f2 is_convergent_in x0 & lim (f1,x0) = lim (f2,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds ( f1 . g <= f . g & f . g <= f2 . g ) ) & ( ( (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) ) or ( (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) ) ) ) implies ( f is_convergent_in x0 & lim (f,x0) = lim (f1,x0) ) ) assume that A1: f1 is_convergent_in x0 and A2: f2 is_convergent_in x0 and A3: lim (f1,x0) = lim (f2,x0) and A4: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) ; ::_thesis: ( for r being Real holds ( not 0 < r or ex g being Real st ( g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & not ( f1 . g <= f . g & f . g <= f2 . g ) ) or ( not ( (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) ) & not ( (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) ) ) ) or ( f is_convergent_in x0 & lim (f,x0) = lim (f1,x0) ) ) given r1 being Real such that A5: 0 < r1 and A6: for g being Real st g in (dom f) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) holds ( f1 . g <= f . g & f . g <= f2 . g ) and A7: ( ( (dom f1) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) c= (dom f2) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) & (dom f) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) c= (dom f1) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) ) or ( (dom f2) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) c= (dom f1) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) & (dom f) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) c= (dom f2) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) ) ) ; ::_thesis: ( f is_convergent_in x0 & lim (f,x0) = lim (f1,x0) ) now__::_thesis:_(_f_is_convergent_in_x0_&_f_is_convergent_in_x0_&_lim_(f,x0)_=_lim_(f1,x0)_) percases ( ( (dom f1) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) c= (dom f2) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) & (dom f) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) c= (dom f1) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) ) or ( (dom f2) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) c= (dom f1) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) & (dom f) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) c= (dom f2) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) ) ) by A7; supposeA8: ( (dom f1) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) c= (dom f2) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) & (dom f) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) c= (dom f1) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) ) ; ::_thesis: ( f is_convergent_in x0 & f is_convergent_in x0 & lim (f,x0) = lim (f1,x0) ) A9: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_f)_\_{x0}_holds_ (_f_/*_s_is_convergent_&_lim_(f_/*_s)_=_lim_(f1,x0)_) let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom f) \ {x0} implies ( f /* s is convergent & lim (f /* s) = lim (f1,x0) ) ) assume that A10: s is convergent and A11: lim s = x0 and A12: rng s c= (dom f) \ {x0} ; ::_thesis: ( f /* s is convergent & lim (f /* s) = lim (f1,x0) ) consider k being Element of NAT such that A13: for n being Element of NAT st k <= n holds ( x0 - r1 < s . n & s . n < x0 + r1 ) by A5, A10, A11, Th7; A14: rng (s ^\ k) c= rng s by VALUED_0:21; then A15: rng (s ^\ k) c= (dom f) \ {x0} by A12, XBOOLE_1:1; now__::_thesis:_for_x_being_set_st_x_in_rng_(s_^\_k)_holds_ x_in_].(x0_-_r1),x0.[_\/_].x0,(x0_+_r1).[ let x be set ; ::_thesis: ( x in rng (s ^\ k) implies x in ].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[ ) assume A16: x in rng (s ^\ k) ; ::_thesis: x in ].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[ then consider n being Element of NAT such that A17: x = (s ^\ k) . n by FUNCT_2:113; A18: k <= n + k by NAT_1:12; then s . (n + k) < x0 + r1 by A13; then A19: (s ^\ k) . n < x0 + r1 by NAT_1:def_3; x0 - r1 < s . (n + k) by A13, A18; then x0 - r1 < (s ^\ k) . n by NAT_1:def_3; then (s ^\ k) . n in { g1 where g1 is Real : ( x0 - r1 < g1 & g1 < x0 + r1 ) } by A19; then A20: (s ^\ k) . n in ].(x0 - r1),(x0 + r1).[ by RCOMP_1:def_2; not (s ^\ k) . n in {x0} by A15, A16, A17, XBOOLE_0:def_5; then x in ].(x0 - r1),(x0 + r1).[ \ {x0} by A17, A20, XBOOLE_0:def_5; hence x in ].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[ by A5, Th4; ::_thesis: verum end; then A21: rng (s ^\ k) c= ].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[ by TARSKI:def_3; A22: rng s c= dom f by A12, XBOOLE_1:1; then rng (s ^\ k) c= dom f by A14, XBOOLE_1:1; then A23: rng (s ^\ k) c= (dom f) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) by A21, XBOOLE_1:19; then A24: rng (s ^\ k) c= (dom f1) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) by A8, XBOOLE_1:1; then A25: rng (s ^\ k) c= (dom f2) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) by A8, XBOOLE_1:1; A26: lim (s ^\ k) = x0 by A10, A11, SEQ_4:20; A27: (dom f2) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) c= dom f2 by XBOOLE_1:17; then A28: rng (s ^\ k) c= dom f2 by A25, XBOOLE_1:1; A29: rng (s ^\ k) c= (dom f2) \ {x0} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (s ^\ k) or x in (dom f2) \ {x0} ) assume A30: x in rng (s ^\ k) ; ::_thesis: x in (dom f2) \ {x0} then not x in {x0} by A15, XBOOLE_0:def_5; hence x in (dom f2) \ {x0} by A28, A30, XBOOLE_0:def_5; ::_thesis: verum end; then A31: lim (f2 /* (s ^\ k)) = lim (f2,x0) by A2, A10, A26, Def4; A32: (dom f1) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) c= dom f1 by XBOOLE_1:17; then A33: rng (s ^\ k) c= dom f1 by A24, XBOOLE_1:1; A34: rng (s ^\ k) c= (dom f1) \ {x0} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (s ^\ k) or x in (dom f1) \ {x0} ) assume A35: x in rng (s ^\ k) ; ::_thesis: x in (dom f1) \ {x0} then not x in {x0} by A15, XBOOLE_0:def_5; hence x in (dom f1) \ {x0} by A33, A35, XBOOLE_0:def_5; ::_thesis: verum end; then A36: lim (f1 /* (s ^\ k)) = lim (f1,x0) by A1, A10, A26, Def4; A37: now__::_thesis:_for_n_being_Element_of_NAT_holds_ (_(f1_/*_(s_^\_k))_._n_<=_(f_/*_(s_^\_k))_._n_&_(f_/*_(s_^\_k))_._n_<=_(f2_/*_(s_^\_k))_._n_) let n be Element of NAT ; ::_thesis: ( (f1 /* (s ^\ k)) . n <= (f /* (s ^\ k)) . n & (f /* (s ^\ k)) . n <= (f2 /* (s ^\ k)) . n ) A38: (s ^\ k) . n in rng (s ^\ k) by VALUED_0:28; then f . ((s ^\ k) . n) <= f2 . ((s ^\ k) . n) by A6, A23; then A39: (f /* (s ^\ k)) . n <= f2 . ((s ^\ k) . n) by A14, A22, FUNCT_2:108, XBOOLE_1:1; f1 . ((s ^\ k) . n) <= f . ((s ^\ k) . n) by A6, A23, A38; then f1 . ((s ^\ k) . n) <= (f /* (s ^\ k)) . n by A14, A22, FUNCT_2:108, XBOOLE_1:1; hence ( (f1 /* (s ^\ k)) . n <= (f /* (s ^\ k)) . n & (f /* (s ^\ k)) . n <= (f2 /* (s ^\ k)) . n ) by A32, A27, A24, A25, A39, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; A40: f2 /* (s ^\ k) is convergent by A2, A3, A10, A26, A29, Def4; A41: f1 /* (s ^\ k) is convergent by A1, A3, A10, A26, A34, Def4; then f /* (s ^\ k) is convergent by A3, A36, A40, A31, A37, SEQ_2:19; then A42: (f /* s) ^\ k is convergent by A12, VALUED_0:27, XBOOLE_1:1; hence f /* s is convergent by SEQ_4:21; ::_thesis: lim (f /* s) = lim (f1,x0) lim (f /* (s ^\ k)) = lim (f1,x0) by A3, A41, A36, A40, A31, A37, SEQ_2:20; then lim ((f /* s) ^\ k) = lim (f1,x0) by A12, VALUED_0:27, XBOOLE_1:1; hence lim (f /* s) = lim (f1,x0) by A42, SEQ_4:22; ::_thesis: verum end; hence f is_convergent_in x0 by A4, Def1; ::_thesis: ( f is_convergent_in x0 & lim (f,x0) = lim (f1,x0) ) hence ( f is_convergent_in x0 & lim (f,x0) = lim (f1,x0) ) by A9, Def4; ::_thesis: verum end; supposeA43: ( (dom f2) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) c= (dom f1) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) & (dom f) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) c= (dom f2) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) ) ; ::_thesis: ( f is_convergent_in x0 & f is_convergent_in x0 & lim (f,x0) = lim (f1,x0) ) A44: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_f)_\_{x0}_holds_ (_f_/*_s_is_convergent_&_lim_(f_/*_s)_=_lim_(f1,x0)_) let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom f) \ {x0} implies ( f /* s is convergent & lim (f /* s) = lim (f1,x0) ) ) assume that A45: s is convergent and A46: lim s = x0 and A47: rng s c= (dom f) \ {x0} ; ::_thesis: ( f /* s is convergent & lim (f /* s) = lim (f1,x0) ) consider k being Element of NAT such that A48: for n being Element of NAT st k <= n holds ( x0 - r1 < s . n & s . n < x0 + r1 ) by A5, A45, A46, Th7; A49: rng (s ^\ k) c= rng s by VALUED_0:21; then A50: rng (s ^\ k) c= (dom f) \ {x0} by A47, XBOOLE_1:1; now__::_thesis:_for_x_being_set_st_x_in_rng_(s_^\_k)_holds_ x_in_].(x0_-_r1),x0.[_\/_].x0,(x0_+_r1).[ let x be set ; ::_thesis: ( x in rng (s ^\ k) implies x in ].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[ ) assume A51: x in rng (s ^\ k) ; ::_thesis: x in ].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[ then consider n being Element of NAT such that A52: x = (s ^\ k) . n by FUNCT_2:113; A53: k <= n + k by NAT_1:12; then s . (n + k) < x0 + r1 by A48; then A54: (s ^\ k) . n < x0 + r1 by NAT_1:def_3; x0 - r1 < s . (n + k) by A48, A53; then x0 - r1 < (s ^\ k) . n by NAT_1:def_3; then (s ^\ k) . n in { g1 where g1 is Real : ( x0 - r1 < g1 & g1 < x0 + r1 ) } by A54; then A55: (s ^\ k) . n in ].(x0 - r1),(x0 + r1).[ by RCOMP_1:def_2; not (s ^\ k) . n in {x0} by A50, A51, A52, XBOOLE_0:def_5; then x in ].(x0 - r1),(x0 + r1).[ \ {x0} by A52, A55, XBOOLE_0:def_5; hence x in ].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[ by A5, Th4; ::_thesis: verum end; then A56: rng (s ^\ k) c= ].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[ by TARSKI:def_3; A57: rng s c= dom f by A47, XBOOLE_1:1; then rng (s ^\ k) c= dom f by A49, XBOOLE_1:1; then A58: rng (s ^\ k) c= (dom f) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) by A56, XBOOLE_1:19; then A59: rng (s ^\ k) c= (dom f2) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) by A43, XBOOLE_1:1; then A60: rng (s ^\ k) c= (dom f1) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) by A43, XBOOLE_1:1; A61: lim (s ^\ k) = x0 by A45, A46, SEQ_4:20; A62: (dom f2) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) c= dom f2 by XBOOLE_1:17; then A63: rng (s ^\ k) c= dom f2 by A59, XBOOLE_1:1; A64: rng (s ^\ k) c= (dom f2) \ {x0} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (s ^\ k) or x in (dom f2) \ {x0} ) assume A65: x in rng (s ^\ k) ; ::_thesis: x in (dom f2) \ {x0} then not x in {x0} by A50, XBOOLE_0:def_5; hence x in (dom f2) \ {x0} by A63, A65, XBOOLE_0:def_5; ::_thesis: verum end; then A66: lim (f2 /* (s ^\ k)) = lim (f2,x0) by A2, A45, A61, Def4; A67: (dom f1) /\ (].(x0 - r1),x0.[ \/ ].x0,(x0 + r1).[) c= dom f1 by XBOOLE_1:17; then A68: rng (s ^\ k) c= dom f1 by A60, XBOOLE_1:1; A69: rng (s ^\ k) c= (dom f1) \ {x0} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (s ^\ k) or x in (dom f1) \ {x0} ) assume A70: x in rng (s ^\ k) ; ::_thesis: x in (dom f1) \ {x0} then not x in {x0} by A50, XBOOLE_0:def_5; hence x in (dom f1) \ {x0} by A68, A70, XBOOLE_0:def_5; ::_thesis: verum end; then A71: lim (f1 /* (s ^\ k)) = lim (f1,x0) by A1, A45, A61, Def4; A72: now__::_thesis:_for_n_being_Element_of_NAT_holds_ (_(f1_/*_(s_^\_k))_._n_<=_(f_/*_(s_^\_k))_._n_&_(f_/*_(s_^\_k))_._n_<=_(f2_/*_(s_^\_k))_._n_) let n be Element of NAT ; ::_thesis: ( (f1 /* (s ^\ k)) . n <= (f /* (s ^\ k)) . n & (f /* (s ^\ k)) . n <= (f2 /* (s ^\ k)) . n ) A73: (s ^\ k) . n in rng (s ^\ k) by VALUED_0:28; then f . ((s ^\ k) . n) <= f2 . ((s ^\ k) . n) by A6, A58; then A74: (f /* (s ^\ k)) . n <= f2 . ((s ^\ k) . n) by A49, A57, FUNCT_2:108, XBOOLE_1:1; f1 . ((s ^\ k) . n) <= f . ((s ^\ k) . n) by A6, A58, A73; then f1 . ((s ^\ k) . n) <= (f /* (s ^\ k)) . n by A49, A57, FUNCT_2:108, XBOOLE_1:1; hence ( (f1 /* (s ^\ k)) . n <= (f /* (s ^\ k)) . n & (f /* (s ^\ k)) . n <= (f2 /* (s ^\ k)) . n ) by A67, A62, A59, A60, A74, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; A75: f2 /* (s ^\ k) is convergent by A2, A3, A45, A61, A64, Def4; A76: f1 /* (s ^\ k) is convergent by A1, A3, A45, A61, A69, Def4; then f /* (s ^\ k) is convergent by A3, A71, A75, A66, A72, SEQ_2:19; then A77: (f /* s) ^\ k is convergent by A47, VALUED_0:27, XBOOLE_1:1; hence f /* s is convergent by SEQ_4:21; ::_thesis: lim (f /* s) = lim (f1,x0) lim (f /* (s ^\ k)) = lim (f1,x0) by A3, A76, A71, A75, A66, A72, SEQ_2:20; then lim ((f /* s) ^\ k) = lim (f1,x0) by A47, VALUED_0:27, XBOOLE_1:1; hence lim (f /* s) = lim (f1,x0) by A77, SEQ_4:22; ::_thesis: verum end; hence f is_convergent_in x0 by A4, Def1; ::_thesis: ( f is_convergent_in x0 & lim (f,x0) = lim (f1,x0) ) hence ( f is_convergent_in x0 & lim (f,x0) = lim (f1,x0) ) by A44, Def4; ::_thesis: verum end; end; end; hence ( f is_convergent_in x0 & lim (f,x0) = lim (f1,x0) ) ; ::_thesis: verum end; theorem :: LIMFUNC3:42 for x0 being Real for f1, f2, f being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_convergent_in x0 & lim (f1,x0) = lim (f2,x0) & ex r being Real st ( 0 < r & ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ c= ((dom f1) /\ (dom f2)) /\ (dom f) & ( for g being Real st g in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ holds ( f1 . g <= f . g & f . g <= f2 . g ) ) ) holds ( f is_convergent_in x0 & lim (f,x0) = lim (f1,x0) ) proof let x0 be Real; ::_thesis: for f1, f2, f being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_convergent_in x0 & lim (f1,x0) = lim (f2,x0) & ex r being Real st ( 0 < r & ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ c= ((dom f1) /\ (dom f2)) /\ (dom f) & ( for g being Real st g in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ holds ( f1 . g <= f . g & f . g <= f2 . g ) ) ) holds ( f is_convergent_in x0 & lim (f,x0) = lim (f1,x0) ) let f1, f2, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is_convergent_in x0 & f2 is_convergent_in x0 & lim (f1,x0) = lim (f2,x0) & ex r being Real st ( 0 < r & ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ c= ((dom f1) /\ (dom f2)) /\ (dom f) & ( for g being Real st g in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ holds ( f1 . g <= f . g & f . g <= f2 . g ) ) ) implies ( f is_convergent_in x0 & lim (f,x0) = lim (f1,x0) ) ) assume that A1: f1 is_convergent_in x0 and A2: f2 is_convergent_in x0 and A3: lim (f1,x0) = lim (f2,x0) ; ::_thesis: ( for r being Real holds ( not 0 < r or not ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ c= ((dom f1) /\ (dom f2)) /\ (dom f) or ex g being Real st ( g in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ & not ( f1 . g <= f . g & f . g <= f2 . g ) ) ) or ( f is_convergent_in x0 & lim (f,x0) = lim (f1,x0) ) ) given r being Real such that A4: 0 < r and A5: ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ c= ((dom f1) /\ (dom f2)) /\ (dom f) and A6: for g being Real st g in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ holds ( f1 . g <= f . g & f . g <= f2 . g ) ; ::_thesis: ( f is_convergent_in x0 & lim (f,x0) = lim (f1,x0) ) A7: (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) = ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ by A5, XBOOLE_1:18, XBOOLE_1:28; A8: ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ c= (dom f1) /\ (dom f2) by A5, XBOOLE_1:18; then A9: (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) = ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ by XBOOLE_1:18, XBOOLE_1:28; A10: (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) = ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ by A8, XBOOLE_1:18, XBOOLE_1:28; for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) by A4, A5, Th5, XBOOLE_1:18; hence ( f is_convergent_in x0 & lim (f,x0) = lim (f1,x0) ) by A1, A2, A3, A4, A6, A7, A9, A10, Th41; ::_thesis: verum end; theorem :: LIMFUNC3:43 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_convergent_in x0 & ex r being Real st ( 0 < r & ( ( (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for g being Real st g in (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f1 . g <= f2 . g ) ) or ( (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for g being Real st g in (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f1 . g <= f2 . g ) ) ) ) holds lim (f1,x0) <= lim (f2,x0) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_convergent_in x0 & ex r being Real st ( 0 < r & ( ( (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for g being Real st g in (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f1 . g <= f2 . g ) ) or ( (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for g being Real st g in (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f1 . g <= f2 . g ) ) ) ) holds lim (f1,x0) <= lim (f2,x0) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_convergent_in x0 & f2 is_convergent_in x0 & ex r being Real st ( 0 < r & ( ( (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for g being Real st g in (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f1 . g <= f2 . g ) ) or ( (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for g being Real st g in (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f1 . g <= f2 . g ) ) ) ) implies lim (f1,x0) <= lim (f2,x0) ) assume that A1: f1 is_convergent_in x0 and A2: f2 is_convergent_in x0 ; ::_thesis: ( for r being Real holds ( not 0 < r or ( not ( (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for g being Real st g in (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f1 . g <= f2 . g ) ) & not ( (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for g being Real st g in (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f1 . g <= f2 . g ) ) ) ) or lim (f1,x0) <= lim (f2,x0) ) A3: lim (f2,x0) = lim (f2,x0) ; given r being Real such that A4: 0 < r and A5: ( ( (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for g being Real st g in (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f1 . g <= f2 . g ) ) or ( (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for g being Real st g in (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f1 . g <= f2 . g ) ) ) ; ::_thesis: lim (f1,x0) <= lim (f2,x0) A6: lim (f1,x0) = lim (f1,x0) ; now__::_thesis:_lim_(f1,x0)_<=_lim_(f2,x0) percases ( ( (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for g being Real st g in (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f1 . g <= f2 . g ) ) or ( (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for g being Real st g in (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f1 . g <= f2 . g ) ) ) by A5; supposeA7: ( (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for g being Real st g in (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f1 . g <= f2 . g ) ) ; ::_thesis: lim (f1,x0) <= lim (f2,x0) defpred S1[ Element of NAT , real number ] means ( x0 - (1 / ($1 + 1)) < $2 & $2 < x0 & $2 in dom f1 ); A8: now__::_thesis:_for_n_being_Element_of_NAT_ex_g1_being_Real_st_S1[n,g1] let n be Element of NAT ; ::_thesis: ex g1 being Real st S1[n,g1] A9: x0 < x0 + 1 by Lm1; x0 - (1 / (n + 1)) < x0 by Lm3; then consider g1, g2 being Real such that A10: x0 - (1 / (n + 1)) < g1 and A11: g1 < x0 and A12: g1 in dom f1 and g2 < x0 + 1 and x0 < g2 and g2 in dom f1 by A1, A9, Def1; take g1 = g1; ::_thesis: S1[n,g1] thus S1[n,g1] by A10, A11, A12; ::_thesis: verum end; consider s being Real_Sequence such that A13: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A8); A14: lim s = x0 by A13, Th6; A15: rng s c= (dom f1) \ {x0} by A13, Th6; A16: s is convergent by A13, Th6; x0 - r < x0 by A4, Lm1; then consider k being Element of NAT such that A17: for n being Element of NAT st k <= n holds x0 - r < s . n by A16, A14, LIMFUNC2:1; A18: lim (s ^\ k) = x0 by A16, A14, SEQ_4:20; rng (s ^\ k) c= rng s by VALUED_0:21; then A19: rng (s ^\ k) c= (dom f1) \ {x0} by A15, XBOOLE_1:1; then A20: lim (f1 /* (s ^\ k)) = lim (f1,x0) by A1, A16, A18, Def4; now__::_thesis:_for_x_being_set_st_x_in_rng_(s_^\_k)_holds_ x_in_(dom_f1)_/\_(].(x0_-_r),x0.[_\/_].x0,(x0_+_r).[) let x be set ; ::_thesis: ( x in rng (s ^\ k) implies x in (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) ) assume x in rng (s ^\ k) ; ::_thesis: x in (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) then consider n being Element of NAT such that A21: (s ^\ k) . n = x by FUNCT_2:113; s . (n + k) < x0 by A13; then A22: (s ^\ k) . n < x0 by NAT_1:def_3; x0 - r < s . (n + k) by A17, NAT_1:12; then x0 - r < (s ^\ k) . n by NAT_1:def_3; then (s ^\ k) . n in { g2 where g2 is Real : ( x0 - r < g2 & g2 < x0 ) } by A22; then (s ^\ k) . n in ].(x0 - r),x0.[ by RCOMP_1:def_2; then A23: (s ^\ k) . n in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ by XBOOLE_0:def_3; s . (n + k) in dom f1 by A13; then (s ^\ k) . n in dom f1 by NAT_1:def_3; hence x in (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) by A21, A23, XBOOLE_0:def_4; ::_thesis: verum end; then A24: rng (s ^\ k) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) by TARSKI:def_3; then A25: rng (s ^\ k) c= (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) by A7, XBOOLE_1:1; A26: now__::_thesis:_for_n_being_Element_of_NAT_holds_(f1_/*_(s_^\_k))_._n_<=_(f2_/*_(s_^\_k))_._n let n be Element of NAT ; ::_thesis: (f1 /* (s ^\ k)) . n <= (f2 /* (s ^\ k)) . n (s ^\ k) . n in rng (s ^\ k) by VALUED_0:28; then f1 . ((s ^\ k) . n) <= f2 . ((s ^\ k) . n) by A7, A24; then f1 . ((s ^\ k) . n) <= (f2 /* (s ^\ k)) . n by A25, FUNCT_2:108, XBOOLE_1:18; hence (f1 /* (s ^\ k)) . n <= (f2 /* (s ^\ k)) . n by A24, FUNCT_2:108, XBOOLE_1:18; ::_thesis: verum end; A27: rng (s ^\ k) c= dom f2 by A25, XBOOLE_1:18; A28: rng (s ^\ k) c= (dom f2) \ {x0} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (s ^\ k) or x in (dom f2) \ {x0} ) assume A29: x in rng (s ^\ k) ; ::_thesis: x in (dom f2) \ {x0} then not x in {x0} by A19, XBOOLE_0:def_5; hence x in (dom f2) \ {x0} by A27, A29, XBOOLE_0:def_5; ::_thesis: verum end; then A30: lim (f2 /* (s ^\ k)) = lim (f2,x0) by A2, A16, A18, Def4; A31: f2 /* (s ^\ k) is convergent by A2, A3, A16, A18, A28, Def4; f1 /* (s ^\ k) is convergent by A1, A6, A16, A18, A19, Def4; hence lim (f1,x0) <= lim (f2,x0) by A20, A31, A30, A26, SEQ_2:18; ::_thesis: verum end; supposeA32: ( (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for g being Real st g in (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f1 . g <= f2 . g ) ) ; ::_thesis: lim (f1,x0) <= lim (f2,x0) defpred S1[ Element of NAT , real number ] means ( x0 - (1 / ($1 + 1)) < $2 & $2 < x0 & $2 in dom f2 ); A33: now__::_thesis:_for_n_being_Element_of_NAT_ex_g1_being_Real_st_S1[n,g1] let n be Element of NAT ; ::_thesis: ex g1 being Real st S1[n,g1] A34: x0 < x0 + 1 by Lm1; x0 - (1 / (n + 1)) < x0 by Lm3; then consider g1, g2 being Real such that A35: x0 - (1 / (n + 1)) < g1 and A36: g1 < x0 and A37: g1 in dom f2 and g2 < x0 + 1 and x0 < g2 and g2 in dom f2 by A2, A34, Def1; take g1 = g1; ::_thesis: S1[n,g1] thus S1[n,g1] by A35, A36, A37; ::_thesis: verum end; consider s being Real_Sequence such that A38: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A33); A39: lim s = x0 by A38, Th6; A40: rng s c= (dom f2) \ {x0} by A38, Th6; A41: s is convergent by A38, Th6; x0 - r < x0 by A4, Lm1; then consider k being Element of NAT such that A42: for n being Element of NAT st k <= n holds x0 - r < s . n by A41, A39, LIMFUNC2:1; A43: lim (s ^\ k) = x0 by A41, A39, SEQ_4:20; rng (s ^\ k) c= rng s by VALUED_0:21; then A44: rng (s ^\ k) c= (dom f2) \ {x0} by A40, XBOOLE_1:1; then A45: lim (f2 /* (s ^\ k)) = lim (f2,x0) by A2, A41, A43, Def4; A46: now__::_thesis:_for_x_being_set_st_x_in_rng_(s_^\_k)_holds_ x_in_(dom_f2)_/\_(].(x0_-_r),x0.[_\/_].x0,(x0_+_r).[) let x be set ; ::_thesis: ( x in rng (s ^\ k) implies x in (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) ) assume x in rng (s ^\ k) ; ::_thesis: x in (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) then consider n being Element of NAT such that A47: (s ^\ k) . n = x by FUNCT_2:113; s . (n + k) < x0 by A38; then A48: (s ^\ k) . n < x0 by NAT_1:def_3; x0 - r < s . (n + k) by A42, NAT_1:12; then x0 - r < (s ^\ k) . n by NAT_1:def_3; then (s ^\ k) . n in { g2 where g2 is Real : ( x0 - r < g2 & g2 < x0 ) } by A48; then (s ^\ k) . n in ].(x0 - r),x0.[ by RCOMP_1:def_2; then A49: (s ^\ k) . n in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ by XBOOLE_0:def_3; s . (n + k) in dom f2 by A38; then (s ^\ k) . n in dom f2 by NAT_1:def_3; hence x in (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) by A47, A49, XBOOLE_0:def_4; ::_thesis: verum end; then A50: rng (s ^\ k) c= (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) by TARSKI:def_3; then A51: rng (s ^\ k) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) by A32, XBOOLE_1:1; A52: now__::_thesis:_for_n_being_Element_of_NAT_holds_(f1_/*_(s_^\_k))_._n_<=_(f2_/*_(s_^\_k))_._n let n be Element of NAT ; ::_thesis: (f1 /* (s ^\ k)) . n <= (f2 /* (s ^\ k)) . n (s ^\ k) . n in rng (s ^\ k) by VALUED_0:28; then f1 . ((s ^\ k) . n) <= f2 . ((s ^\ k) . n) by A32, A46; then f1 . ((s ^\ k) . n) <= (f2 /* (s ^\ k)) . n by A50, FUNCT_2:108, XBOOLE_1:18; hence (f1 /* (s ^\ k)) . n <= (f2 /* (s ^\ k)) . n by A51, FUNCT_2:108, XBOOLE_1:18; ::_thesis: verum end; A53: rng (s ^\ k) c= dom f1 by A51, XBOOLE_1:18; A54: rng (s ^\ k) c= (dom f1) \ {x0} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (s ^\ k) or x in (dom f1) \ {x0} ) assume A55: x in rng (s ^\ k) ; ::_thesis: x in (dom f1) \ {x0} then not x in {x0} by A44, XBOOLE_0:def_5; hence x in (dom f1) \ {x0} by A53, A55, XBOOLE_0:def_5; ::_thesis: verum end; then A56: lim (f1 /* (s ^\ k)) = lim (f1,x0) by A1, A41, A43, Def4; A57: f1 /* (s ^\ k) is convergent by A1, A6, A41, A43, A54, Def4; f2 /* (s ^\ k) is convergent by A2, A3, A41, A43, A44, Def4; hence lim (f1,x0) <= lim (f2,x0) by A45, A57, A56, A52, SEQ_2:18; ::_thesis: verum end; end; end; hence lim (f1,x0) <= lim (f2,x0) ; ::_thesis: verum end; theorem :: LIMFUNC3:44 for x0 being Real for f being PartFunc of REAL,REAL st ( f is_divergent_to+infty_in x0 or f is_divergent_to-infty_in x0 ) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f & f . g1 <> 0 & f . g2 <> 0 ) ) holds ( f ^ is_convergent_in x0 & lim ((f ^),x0) = 0 ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st ( f is_divergent_to+infty_in x0 or f is_divergent_to-infty_in x0 ) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f & f . g1 <> 0 & f . g2 <> 0 ) ) holds ( f ^ is_convergent_in x0 & lim ((f ^),x0) = 0 ) let f be PartFunc of REAL,REAL; ::_thesis: ( ( f is_divergent_to+infty_in x0 or f is_divergent_to-infty_in x0 ) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f & f . g1 <> 0 & f . g2 <> 0 ) ) implies ( f ^ is_convergent_in x0 & lim ((f ^),x0) = 0 ) ) A1: (dom f) \ (f " {0}) = dom (f ^) by RFUNCT_1:def_2; assume A2: ( f is_divergent_to+infty_in x0 or f is_divergent_to-infty_in x0 ) ; ::_thesis: ( ex r1, r2 being Real st ( r1 < x0 & x0 < r2 & ( for g1, g2 being Real holds ( not r1 < g1 or not g1 < x0 or not g1 in dom f or not g2 < r2 or not x0 < g2 or not g2 in dom f or not f . g1 <> 0 or not f . g2 <> 0 ) ) ) or ( f ^ is_convergent_in x0 & lim ((f ^),x0) = 0 ) ) A3: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(f_^))_\_{x0}_holds_ (_(f_^)_/*_seq_is_convergent_&_lim_((f_^)_/*_seq)_=_0_) let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f ^)) \ {x0} implies ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) ) assume that A4: seq is convergent and A5: lim seq = x0 and A6: rng seq c= (dom (f ^)) \ {x0} ; ::_thesis: ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) rng seq c= dom (f ^) by A6, XBOOLE_1:1; then A7: rng seq c= dom f by A1, XBOOLE_1:1; A8: rng seq c= (dom f) \ {x0} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng seq or x in (dom f) \ {x0} ) assume A9: x in rng seq ; ::_thesis: x in (dom f) \ {x0} then not x in {x0} by A6, XBOOLE_0:def_5; hence x in (dom f) \ {x0} by A7, A9, XBOOLE_0:def_5; ::_thesis: verum end; now__::_thesis:_(_(f_^)_/*_seq_is_convergent_&_lim_((f_^)_/*_seq)_=_0_) percases ( f is_divergent_to+infty_in x0 or f is_divergent_to-infty_in x0 ) by A2; suppose f is_divergent_to+infty_in x0 ; ::_thesis: ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) then A10: f /* seq is divergent_to+infty by A4, A5, A8, Def2; then A11: lim ((f /* seq) ") = 0 by LIMFUNC1:34; (f /* seq) " is convergent by A10, LIMFUNC1:34; hence ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) by A6, A11, RFUNCT_2:12, XBOOLE_1:1; ::_thesis: verum end; suppose f is_divergent_to-infty_in x0 ; ::_thesis: ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) then A12: f /* seq is divergent_to-infty by A4, A5, A8, Def3; then A13: lim ((f /* seq) ") = 0 by LIMFUNC1:34; (f /* seq) " is convergent by A12, LIMFUNC1:34; hence ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) by A6, A13, RFUNCT_2:12, XBOOLE_1:1; ::_thesis: verum end; end; end; hence ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) ; ::_thesis: verum end; assume A14: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f & f . g1 <> 0 & f . g2 <> 0 ) ; ::_thesis: ( f ^ is_convergent_in x0 & lim ((f ^),x0) = 0 ) now__::_thesis:_for_r1,_r2_being_Real_st_r1_<_x0_&_x0_<_r2_holds_ ex_g1,_g2_being_Real_st_ (_r1_<_g1_&_g1_<_x0_&_g1_in_dom_(f_^)_&_g2_<_r2_&_x0_<_g2_&_g2_in_dom_(f_^)_) let r1, r2 be Real; ::_thesis: ( r1 < x0 & x0 < r2 implies ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) & g2 < r2 & x0 < g2 & g2 in dom (f ^) ) ) assume that A15: r1 < x0 and A16: x0 < r2 ; ::_thesis: ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) & g2 < r2 & x0 < g2 & g2 in dom (f ^) ) consider g1, g2 being Real such that A17: r1 < g1 and A18: g1 < x0 and A19: g1 in dom f and A20: g2 < r2 and A21: x0 < g2 and A22: g2 in dom f and A23: f . g1 <> 0 and A24: f . g2 <> 0 by A14, A15, A16; take g1 = g1; ::_thesis: ex g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) & g2 < r2 & x0 < g2 & g2 in dom (f ^) ) take g2 = g2; ::_thesis: ( r1 < g1 & g1 < x0 & g1 in dom (f ^) & g2 < r2 & x0 < g2 & g2 in dom (f ^) ) not f . g2 in {0} by A24, TARSKI:def_1; then A25: not g2 in f " {0} by FUNCT_1:def_7; not f . g1 in {0} by A23, TARSKI:def_1; then not g1 in f " {0} by FUNCT_1:def_7; hence ( r1 < g1 & g1 < x0 & g1 in dom (f ^) & g2 < r2 & x0 < g2 & g2 in dom (f ^) ) by A1, A17, A18, A19, A20, A21, A22, A25, XBOOLE_0:def_5; ::_thesis: verum end; hence f ^ is_convergent_in x0 by A3, Def1; ::_thesis: lim ((f ^),x0) = 0 hence lim ((f ^),x0) = 0 by A3, Def4; ::_thesis: verum end; theorem :: LIMFUNC3:45 for x0 being Real for f being PartFunc of REAL,REAL st f is_convergent_in x0 & lim (f,x0) = 0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f & f . g1 <> 0 & f . g2 <> 0 ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds 0 <= f . g ) ) holds f ^ is_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_convergent_in x0 & lim (f,x0) = 0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f & f . g1 <> 0 & f . g2 <> 0 ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds 0 <= f . g ) ) holds f ^ is_divergent_to+infty_in x0 let f be PartFunc of REAL,REAL; ::_thesis: ( f is_convergent_in x0 & lim (f,x0) = 0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f & f . g1 <> 0 & f . g2 <> 0 ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds 0 <= f . g ) ) implies f ^ is_divergent_to+infty_in x0 ) assume that A1: f is_convergent_in x0 and A2: lim (f,x0) = 0 and A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f & f . g1 <> 0 & f . g2 <> 0 ) ; ::_thesis: ( for r being Real holds ( not 0 < r or ex g being Real st ( g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & not 0 <= f . g ) ) or f ^ is_divergent_to+infty_in x0 ) given r being Real such that A4: 0 < r and A5: for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds 0 <= f . g ; ::_thesis: f ^ is_divergent_to+infty_in x0 thus for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) & g2 < r2 & x0 < g2 & g2 in dom (f ^) ) :: according to LIMFUNC3:def_2 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom (f ^)) \ {x0} holds (f ^) /* seq is divergent_to+infty proof let r1, r2 be Real; ::_thesis: ( r1 < x0 & x0 < r2 implies ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) & g2 < r2 & x0 < g2 & g2 in dom (f ^) ) ) assume that A6: r1 < x0 and A7: x0 < r2 ; ::_thesis: ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) & g2 < r2 & x0 < g2 & g2 in dom (f ^) ) consider g1, g2 being Real such that A8: r1 < g1 and A9: g1 < x0 and A10: g1 in dom f and A11: g2 < r2 and A12: x0 < g2 and A13: g2 in dom f and A14: f . g1 <> 0 and A15: f . g2 <> 0 by A3, A6, A7; not f . g2 in {0} by A15, TARSKI:def_1; then not g2 in f " {0} by FUNCT_1:def_7; then A16: g2 in (dom f) \ (f " {0}) by A13, XBOOLE_0:def_5; take g1 ; ::_thesis: ex g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) & g2 < r2 & x0 < g2 & g2 in dom (f ^) ) take g2 ; ::_thesis: ( r1 < g1 & g1 < x0 & g1 in dom (f ^) & g2 < r2 & x0 < g2 & g2 in dom (f ^) ) not f . g1 in {0} by A14, TARSKI:def_1; then not g1 in f " {0} by FUNCT_1:def_7; then g1 in (dom f) \ (f " {0}) by A10, XBOOLE_0:def_5; hence ( r1 < g1 & g1 < x0 & g1 in dom (f ^) & g2 < r2 & x0 < g2 & g2 in dom (f ^) ) by A8, A9, A11, A12, A16, RFUNCT_1:def_2; ::_thesis: verum end; let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f ^)) \ {x0} implies (f ^) /* s is divergent_to+infty ) assume that A17: s is convergent and A18: lim s = x0 and A19: rng s c= (dom (f ^)) \ {x0} ; ::_thesis: (f ^) /* s is divergent_to+infty consider k being Element of NAT such that A20: for n being Element of NAT st k <= n holds ( x0 - r < s . n & s . n < x0 + r ) by A4, A17, A18, Th7; A21: rng s c= dom (f ^) by A19, XBOOLE_1:1; A22: dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def_2; then A23: (f /* (s ^\ k)) " = ((f /* s) ^\ k) " by A21, VALUED_0:27, XBOOLE_1:1 .= ((f /* s) ") ^\ k by SEQM_3:18 .= ((f ^) /* s) ^\ k by A19, RFUNCT_2:12, XBOOLE_1:1 ; A24: rng (s ^\ k) c= rng s by VALUED_0:21; A25: rng s c= dom f by A21, A22, XBOOLE_1:1; then A26: rng (s ^\ k) c= dom f by A24, XBOOLE_1:1; A27: rng (s ^\ k) c= (dom (f ^)) \ {x0} by A19, A24, XBOOLE_1:1; A28: rng (s ^\ k) c= (dom f) \ {x0} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (s ^\ k) or x in (dom f) \ {x0} ) assume A29: x in rng (s ^\ k) ; ::_thesis: x in (dom f) \ {x0} then not x in {x0} by A27, XBOOLE_0:def_5; hence x in (dom f) \ {x0} by A26, A29, XBOOLE_0:def_5; ::_thesis: verum end; A30: lim (s ^\ k) = x0 by A17, A18, SEQ_4:20; then A31: lim (f /* (s ^\ k)) = 0 by A1, A2, A17, A28, Def4; A32: f /* (s ^\ k) is non-zero by A21, A24, RFUNCT_2:11, XBOOLE_1:1; now__::_thesis:_for_n_being_Element_of_NAT_holds_0_<_(f_/*_(s_^\_k))_._n let n be Element of NAT ; ::_thesis: 0 < (f /* (s ^\ k)) . n A33: k <= n + k by NAT_1:12; then s . (n + k) < x0 + r by A20; then A34: (s ^\ k) . n < x0 + r by NAT_1:def_3; x0 - r < s . (n + k) by A20, A33; then x0 - r < (s ^\ k) . n by NAT_1:def_3; then (s ^\ k) . n in { g1 where g1 is Real : ( x0 - r < g1 & g1 < x0 + r ) } by A34; then A35: (s ^\ k) . n in ].(x0 - r),(x0 + r).[ by RCOMP_1:def_2; A36: (s ^\ k) . n in rng (s ^\ k) by VALUED_0:28; then not (s ^\ k) . n in {x0} by A27, XBOOLE_0:def_5; then (s ^\ k) . n in ].(x0 - r),(x0 + r).[ \ {x0} by A35, XBOOLE_0:def_5; then (s ^\ k) . n in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ by A4, Th4; then (s ^\ k) . n in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) by A26, A36, XBOOLE_0:def_4; then A37: 0 <= f . ((s ^\ k) . n) by A5; (f /* (s ^\ k)) . n <> 0 by A32, SEQ_1:5; hence 0 < (f /* (s ^\ k)) . n by A25, A24, A37, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; then A38: for n being Element of NAT st 0 <= n holds 0 < (f /* (s ^\ k)) . n ; f /* (s ^\ k) is convergent by A1, A2, A17, A30, A28, Def4; then (f /* (s ^\ k)) " is divergent_to+infty by A31, A38, LIMFUNC1:35; hence (f ^) /* s is divergent_to+infty by A23, LIMFUNC1:7; ::_thesis: verum end; theorem :: LIMFUNC3:46 for x0 being Real for f being PartFunc of REAL,REAL st f is_convergent_in x0 & lim (f,x0) = 0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f & f . g1 <> 0 & f . g2 <> 0 ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f . g <= 0 ) ) holds f ^ is_divergent_to-infty_in x0 proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_convergent_in x0 & lim (f,x0) = 0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f & f . g1 <> 0 & f . g2 <> 0 ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f . g <= 0 ) ) holds f ^ is_divergent_to-infty_in x0 let f be PartFunc of REAL,REAL; ::_thesis: ( f is_convergent_in x0 & lim (f,x0) = 0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f & f . g1 <> 0 & f . g2 <> 0 ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f . g <= 0 ) ) implies f ^ is_divergent_to-infty_in x0 ) assume that A1: f is_convergent_in x0 and A2: lim (f,x0) = 0 and A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f & f . g1 <> 0 & f . g2 <> 0 ) ; ::_thesis: ( for r being Real holds ( not 0 < r or ex g being Real st ( g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & not f . g <= 0 ) ) or f ^ is_divergent_to-infty_in x0 ) given r being Real such that A4: 0 < r and A5: for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f . g <= 0 ; ::_thesis: f ^ is_divergent_to-infty_in x0 thus for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) & g2 < r2 & x0 < g2 & g2 in dom (f ^) ) :: according to LIMFUNC3:def_3 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom (f ^)) \ {x0} holds (f ^) /* seq is divergent_to-infty proof let r1, r2 be Real; ::_thesis: ( r1 < x0 & x0 < r2 implies ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) & g2 < r2 & x0 < g2 & g2 in dom (f ^) ) ) assume that A6: r1 < x0 and A7: x0 < r2 ; ::_thesis: ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) & g2 < r2 & x0 < g2 & g2 in dom (f ^) ) consider g1, g2 being Real such that A8: r1 < g1 and A9: g1 < x0 and A10: g1 in dom f and A11: g2 < r2 and A12: x0 < g2 and A13: g2 in dom f and A14: f . g1 <> 0 and A15: f . g2 <> 0 by A3, A6, A7; not f . g2 in {0} by A15, TARSKI:def_1; then not g2 in f " {0} by FUNCT_1:def_7; then A16: g2 in (dom f) \ (f " {0}) by A13, XBOOLE_0:def_5; take g1 ; ::_thesis: ex g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) & g2 < r2 & x0 < g2 & g2 in dom (f ^) ) take g2 ; ::_thesis: ( r1 < g1 & g1 < x0 & g1 in dom (f ^) & g2 < r2 & x0 < g2 & g2 in dom (f ^) ) not f . g1 in {0} by A14, TARSKI:def_1; then not g1 in f " {0} by FUNCT_1:def_7; then g1 in (dom f) \ (f " {0}) by A10, XBOOLE_0:def_5; hence ( r1 < g1 & g1 < x0 & g1 in dom (f ^) & g2 < r2 & x0 < g2 & g2 in dom (f ^) ) by A8, A9, A11, A12, A16, RFUNCT_1:def_2; ::_thesis: verum end; let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f ^)) \ {x0} implies (f ^) /* s is divergent_to-infty ) assume that A17: s is convergent and A18: lim s = x0 and A19: rng s c= (dom (f ^)) \ {x0} ; ::_thesis: (f ^) /* s is divergent_to-infty consider k being Element of NAT such that A20: for n being Element of NAT st k <= n holds ( x0 - r < s . n & s . n < x0 + r ) by A4, A17, A18, Th7; A21: rng s c= dom (f ^) by A19, XBOOLE_1:1; A22: dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def_2; then A23: (f /* (s ^\ k)) " = ((f /* s) ^\ k) " by A21, VALUED_0:27, XBOOLE_1:1 .= ((f /* s) ") ^\ k by SEQM_3:18 .= ((f ^) /* s) ^\ k by A19, RFUNCT_2:12, XBOOLE_1:1 ; A24: rng (s ^\ k) c= rng s by VALUED_0:21; A25: rng s c= dom f by A21, A22, XBOOLE_1:1; then A26: rng (s ^\ k) c= dom f by A24, XBOOLE_1:1; A27: rng (s ^\ k) c= (dom (f ^)) \ {x0} by A19, A24, XBOOLE_1:1; A28: rng (s ^\ k) c= (dom f) \ {x0} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (s ^\ k) or x in (dom f) \ {x0} ) assume A29: x in rng (s ^\ k) ; ::_thesis: x in (dom f) \ {x0} then not x in {x0} by A27, XBOOLE_0:def_5; hence x in (dom f) \ {x0} by A26, A29, XBOOLE_0:def_5; ::_thesis: verum end; A30: lim (s ^\ k) = x0 by A17, A18, SEQ_4:20; then A31: lim (f /* (s ^\ k)) = 0 by A1, A2, A17, A28, Def4; A32: f /* (s ^\ k) is non-zero by A21, A24, RFUNCT_2:11, XBOOLE_1:1; now__::_thesis:_for_n_being_Element_of_NAT_holds_(f_/*_(s_^\_k))_._n_<_0 let n be Element of NAT ; ::_thesis: (f /* (s ^\ k)) . n < 0 A33: k <= n + k by NAT_1:12; then s . (n + k) < x0 + r by A20; then A34: (s ^\ k) . n < x0 + r by NAT_1:def_3; x0 - r < s . (n + k) by A20, A33; then x0 - r < (s ^\ k) . n by NAT_1:def_3; then (s ^\ k) . n in { g1 where g1 is Real : ( x0 - r < g1 & g1 < x0 + r ) } by A34; then A35: (s ^\ k) . n in ].(x0 - r),(x0 + r).[ by RCOMP_1:def_2; A36: (s ^\ k) . n in rng (s ^\ k) by VALUED_0:28; then not (s ^\ k) . n in {x0} by A27, XBOOLE_0:def_5; then (s ^\ k) . n in ].(x0 - r),(x0 + r).[ \ {x0} by A35, XBOOLE_0:def_5; then (s ^\ k) . n in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ by A4, Th4; then (s ^\ k) . n in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) by A26, A36, XBOOLE_0:def_4; then A37: f . ((s ^\ k) . n) <= 0 by A5; (f /* (s ^\ k)) . n <> 0 by A32, SEQ_1:5; hence (f /* (s ^\ k)) . n < 0 by A25, A24, A37, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; then A38: for n being Element of NAT st 0 <= n holds (f /* (s ^\ k)) . n < 0 ; f /* (s ^\ k) is convergent by A1, A2, A17, A30, A28, Def4; then (f /* (s ^\ k)) " is divergent_to-infty by A31, A38, LIMFUNC1:36; hence (f ^) /* s is divergent_to-infty by A23, LIMFUNC1:7; ::_thesis: verum end; theorem :: LIMFUNC3:47 for x0 being Real for f being PartFunc of REAL,REAL st f is_convergent_in x0 & lim (f,x0) = 0 & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds 0 < f . g ) ) holds f ^ is_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_convergent_in x0 & lim (f,x0) = 0 & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds 0 < f . g ) ) holds f ^ is_divergent_to+infty_in x0 let f be PartFunc of REAL,REAL; ::_thesis: ( f is_convergent_in x0 & lim (f,x0) = 0 & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds 0 < f . g ) ) implies f ^ is_divergent_to+infty_in x0 ) assume that A1: f is_convergent_in x0 and A2: lim (f,x0) = 0 ; ::_thesis: ( for r being Real holds ( not 0 < r or ex g being Real st ( g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & not 0 < f . g ) ) or f ^ is_divergent_to+infty_in x0 ) A3: f is_right_convergent_in x0 by A1, Th29; given r being Real such that A4: 0 < r and A5: for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds 0 < f . g ; ::_thesis: f ^ is_divergent_to+infty_in x0 A6: now__::_thesis:_for_g_being_Real_st_g_in_(dom_f)_/\_].x0,(x0_+_r).[_holds_ 0_<_f_._g let g be Real; ::_thesis: ( g in (dom f) /\ ].x0,(x0 + r).[ implies 0 < f . g ) assume A7: g in (dom f) /\ ].x0,(x0 + r).[ ; ::_thesis: 0 < f . g then g in ].x0,(x0 + r).[ by XBOOLE_0:def_4; then A8: g in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ by XBOOLE_0:def_3; g in dom f by A7, XBOOLE_0:def_4; then g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) by A8, XBOOLE_0:def_4; hence 0 < f . g by A5; ::_thesis: verum end; A9: now__::_thesis:_for_g_being_Real_st_g_in_(dom_f)_/\_].(x0_-_r),x0.[_holds_ 0_<_f_._g let g be Real; ::_thesis: ( g in (dom f) /\ ].(x0 - r),x0.[ implies 0 < f . g ) assume A10: g in (dom f) /\ ].(x0 - r),x0.[ ; ::_thesis: 0 < f . g then g in ].(x0 - r),x0.[ by XBOOLE_0:def_4; then A11: g in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ by XBOOLE_0:def_3; g in dom f by A10, XBOOLE_0:def_4; then g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) by A11, XBOOLE_0:def_4; hence 0 < f . g by A5; ::_thesis: verum end; lim_right (f,x0) = 0 by A1, A2, Th29; then A12: f ^ is_right_divergent_to+infty_in x0 by A3, A4, A6, LIMFUNC2:73; A13: f is_left_convergent_in x0 by A1, Th29; lim_left (f,x0) = 0 by A1, A2, Th29; then f ^ is_left_divergent_to+infty_in x0 by A13, A4, A9, LIMFUNC2:71; hence f ^ is_divergent_to+infty_in x0 by A12, Th12; ::_thesis: verum end; theorem :: LIMFUNC3:48 for x0 being Real for f being PartFunc of REAL,REAL st f is_convergent_in x0 & lim (f,x0) = 0 & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f . g < 0 ) ) holds f ^ is_divergent_to-infty_in x0 proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_convergent_in x0 & lim (f,x0) = 0 & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f . g < 0 ) ) holds f ^ is_divergent_to-infty_in x0 let f be PartFunc of REAL,REAL; ::_thesis: ( f is_convergent_in x0 & lim (f,x0) = 0 & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f . g < 0 ) ) implies f ^ is_divergent_to-infty_in x0 ) assume that A1: f is_convergent_in x0 and A2: lim (f,x0) = 0 ; ::_thesis: ( for r being Real holds ( not 0 < r or ex g being Real st ( g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & not f . g < 0 ) ) or f ^ is_divergent_to-infty_in x0 ) A3: f is_right_convergent_in x0 by A1, Th29; given r being Real such that A4: 0 < r and A5: for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds f . g < 0 ; ::_thesis: f ^ is_divergent_to-infty_in x0 A6: now__::_thesis:_for_g_being_Real_st_g_in_(dom_f)_/\_].x0,(x0_+_r).[_holds_ f_._g_<_0 let g be Real; ::_thesis: ( g in (dom f) /\ ].x0,(x0 + r).[ implies f . g < 0 ) assume A7: g in (dom f) /\ ].x0,(x0 + r).[ ; ::_thesis: f . g < 0 then g in ].x0,(x0 + r).[ by XBOOLE_0:def_4; then A8: g in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ by XBOOLE_0:def_3; g in dom f by A7, XBOOLE_0:def_4; then g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) by A8, XBOOLE_0:def_4; hence f . g < 0 by A5; ::_thesis: verum end; A9: now__::_thesis:_for_g_being_Real_st_g_in_(dom_f)_/\_].(x0_-_r),x0.[_holds_ f_._g_<_0 let g be Real; ::_thesis: ( g in (dom f) /\ ].(x0 - r),x0.[ implies f . g < 0 ) assume A10: g in (dom f) /\ ].(x0 - r),x0.[ ; ::_thesis: f . g < 0 then g in ].(x0 - r),x0.[ by XBOOLE_0:def_4; then A11: g in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ by XBOOLE_0:def_3; g in dom f by A10, XBOOLE_0:def_4; then g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) by A11, XBOOLE_0:def_4; hence f . g < 0 by A5; ::_thesis: verum end; lim_right (f,x0) = 0 by A1, A2, Th29; then A12: f ^ is_right_divergent_to-infty_in x0 by A3, A4, A6, LIMFUNC2:74; A13: f is_left_convergent_in x0 by A1, Th29; lim_left (f,x0) = 0 by A1, A2, Th29; then f ^ is_left_divergent_to-infty_in x0 by A13, A4, A9, LIMFUNC2:72; hence f ^ is_divergent_to-infty_in x0 by A12, Th13; ::_thesis: verum end;