:: LIMFUNC2 semantic presentation begin Lm1: for r, g, r1 being real number st 0 < g & r <= r1 holds ( r - g < r1 & r < r1 + g ) proof let r, g, 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 Subset of REAL st rng seq c= (dom (f1 (#) f2)) /\ X holds ( rng seq c= dom (f1 (#) f2) & rng seq c= X & 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 Subset of REAL st rng seq c= (dom (f1 (#) f2)) /\ X holds ( rng seq c= dom (f1 (#) f2) & rng seq c= X & 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 Subset of REAL st rng seq c= (dom (f1 (#) f2)) /\ X holds ( rng seq c= dom (f1 (#) f2) & rng seq c= X & 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 Subset of REAL; ::_thesis: ( rng seq c= (dom (f1 (#) f2)) /\ X implies ( rng seq c= dom (f1 (#) f2) & rng seq c= X & 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) & rng seq c= X & 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 ) A2: (dom (f1 (#) f2)) /\ X c= X by XBOOLE_1:17; (dom (f1 (#) f2)) /\ X c= dom (f1 (#) f2) by XBOOLE_1:17; hence A3: ( rng seq c= dom (f1 (#) f2) & rng seq c= X ) by A1, A2, 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 A4: 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 A5: dom (f1 (#) f2) c= dom f2 by XBOOLE_1:17; dom (f1 (#) f2) c= dom f1 by A4, XBOOLE_1:17; hence ( rng seq c= dom f1 & rng seq c= dom f2 ) by A3, A5, XBOOLE_1:1; ::_thesis: ( rng seq c= (dom f1) /\ X & rng seq c= (dom f2) /\ X ) hence ( rng seq c= (dom f1) /\ X & rng seq c= (dom f2) /\ X ) by A3, XBOOLE_1:19; ::_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 Subset of REAL st rng seq c= (dom (f1 + f2)) /\ X holds ( rng seq c= dom (f1 + f2) & rng seq c= X & dom (f1 + f2) = (dom f1) /\ (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 Subset of REAL st rng seq c= (dom (f1 + f2)) /\ X holds ( rng seq c= dom (f1 + f2) & rng seq c= X & dom (f1 + f2) = (dom f1) /\ (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 Subset of REAL st rng seq c= (dom (f1 + f2)) /\ X holds ( rng seq c= dom (f1 + f2) & rng seq c= X & dom (f1 + f2) = (dom f1) /\ (dom f2) & rng seq c= (dom f1) /\ X & rng seq c= (dom f2) /\ X ) let X be Subset of REAL; ::_thesis: ( rng seq c= (dom (f1 + f2)) /\ X implies ( rng seq c= dom (f1 + f2) & rng seq c= X & dom (f1 + f2) = (dom f1) /\ (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) & rng seq c= X & dom (f1 + f2) = (dom f1) /\ (dom f2) & rng seq c= (dom f1) /\ X & rng seq c= (dom f2) /\ X ) A2: (dom (f1 + f2)) /\ X c= X by XBOOLE_1:17; (dom (f1 + f2)) /\ X c= dom (f1 + f2) by XBOOLE_1:17; hence A3: ( rng seq c= dom (f1 + f2) & rng seq c= X ) by A1, A2, XBOOLE_1:1; ::_thesis: ( dom (f1 + f2) = (dom f1) /\ (dom f2) & rng seq c= (dom f1) /\ X & rng seq c= (dom f2) /\ X ) thus A4: dom (f1 + f2) = (dom f1) /\ (dom f2) by VALUED_1:def_1; ::_thesis: ( rng seq c= (dom f1) /\ X & rng seq c= (dom f2) /\ X ) then dom (f1 + f2) c= dom f2 by XBOOLE_1:17; then A5: rng seq c= dom f2 by A3, XBOOLE_1:1; dom (f1 + f2) c= dom f1 by A4, XBOOLE_1:17; then rng seq c= dom f1 by A3, XBOOLE_1:1; hence ( rng seq c= (dom f1) /\ X & rng seq c= (dom f2) /\ X ) by A3, A5, XBOOLE_1:19; ::_thesis: verum end; theorem Th1: :: LIMFUNC2:1 for r being Real for seq being Real_Sequence st seq is convergent & r < lim seq holds ex n being Element of NAT st for k being Element of NAT st n <= k holds r < seq . k proof let r be Real; ::_thesis: for seq being Real_Sequence st seq is convergent & r < lim seq holds ex n being Element of NAT st for k being Element of NAT st n <= k holds r < seq . k let seq be Real_Sequence; ::_thesis: ( seq is convergent & r < lim seq implies ex n being Element of NAT st for k being Element of NAT st n <= k holds r < seq . k ) assume that A1: seq is convergent and A2: r < lim seq ; ::_thesis: ex n being Element of NAT st for k being Element of NAT st n <= k holds r < seq . k reconsider s = NAT --> r as Real_Sequence ; A3: seq - s is convergent by A1, SEQ_2:11; s . 0 = r by FUNCOP_1:7; then lim s = r by SEQ_4:25; then lim (seq - s) = (lim seq) - r by A1, SEQ_2:12; then consider n being Element of NAT such that A4: for k being Element of NAT st n <= k holds 0 < (seq - s) . k by A2, A3, LIMFUNC1:4, XREAL_1:50; take n ; ::_thesis: for k being Element of NAT st n <= k holds r < seq . k let k be Element of NAT ; ::_thesis: ( n <= k implies r < seq . k ) assume n <= k ; ::_thesis: r < seq . k then 0 < (seq - s) . k by A4; then 0 < (seq . k) - (s . k) by RFUNCT_2:1; then 0 < (seq . k) - r by FUNCOP_1:7; then 0 + r < seq . k by XREAL_1:20; hence r < seq . k ; ::_thesis: verum end; theorem Th2: :: LIMFUNC2:2 for r being Real for seq being Real_Sequence st seq is convergent & lim seq < r holds ex n being Element of NAT st for k being Element of NAT st n <= k holds seq . k < r proof let r be Real; ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq < r holds ex n being Element of NAT st for k being Element of NAT st n <= k holds seq . k < r let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq < r implies ex n being Element of NAT st for k being Element of NAT st n <= k holds seq . k < r ) assume that A1: seq is convergent and A2: lim seq < r ; ::_thesis: ex n being Element of NAT st for k being Element of NAT st n <= k holds seq . k < r reconsider s = NAT --> r as Real_Sequence ; A3: s - seq is convergent by A1, SEQ_2:11; s . 0 = r by FUNCOP_1:7; then lim s = r by SEQ_4:25; then lim (s - seq) = r - (lim seq) by A1, SEQ_2:12; then consider n being Element of NAT such that A4: for k being Element of NAT st n <= k holds 0 < (s - seq) . k by A2, A3, LIMFUNC1:4, XREAL_1:50; take n ; ::_thesis: for k being Element of NAT st n <= k holds seq . k < r let k be Element of NAT ; ::_thesis: ( n <= k implies seq . k < r ) assume n <= k ; ::_thesis: seq . k < r then 0 < (s - seq) . k by A4; then 0 < (s . k) - (seq . k) by RFUNCT_2:1; then 0 < r - (seq . k) by FUNCOP_1:7; then 0 + (seq . k) < r by XREAL_1:20; hence seq . k < r ; ::_thesis: verum end; Lm5: for x0 being Real for seq being Real_Sequence for f being PartFunc of REAL,REAL st ( for g1 being Real ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds f . r1 < g1 ) ) ) & seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (right_open_halfline x0) holds f /* seq is divergent_to-infty proof let x0 be Real; ::_thesis: for seq being Real_Sequence for f being PartFunc of REAL,REAL st ( for g1 being Real ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds f . r1 < g1 ) ) ) & seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (right_open_halfline x0) holds f /* seq is divergent_to-infty let seq be Real_Sequence; ::_thesis: for f being PartFunc of REAL,REAL st ( for g1 being Real ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds f . r1 < g1 ) ) ) & seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (right_open_halfline x0) holds f /* seq is divergent_to-infty let f be PartFunc of REAL,REAL; ::_thesis: ( ( for g1 being Real ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds f . r1 < g1 ) ) ) & seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (right_open_halfline x0) implies f /* seq is divergent_to-infty ) assume that A1: for g1 being Real ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds f . r1 < g1 ) ) and A2: seq is convergent and A3: lim seq = x0 and A4: rng seq c= (dom f) /\ (right_open_halfline x0) ; ::_thesis: f /* seq is divergent_to-infty A5: (dom f) /\ (right_open_halfline x0) c= dom f by XBOOLE_1:17; now__::_thesis:_for_g1_being_Real_ex_n_being_Element_of_NAT_st_ for_k_being_Element_of_NAT_st_n_<=_k_holds_ (f_/*_seq)_._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 /* seq) . k < g1 consider r being Real such that A6: x0 < r and A7: for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds f . r1 < g1 by A1; consider n being Element of NAT such that A8: for k being Element of NAT st n <= k holds seq . k < r by A2, A3, A6, Th2; take n = n; ::_thesis: for k being Element of NAT st n <= k holds (f /* seq) . k < g1 let k be Element of NAT ; ::_thesis: ( n <= k implies (f /* seq) . k < g1 ) assume A9: n <= k ; ::_thesis: (f /* seq) . k < g1 A10: seq . k in rng seq by VALUED_0:28; then seq . k in right_open_halfline x0 by A4, XBOOLE_0:def_4; then seq . k in { g2 where g2 is Real : x0 < g2 } by XXREAL_1:230; then A11: ex g2 being Real st ( g2 = seq . k & x0 < g2 ) ; seq . k in dom f by A4, A10, XBOOLE_0:def_4; then f . (seq . k) < g1 by A7, A8, A9, A11; hence (f /* seq) . k < g1 by A4, A5, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; hence f /* seq is divergent_to-infty by LIMFUNC1:def_5; ::_thesis: verum end; theorem Th3: :: LIMFUNC2:3 for r2, r1 being Real for f being PartFunc of REAL,REAL st 0 < r2 & ].(r1 - r2),r1.[ c= dom f holds for r being Real st r < r1 holds ex g being Real st ( r < g & g < r1 & g in dom f ) proof let r2, r1 be Real; ::_thesis: for f being PartFunc of REAL,REAL st 0 < r2 & ].(r1 - r2),r1.[ c= dom f holds for r being Real st r < r1 holds ex g being Real st ( r < g & g < r1 & g in dom f ) let f be PartFunc of REAL,REAL; ::_thesis: ( 0 < r2 & ].(r1 - r2),r1.[ c= dom f implies for r being Real st r < r1 holds ex g being Real st ( r < g & g < r1 & g in dom f ) ) assume that A1: 0 < r2 and A2: ].(r1 - r2),r1.[ c= dom f ; ::_thesis: for r being Real st r < r1 holds ex g being Real st ( r < g & g < r1 & g in dom f ) let r be Real; ::_thesis: ( r < r1 implies ex g being Real st ( r < g & g < r1 & g in dom f ) ) assume A3: r < r1 ; ::_thesis: ex g being Real st ( r < g & g < r1 & g in dom f ) now__::_thesis:_ex_g_being_Real_st_ (_r_<_g_&_g_<_r1_&_g_in_dom_f_) percases ( r1 - r2 <= r or r <= r1 - r2 ) ; supposeA4: r1 - r2 <= r ; ::_thesis: ex g being Real st ( r < g & g < r1 & g in dom f ) consider g being real number such that A5: r < g and A6: g < r1 by A3, XREAL_1:5; reconsider g = g as Real by XREAL_0:def_1; take g = g; ::_thesis: ( r < g & g < r1 & g in dom f ) thus ( r < g & g < r1 ) by A5, A6; ::_thesis: g in dom f r1 - r2 < g by A4, A5, XXREAL_0:2; then g in { g2 where g2 is Real : ( r1 - r2 < g2 & g2 < r1 ) } by A6; then g in ].(r1 - r2),r1.[ by RCOMP_1:def_2; hence g in dom f by A2; ::_thesis: verum end; supposeA7: r <= r1 - r2 ; ::_thesis: ex g being Real st ( r < g & g < r1 & g in dom f ) r1 - r2 < r1 by A1, Lm1; then consider g being real number such that A8: r1 - r2 < g and A9: g < r1 by XREAL_1:5; reconsider g = g as Real by XREAL_0:def_1; take g = g; ::_thesis: ( r < g & g < r1 & g in dom f ) thus ( r < g & g < r1 ) by A7, A8, A9, XXREAL_0:2; ::_thesis: g in dom f g in { g2 where g2 is Real : ( r1 - r2 < g2 & g2 < r1 ) } by A8, A9; then g in ].(r1 - r2),r1.[ by RCOMP_1:def_2; hence g in dom f by A2; ::_thesis: verum end; end; end; hence ex g being Real st ( r < g & g < r1 & g in dom f ) ; ::_thesis: verum end; theorem Th4: :: LIMFUNC2:4 for r2, r1 being Real for f being PartFunc of REAL,REAL st 0 < r2 & ].r1,(r1 + r2).[ c= dom f holds for r being Real st r1 < r holds ex g being Real st ( g < r & r1 < g & g in dom f ) proof let r2, r1 be Real; ::_thesis: for f being PartFunc of REAL,REAL st 0 < r2 & ].r1,(r1 + r2).[ c= dom f holds for r being Real st r1 < r holds ex g being Real st ( g < r & r1 < g & g in dom f ) let f be PartFunc of REAL,REAL; ::_thesis: ( 0 < r2 & ].r1,(r1 + r2).[ c= dom f implies for r being Real st r1 < r holds ex g being Real st ( g < r & r1 < g & g in dom f ) ) assume that A1: 0 < r2 and A2: ].r1,(r1 + r2).[ c= dom f ; ::_thesis: for r being Real st r1 < r holds ex g being Real st ( g < r & r1 < g & g in dom f ) let r be Real; ::_thesis: ( r1 < r implies ex g being Real st ( g < r & r1 < g & g in dom f ) ) assume A3: r1 < r ; ::_thesis: ex g being Real st ( g < r & r1 < g & g in dom f ) now__::_thesis:_ex_g_being_Real_st_ (_g_<_r_&_r1_<_g_&_g_in_dom_f_) percases ( r1 + r2 <= r or r <= r1 + r2 ) ; supposeA4: r1 + r2 <= r ; ::_thesis: ex g being Real st ( g < r & r1 < g & g in dom f ) r1 < r1 + r2 by A1, Lm1; then consider g being real number such that A5: r1 < g and A6: g < r1 + r2 by XREAL_1:5; reconsider g = g as Real by XREAL_0:def_1; take g = g; ::_thesis: ( g < r & r1 < g & g in dom f ) thus ( g < r & r1 < g ) by A4, A5, A6, XXREAL_0:2; ::_thesis: g in dom f g in { g2 where g2 is Real : ( r1 < g2 & g2 < r1 + r2 ) } by A5, A6; then g in ].r1,(r1 + r2).[ by RCOMP_1:def_2; hence g in dom f by A2; ::_thesis: verum end; supposeA7: r <= r1 + r2 ; ::_thesis: ex g being Real st ( g < r & r1 < g & g in dom f ) consider g being real number such that A8: r1 < g and A9: g < r by A3, XREAL_1:5; reconsider g = g as Real by XREAL_0:def_1; take g = g; ::_thesis: ( g < r & r1 < g & g in dom f ) thus ( g < r & r1 < g ) by A8, A9; ::_thesis: g in dom f g < r1 + r2 by A7, A9, XXREAL_0:2; then g in { g2 where g2 is Real : ( r1 < g2 & g2 < r1 + r2 ) } by A8; then g in ].r1,(r1 + r2).[ by RCOMP_1:def_2; hence g in dom f by A2; ::_thesis: verum end; end; end; hence ex g being Real st ( g < r & r1 < g & g in dom f ) ; ::_thesis: verum end; theorem Th5: :: LIMFUNC2:5 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 & rng seq c= (dom f) /\ (left_open_halfline 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 & rng seq c= (dom f) /\ (left_open_halfline 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 & rng seq c= (dom f) /\ (left_open_halfline 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 & rng seq c= (dom f) /\ (left_open_halfline x0) ) ) deffunc H1( Element of NAT ) -> Element of REAL = 1 / (\$1 + 1); consider s1 being Real_Sequence such that A1: for n being Element of NAT holds s1 . n = H1(n) from SEQ_1:sch_1(); reconsider s2 = NAT --> x0 as Real_Sequence ; A2: s1 is convergent by A1, SEQ_4:30; then A3: s2 - s1 is convergent by SEQ_2:11; assume A4: 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 & rng seq c= (dom f) /\ (left_open_halfline x0) ) A5: now__::_thesis:_for_n_being_Element_of_NAT_holds_ (_(s2_-_s1)_._n_<=_seq_._n_&_seq_._n_<=_s2_._n_) let n be Element of NAT ; ::_thesis: ( (s2 - s1) . n <= seq . n & seq . n <= s2 . n ) A6: (s2 - s1) . n = (s2 . n) - (s1 . n) by RFUNCT_2:1 .= x0 - (s1 . n) by FUNCOP_1:7 .= x0 - (1 / (n + 1)) by A1 ; seq . n <= x0 by A4; hence ( (s2 - s1) . n <= seq . n & seq . n <= s2 . n ) by A4, A6, FUNCOP_1:7; ::_thesis: verum end; s2 . 0 = x0 by FUNCOP_1:7; then A7: lim s2 = x0 by SEQ_4:25; lim s1 = 0 by A1, SEQ_4:30; then A8: lim (s2 - s1) = x0 - 0 by A7, A2, SEQ_2:12 .= x0 ; hence seq is convergent by A7, A3, A5, SEQ_2:19; ::_thesis: ( lim seq = x0 & rng seq c= dom f & rng seq c= (dom f) /\ (left_open_halfline x0) ) thus lim seq = x0 by A7, A3, A8, A5, SEQ_2:20; ::_thesis: ( rng seq c= dom f & rng seq c= (dom f) /\ (left_open_halfline x0) ) now__::_thesis:_for_x_being_set_st_x_in_rng_seq_holds_ x_in_dom_f let x be set ; ::_thesis: ( x in rng seq implies 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 A4; ::_thesis: verum end; hence A9: rng seq c= dom f by TARSKI:def_3; ::_thesis: rng seq c= (dom f) /\ (left_open_halfline x0) now__::_thesis:_for_x_being_set_st_x_in_rng_seq_holds_ x_in_left_open_halfline_x0 let x be set ; ::_thesis: ( x in rng seq implies x in left_open_halfline x0 ) assume x in rng seq ; ::_thesis: x in left_open_halfline x0 then consider n being Element of NAT such that A10: x = seq . n by FUNCT_2:113; seq . n < x0 by A4; then seq . n in { g2 where g2 is Real : g2 < x0 } ; hence x in left_open_halfline x0 by A10, XXREAL_1:229; ::_thesis: verum end; then rng seq c= left_open_halfline x0 by TARSKI:def_3; hence rng seq c= (dom f) /\ (left_open_halfline x0) by A9, XBOOLE_1:19; ::_thesis: verum end; theorem Th6: :: LIMFUNC2:6 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 < seq . n & seq . n < x0 + (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) /\ (right_open_halfline 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 < seq . n & seq . n < x0 + (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) /\ (right_open_halfline x0) ) let seq be Real_Sequence; ::_thesis: for f being PartFunc of REAL,REAL st ( for n being Element of NAT holds ( x0 < seq . n & seq . n < x0 + (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) /\ (right_open_halfline x0) ) let f be PartFunc of REAL,REAL; ::_thesis: ( ( for n being Element of NAT holds ( x0 < seq . n & seq . n < x0 + (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) /\ (right_open_halfline x0) ) ) deffunc H1( Element of NAT ) -> Element of REAL = 1 / (\$1 + 1); consider s1 being Real_Sequence such that A1: for n being Element of NAT holds s1 . n = H1(n) from SEQ_1:sch_1(); reconsider s2 = NAT --> x0 as Real_Sequence ; A2: s1 is convergent by A1, SEQ_4:30; then A3: s2 + s1 is convergent by SEQ_2:5; assume A4: for n being Element of NAT holds ( x0 < seq . n & seq . n < x0 + (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) /\ (right_open_halfline x0) ) A5: now__::_thesis:_for_n_being_Element_of_NAT_holds_ (_s2_._n_<=_seq_._n_&_seq_._n_<=_(s2_+_s1)_._n_) let n be Element of NAT ; ::_thesis: ( s2 . n <= seq . n & seq . n <= (s2 + s1) . n ) A6: (s2 + s1) . n = (s2 . n) + (s1 . n) by SEQ_1:7 .= x0 + (s1 . n) by FUNCOP_1:7 .= x0 + (1 / (n + 1)) by A1 ; x0 <= seq . n by A4; hence ( s2 . n <= seq . n & seq . n <= (s2 + s1) . n ) by A4, A6, FUNCOP_1:7; ::_thesis: verum end; s2 . 0 = x0 by FUNCOP_1:7; then A7: lim s2 = x0 by SEQ_4:25; lim s1 = 0 by A1, SEQ_4:30; then A8: lim (s2 + s1) = x0 + 0 by A7, A2, SEQ_2:6 .= x0 ; hence seq is convergent by A7, A3, A5, SEQ_2:19; ::_thesis: ( lim seq = x0 & rng seq c= dom f & rng seq c= (dom f) /\ (right_open_halfline x0) ) thus lim seq = x0 by A7, A3, A8, A5, SEQ_2:20; ::_thesis: ( rng seq c= dom f & rng seq c= (dom f) /\ (right_open_halfline x0) ) now__::_thesis:_for_x_being_set_st_x_in_rng_seq_holds_ x_in_dom_f let x be set ; ::_thesis: ( x in rng seq implies 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 A4; ::_thesis: verum end; hence A9: rng seq c= dom f by TARSKI:def_3; ::_thesis: rng seq c= (dom f) /\ (right_open_halfline x0) now__::_thesis:_for_x_being_set_st_x_in_rng_seq_holds_ x_in_right_open_halfline_x0 let x be set ; ::_thesis: ( x in rng seq implies x in right_open_halfline x0 ) assume x in rng seq ; ::_thesis: x in right_open_halfline x0 then consider n being Element of NAT such that A10: x = seq . n by FUNCT_2:113; x0 < seq . n by A4; then seq . n in { g2 where g2 is Real : x0 < g2 } ; hence x in right_open_halfline x0 by A10, XXREAL_1:230; ::_thesis: verum end; then rng seq c= right_open_halfline x0 by TARSKI:def_3; hence rng seq c= (dom f) /\ (right_open_halfline x0) by A9, XBOOLE_1:19; ::_thesis: verum end; definition let f be PartFunc of REAL,REAL; let x0 be Real; predf is_left_convergent_in x0 means :Def1: :: LIMFUNC2:def 1 ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g 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) /\ (left_open_halfline x0) holds ( f /* seq is convergent & lim (f /* seq) = g ) ); predf is_left_divergent_to+infty_in x0 means :Def2: :: LIMFUNC2:def 2 ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (left_open_halfline x0) holds f /* seq is divergent_to+infty ) ); predf is_left_divergent_to-infty_in x0 means :Def3: :: LIMFUNC2:def 3 ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (left_open_halfline x0) holds f /* seq is divergent_to-infty ) ); predf is_right_convergent_in x0 means :Def4: :: LIMFUNC2:def 4 ( ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g 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) /\ (right_open_halfline x0) holds ( f /* seq is convergent & lim (f /* seq) = g ) ); predf is_right_divergent_to+infty_in x0 means :Def5: :: LIMFUNC2:def 5 ( ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) & ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (right_open_halfline x0) holds f /* seq is divergent_to+infty ) ); predf is_right_divergent_to-infty_in x0 means :Def6: :: LIMFUNC2:def 6 ( ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) & ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (right_open_halfline x0) holds f /* seq is divergent_to-infty ) ); end; :: deftheorem Def1 defines is_left_convergent_in LIMFUNC2:def_1_:_ for f being PartFunc of REAL,REAL for x0 being Real holds ( f is_left_convergent_in x0 iff ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g 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) /\ (left_open_halfline x0) holds ( f /* seq is convergent & lim (f /* seq) = g ) ) ); :: deftheorem Def2 defines is_left_divergent_to+infty_in LIMFUNC2:def_2_:_ for f being PartFunc of REAL,REAL for x0 being Real holds ( f is_left_divergent_to+infty_in x0 iff ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (left_open_halfline x0) holds f /* seq is divergent_to+infty ) ) ); :: deftheorem Def3 defines is_left_divergent_to-infty_in LIMFUNC2:def_3_:_ for f being PartFunc of REAL,REAL for x0 being Real holds ( f is_left_divergent_to-infty_in x0 iff ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (left_open_halfline x0) holds f /* seq is divergent_to-infty ) ) ); :: deftheorem Def4 defines is_right_convergent_in LIMFUNC2:def_4_:_ for f being PartFunc of REAL,REAL for x0 being Real holds ( f is_right_convergent_in x0 iff ( ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g 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) /\ (right_open_halfline x0) holds ( f /* seq is convergent & lim (f /* seq) = g ) ) ); :: deftheorem Def5 defines is_right_divergent_to+infty_in LIMFUNC2:def_5_:_ for f being PartFunc of REAL,REAL for x0 being Real holds ( f is_right_divergent_to+infty_in x0 iff ( ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) & ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (right_open_halfline x0) holds f /* seq is divergent_to+infty ) ) ); :: deftheorem Def6 defines is_right_divergent_to-infty_in LIMFUNC2:def_6_:_ for f being PartFunc of REAL,REAL for x0 being Real holds ( f is_right_divergent_to-infty_in x0 iff ( ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) & ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (right_open_halfline x0) holds f /* seq is divergent_to-infty ) ) ); theorem :: LIMFUNC2:7 for x0 being Real for f being PartFunc of REAL,REAL holds ( f is_left_convergent_in x0 iff ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ex g being Real st for g1 being Real st 0 < g1 holds ex r being Real st ( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & 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_left_convergent_in x0 iff ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ex g being Real st for g1 being Real st 0 < g1 holds ex r being Real st ( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_left_convergent_in x0 iff ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ex g being Real st for g1 being Real st 0 < g1 holds ex r being Real st ( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ) ) thus ( f is_left_convergent_in x0 implies ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ex g being Real st for g1 being Real st 0 < g1 holds ex r being Real st ( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ) ) ::_thesis: ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ex g being Real st for g1 being Real st 0 < g1 holds ex r being Real st ( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) implies f is_left_convergent_in x0 ) proof assume that A1: f is_left_convergent_in x0 and A2: ( ex r being Real st ( r < x0 & ( for g being Real holds ( not r < g or not g < x0 or not g in dom f ) ) ) or for g being Real ex g1 being Real st ( 0 < g1 & ( for r being Real st r < x0 holds ex r1 being Real st ( r < r1 & r1 < x0 & 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) /\ (left_open_halfline 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 r being Real st r < x0 holds ex r1 being Real st ( r < r1 & r1 < x0 & r1 in dom f & abs ((f . r1) - g) >= g1 ) by A1, A2, Def1; defpred S1[ Element of NAT , real number ] means ( x0 - (1 / (\$1 + 1)) < \$2 & \$2 < x0 & \$2 in dom f & abs ((f . \$2) - g) >= g1 ); A6: now__::_thesis:_for_n_being_Element_of_NAT_ex_g2_being_Real_st_S1[n,g2] let n be Element of NAT ; ::_thesis: ex g2 being Real st S1[n,g2] x0 - (1 / (n + 1)) < x0 by Lm3; then consider g2 being Real such that A7: x0 - (1 / (n + 1)) < g2 and A8: g2 < x0 and A9: g2 in dom f and A10: abs ((f . g2) - g) >= g1 by A5; take g2 = g2; ::_thesis: S1[n,g2] thus S1[n,g2] by A7, A8, A9, A10; ::_thesis: verum end; consider s being Real_Sequence such that A11: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A6); A12: rng s c= (dom f) /\ (left_open_halfline x0) by A11, Th5; A13: lim s = x0 by A11, Th5; A14: s is convergent by A11, Th5; then A15: lim (f /* s) = g by A3, A13, A12; f /* s is convergent by A3, A14, A13, A12; then consider n being Element of NAT such that A16: for k being Element of NAT st n <= k holds abs (((f /* s) . k) - g) < g1 by A4, A15, SEQ_2:def_7; A17: abs (((f /* s) . n) - g) < g1 by A16; rng s c= dom f by A11, Th5; then abs ((f . (s . n)) - g) < g1 by A17, FUNCT_2:108; hence contradiction by A11; ::_thesis: verum end; assume A18: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ; ::_thesis: ( for g being Real ex g1 being Real st ( 0 < g1 & ( for r being Real holds ( not r < x0 or ex r1 being Real st ( r < r1 & r1 < x0 & r1 in dom f & not abs ((f . r1) - g) < g1 ) ) ) ) or f is_left_convergent_in x0 ) given g being Real such that A19: for g1 being Real st 0 < g1 holds ex r being Real st ( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ; ::_thesis: f is_left_convergent_in x0 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)_=_g_) 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) = g ) ) assume that A20: s is convergent and A21: lim s = x0 and A22: rng s c= (dom f) /\ (left_open_halfline x0) ; ::_thesis: ( f /* s is convergent & lim (f /* s) = g ) A23: (dom f) /\ (left_open_halfline x0) c= dom f by XBOOLE_1:17; A24: 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 A25: 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 r being Real such that A26: r < x0 and A27: for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds abs ((f . r1) - g) < g1 by A19, A25; consider n being Element of NAT such that A28: for k being Element of NAT st n <= k holds r < s . k by A20, A21, A26, Th1; 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 A29: n <= k ; ::_thesis: abs (((f /* s) . k) - g) < g1 A30: s . k in rng s by VALUED_0:28; then s . k in left_open_halfline x0 by A22, XBOOLE_0:def_4; then s . k in { g2 where g2 is Real : g2 < x0 } by XXREAL_1:229; then A31: ex g2 being Real st ( g2 = s . k & g2 < x0 ) ; s . k in dom f by A22, A30, XBOOLE_0:def_4; then abs ((f . (s . k)) - g) < g1 by A27, A28, A29, A31; hence abs (((f /* s) . k) - g) < g1 by A22, A23, 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 A24, SEQ_2:def_7; ::_thesis: verum end; hence f is_left_convergent_in x0 by A18, Def1; ::_thesis: verum end; theorem :: LIMFUNC2:8 for x0 being Real for f being PartFunc of REAL,REAL holds ( f is_left_divergent_to+infty_in x0 iff ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ( for g1 being Real ex r being Real st ( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds g1 < f . r1 ) ) ) ) ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL holds ( f is_left_divergent_to+infty_in x0 iff ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ( for g1 being Real ex r being Real st ( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds g1 < f . r1 ) ) ) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_left_divergent_to+infty_in x0 iff ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ( for g1 being Real ex r being Real st ( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds g1 < f . r1 ) ) ) ) ) thus ( f is_left_divergent_to+infty_in x0 implies ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ( for g1 being Real ex r being Real st ( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds g1 < f . r1 ) ) ) ) ) ::_thesis: ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ( for g1 being Real ex r being Real st ( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds g1 < f . r1 ) ) ) implies f is_left_divergent_to+infty_in x0 ) proof assume that A1: f is_left_divergent_to+infty_in x0 and A2: ( ex r being Real st ( r < x0 & ( for g being Real holds ( not r < g or not g < x0 or not g in dom f ) ) ) or ex g1 being Real st for r being Real st r < x0 holds ex r1 being Real st ( r < r1 & r1 < x0 & r1 in dom f & g1 >= f . r1 ) ) ; ::_thesis: contradiction consider g1 being Real such that A3: for r being Real st r < x0 holds ex r1 being Real st ( r < r1 & r1 < x0 & r1 in dom f & g1 >= f . r1 ) by A1, A2, Def2; defpred S1[ Element of NAT , real number ] means ( x0 - (1 / (\$1 + 1)) < \$2 & \$2 < x0 & \$2 in dom f & f . \$2 <= g1 ); A4: now__::_thesis:_for_n_being_Element_of_NAT_ex_g2_being_Real_st_S1[n,g2] let n be Element of NAT ; ::_thesis: ex g2 being Real st S1[n,g2] x0 - (1 / (n + 1)) < x0 by Lm3; then consider g2 being Real such that A5: x0 - (1 / (n + 1)) < g2 and A6: g2 < x0 and A7: g2 in dom f and A8: f . g2 <= g1 by A3; take g2 = g2; ::_thesis: S1[n,g2] thus S1[n,g2] by A5, A6, A7, A8; ::_thesis: verum end; consider s being Real_Sequence such that A9: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A4); A10: rng s c= (dom f) /\ (left_open_halfline x0) by A9, Th5; A11: lim s = x0 by A9, Th5; s is convergent by A9, Th5; then f /* s is divergent_to+infty by A1, A11, A10, Def2; then consider n being Element of NAT such that A12: for k being Element of NAT st n <= k holds g1 < (f /* s) . k by LIMFUNC1:def_4; A13: g1 < (f /* s) . n by A12; rng s c= dom f by A9, Th5; then g1 < f . (s . n) by A13, FUNCT_2:108; hence contradiction by A9; ::_thesis: verum end; assume that A14: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) and A15: for g1 being Real ex r being Real st ( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds g1 < f . r1 ) ) ; ::_thesis: f is_left_divergent_to+infty_in x0 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 A16: s is convergent and A17: lim s = x0 and A18: rng s c= (dom f) /\ (left_open_halfline x0) ; ::_thesis: f /* s is divergent_to+infty A19: (dom f) /\ (left_open_halfline x0) c= dom f by XBOOLE_1:17; 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 r being Real such that A20: r < x0 and A21: for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds g1 < f . r1 by A15; consider n being Element of NAT such that A22: for k being Element of NAT st n <= k holds r < s . k by A16, A17, A20, Th1; 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 A23: n <= k ; ::_thesis: g1 < (f /* s) . k A24: s . k in rng s by VALUED_0:28; then s . k in left_open_halfline x0 by A18, XBOOLE_0:def_4; then s . k in { g2 where g2 is Real : g2 < x0 } by XXREAL_1:229; then A25: ex g2 being Real st ( g2 = s . k & g2 < x0 ) ; s . k in dom f by A18, A24, XBOOLE_0:def_4; then g1 < f . (s . k) by A21, A22, A23, A25; hence g1 < (f /* s) . k by A18, A19, 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_left_divergent_to+infty_in x0 by A14, Def2; ::_thesis: verum end; theorem :: LIMFUNC2:9 for x0 being Real for f being PartFunc of REAL,REAL holds ( f is_left_divergent_to-infty_in x0 iff ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ( for g1 being Real ex r being Real st ( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds f . r1 < g1 ) ) ) ) ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL holds ( f is_left_divergent_to-infty_in x0 iff ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ( for g1 being Real ex r being Real st ( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds f . r1 < g1 ) ) ) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_left_divergent_to-infty_in x0 iff ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ( for g1 being Real ex r being Real st ( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds f . r1 < g1 ) ) ) ) ) thus ( f is_left_divergent_to-infty_in x0 implies ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ( for g1 being Real ex r being Real st ( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds f . r1 < g1 ) ) ) ) ) ::_thesis: ( ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ( for g1 being Real ex r being Real st ( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds f . r1 < g1 ) ) ) implies f is_left_divergent_to-infty_in x0 ) proof assume that A1: f is_left_divergent_to-infty_in x0 and A2: ( ex r being Real st ( r < x0 & ( for g being Real holds ( not r < g or not g < x0 or not g in dom f ) ) ) or ex g1 being Real st for r being Real st r < x0 holds ex r1 being Real st ( r < r1 & r1 < x0 & r1 in dom f & g1 <= f . r1 ) ) ; ::_thesis: contradiction consider g1 being Real such that A3: for r being Real st r < x0 holds ex r1 being Real st ( r < r1 & r1 < x0 & r1 in dom f & g1 <= f . r1 ) by A1, A2, Def3; defpred S1[ Element of NAT , real number ] means ( x0 - (1 / (\$1 + 1)) < \$2 & \$2 < x0 & \$2 in dom f & g1 <= f . \$2 ); A4: now__::_thesis:_for_n_being_Element_of_NAT_ex_g2_being_Real_st_S1[n,g2] let n be Element of NAT ; ::_thesis: ex g2 being Real st S1[n,g2] x0 - (1 / (n + 1)) < x0 by Lm3; then consider g2 being Real such that A5: x0 - (1 / (n + 1)) < g2 and A6: g2 < x0 and A7: g2 in dom f and A8: g1 <= f . g2 by A3; take g2 = g2; ::_thesis: S1[n,g2] thus S1[n,g2] by A5, A6, A7, A8; ::_thesis: verum end; consider s being Real_Sequence such that A9: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A4); A10: rng s c= (dom f) /\ (left_open_halfline x0) by A9, Th5; A11: lim s = x0 by A9, Th5; s is convergent by A9, Th5; then f /* s is divergent_to-infty by A1, A11, A10, Def3; then consider n being Element of NAT such that A12: for k being Element of NAT st n <= k holds (f /* s) . k < g1 by LIMFUNC1:def_5; A13: (f /* s) . n < g1 by A12; rng s c= dom f by A9, Th5; then f . (s . n) < g1 by A13, FUNCT_2:108; hence contradiction by A9; ::_thesis: verum end; assume that A14: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) and A15: for g1 being Real ex r being Real st ( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds f . r1 < g1 ) ) ; ::_thesis: f is_left_divergent_to-infty_in x0 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 A16: s is convergent and A17: lim s = x0 and A18: rng s c= (dom f) /\ (left_open_halfline x0) ; ::_thesis: f /* s is divergent_to-infty A19: (dom f) /\ (left_open_halfline x0) c= dom f by XBOOLE_1:17; 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 r being Real such that A20: r < x0 and A21: for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds f . r1 < g1 by A15; consider n being Element of NAT such that A22: for k being Element of NAT st n <= k holds r < s . k by A16, A17, A20, Th1; 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 A23: n <= k ; ::_thesis: (f /* s) . k < g1 A24: s . k in rng s by VALUED_0:28; then s . k in left_open_halfline x0 by A18, XBOOLE_0:def_4; then s . k in { g2 where g2 is Real : g2 < x0 } by XXREAL_1:229; then A25: ex g2 being Real st ( g2 = s . k & g2 < x0 ) ; s . k in dom f by A18, A24, XBOOLE_0:def_4; then f . (s . k) < g1 by A21, A22, A23, A25; hence (f /* s) . k < g1 by A18, A19, 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_left_divergent_to-infty_in x0 by A14, Def3; ::_thesis: verum end; theorem :: LIMFUNC2:10 for x0 being Real for f being PartFunc of REAL,REAL holds ( f is_right_convergent_in x0 iff ( ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) & ex g being Real st for g1 being Real st 0 < g1 holds ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & 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_right_convergent_in x0 iff ( ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) & ex g being Real st for g1 being Real st 0 < g1 holds ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_right_convergent_in x0 iff ( ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) & ex g being Real st for g1 being Real st 0 < g1 holds ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ) ) thus ( f is_right_convergent_in x0 implies ( ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) & ex g being Real st for g1 being Real st 0 < g1 holds ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ) ) ::_thesis: ( ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) & ex g being Real st for g1 being Real st 0 < g1 holds ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) implies f is_right_convergent_in x0 ) proof assume that A1: f is_right_convergent_in x0 and A2: ( ex r being Real st ( x0 < r & ( for g being Real holds ( not g < r or not x0 < g or not g in dom f ) ) ) or for g being Real ex g1 being Real st ( 0 < g1 & ( for r being Real st x0 < r holds ex r1 being Real st ( r1 < r & x0 < r1 & 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) /\ (right_open_halfline x0) holds ( f /* seq is convergent & lim (f /* seq) = g ) by A1, Def4; consider g1 being Real such that A4: 0 < g1 and A5: for r being Real st x0 < r holds ex r1 being Real st ( r1 < r & x0 < r1 & r1 in dom f & abs ((f . r1) - g) >= g1 ) by A1, A2, Def4; defpred S1[ Element of NAT , real number ] means ( x0 < \$2 & \$2 < x0 + (1 / (\$1 + 1)) & \$2 in dom f & g1 <= abs ((f . \$2) - g) ); A6: now__::_thesis:_for_n_being_Element_of_NAT_ex_r1_being_Real_st_S1[n,r1] let n be Element of NAT ; ::_thesis: ex r1 being Real st S1[n,r1] x0 < x0 + (1 / (n + 1)) by Lm3; then consider r1 being Real such that A7: r1 < x0 + (1 / (n + 1)) and A8: x0 < r1 and A9: r1 in dom f and A10: g1 <= abs ((f . r1) - g) by A5; take r1 = r1; ::_thesis: S1[n,r1] thus S1[n,r1] by A7, A8, A9, A10; ::_thesis: verum end; consider s being Real_Sequence such that A11: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A6); A12: rng s c= (dom f) /\ (right_open_halfline x0) by A11, Th6; A13: lim s = x0 by A11, Th6; A14: s is convergent by A11, Th6; then A15: lim (f /* s) = g by A3, A13, A12; f /* s is convergent by A3, A14, A13, A12; then consider n being Element of NAT such that A16: for k being Element of NAT st n <= k holds abs (((f /* s) . k) - g) < g1 by A4, A15, SEQ_2:def_7; A17: abs (((f /* s) . n) - g) < g1 by A16; rng s c= dom f by A11, Th6; then abs ((f . (s . n)) - g) < g1 by A17, FUNCT_2:108; hence contradiction by A11; ::_thesis: verum end; assume A18: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ; ::_thesis: ( for g being Real ex g1 being Real st ( 0 < g1 & ( for r being Real holds ( not x0 < r or ex r1 being Real st ( r1 < r & x0 < r1 & r1 in dom f & not abs ((f . r1) - g) < g1 ) ) ) ) or f is_right_convergent_in x0 ) given g being Real such that A19: for g1 being Real st 0 < g1 holds ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ; ::_thesis: f is_right_convergent_in x0 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)_=_g_) 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) = g ) ) assume that A20: s is convergent and A21: lim s = x0 and A22: rng s c= (dom f) /\ (right_open_halfline x0) ; ::_thesis: ( f /* s is convergent & lim (f /* s) = g ) A23: (dom f) /\ (right_open_halfline x0) c= dom f by XBOOLE_1:17; A24: 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 A25: 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 r being Real such that A26: x0 < r and A27: for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 by A19, A25; consider n being Element of NAT such that A28: for k being Element of NAT st n <= k holds s . k < r by A20, A21, A26, Th2; 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 A29: n <= k ; ::_thesis: abs (((f /* s) . k) - g) < g1 A30: s . k in rng s by VALUED_0:28; then s . k in right_open_halfline x0 by A22, XBOOLE_0:def_4; then s . k in { g2 where g2 is Real : x0 < g2 } by XXREAL_1:230; then A31: ex g2 being Real st ( g2 = s . k & x0 < g2 ) ; s . k in dom f by A22, A30, XBOOLE_0:def_4; then abs ((f . (s . k)) - g) < g1 by A27, A28, A29, A31; hence abs (((f /* s) . k) - g) < g1 by A22, A23, 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 A24, SEQ_2:def_7; ::_thesis: verum end; hence f is_right_convergent_in x0 by A18, Def4; ::_thesis: verum end; theorem :: LIMFUNC2:11 for x0 being Real for f being PartFunc of REAL,REAL holds ( f is_right_divergent_to+infty_in x0 iff ( ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) & ( for g1 being Real ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds g1 < f . r1 ) ) ) ) ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL holds ( f is_right_divergent_to+infty_in x0 iff ( ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) & ( for g1 being Real ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds g1 < f . r1 ) ) ) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_right_divergent_to+infty_in x0 iff ( ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) & ( for g1 being Real ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds g1 < f . r1 ) ) ) ) ) thus ( f is_right_divergent_to+infty_in x0 implies ( ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) & ( for g1 being Real ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds g1 < f . r1 ) ) ) ) ) ::_thesis: ( ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) & ( for g1 being Real ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds g1 < f . r1 ) ) ) implies f is_right_divergent_to+infty_in x0 ) proof assume that A1: f is_right_divergent_to+infty_in x0 and A2: ( ex r being Real st ( x0 < r & ( for g being Real holds ( not g < r or not x0 < g or not g in dom f ) ) ) or ex g1 being Real st for r being Real st x0 < r holds ex r1 being Real st ( r1 < r & x0 < r1 & r1 in dom f & f . r1 <= g1 ) ) ; ::_thesis: contradiction consider g1 being Real such that A3: for r being Real st x0 < r holds ex r1 being Real st ( r1 < r & x0 < r1 & r1 in dom f & f . r1 <= g1 ) by A1, A2, Def5; defpred S1[ Element of NAT , real number ] means ( x0 < \$2 & \$2 < x0 + (1 / (\$1 + 1)) & \$2 in dom f & f . \$2 <= g1 ); A4: now__::_thesis:_for_n_being_Element_of_NAT_ex_r1_being_Real_st_S1[n,r1] let n be Element of NAT ; ::_thesis: ex r1 being Real st S1[n,r1] x0 < x0 + (1 / (n + 1)) by Lm3; then consider r1 being Real such that A5: r1 < x0 + (1 / (n + 1)) and A6: x0 < r1 and A7: r1 in dom f and A8: f . r1 <= g1 by A3; take r1 = r1; ::_thesis: S1[n,r1] thus S1[n,r1] by A5, A6, A7, A8; ::_thesis: verum end; consider s being Real_Sequence such that A9: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A4); A10: rng s c= (dom f) /\ (right_open_halfline x0) by A9, Th6; A11: lim s = x0 by A9, Th6; s is convergent by A9, Th6; then f /* s is divergent_to+infty by A1, A11, A10, Def5; then consider n being Element of NAT such that A12: for k being Element of NAT st n <= k holds g1 < (f /* s) . k by LIMFUNC1:def_4; A13: g1 < (f /* s) . n by A12; rng s c= dom f by A9, Th6; then g1 < f . (s . n) by A13, FUNCT_2:108; hence contradiction by A9; ::_thesis: verum end; assume that A14: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) and A15: for g1 being Real ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds g1 < f . r1 ) ) ; ::_thesis: f is_right_divergent_to+infty_in x0 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 A16: s is convergent and A17: lim s = x0 and A18: rng s c= (dom f) /\ (right_open_halfline x0) ; ::_thesis: f /* s is divergent_to+infty A19: (dom f) /\ (right_open_halfline x0) c= dom f by XBOOLE_1:17; 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 r being Real such that A20: x0 < r and A21: for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds g1 < f . r1 by A15; consider n being Element of NAT such that A22: for k being Element of NAT st n <= k holds s . k < r by A16, A17, A20, Th2; 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 A23: n <= k ; ::_thesis: g1 < (f /* s) . k A24: s . k in rng s by VALUED_0:28; then s . k in right_open_halfline x0 by A18, XBOOLE_0:def_4; then s . k in { g2 where g2 is Real : x0 < g2 } by XXREAL_1:230; then A25: ex g2 being Real st ( g2 = s . k & x0 < g2 ) ; s . k in dom f by A18, A24, XBOOLE_0:def_4; then g1 < f . (s . k) by A21, A22, A23, A25; hence g1 < (f /* s) . k by A18, A19, 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_right_divergent_to+infty_in x0 by A14, Def5; ::_thesis: verum end; theorem :: LIMFUNC2:12 for x0 being Real for f being PartFunc of REAL,REAL holds ( f is_right_divergent_to-infty_in x0 iff ( ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) & ( for g1 being Real ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds f . r1 < g1 ) ) ) ) ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL holds ( f is_right_divergent_to-infty_in x0 iff ( ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) & ( for g1 being Real ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds f . r1 < g1 ) ) ) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_right_divergent_to-infty_in x0 iff ( ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) & ( for g1 being Real ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds f . r1 < g1 ) ) ) ) ) thus ( f is_right_divergent_to-infty_in x0 implies ( ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) & ( for g1 being Real ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds f . r1 < g1 ) ) ) ) ) ::_thesis: ( ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) & ( for g1 being Real ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds f . r1 < g1 ) ) ) implies f is_right_divergent_to-infty_in x0 ) proof assume that A1: f is_right_divergent_to-infty_in x0 and A2: ( ex r being Real st ( x0 < r & ( for g being Real holds ( not g < r or not x0 < g or not g in dom f ) ) ) or ex g1 being Real st for r being Real st x0 < r holds ex r1 being Real st ( r1 < r & x0 < r1 & r1 in dom f & g1 <= f . r1 ) ) ; ::_thesis: contradiction consider g1 being Real such that A3: for r being Real st x0 < r holds ex r1 being Real st ( r1 < r & x0 < r1 & r1 in dom f & g1 <= f . r1 ) by A1, A2, Def6; defpred S1[ Element of NAT , real number ] means ( x0 < \$2 & \$2 < x0 + (1 / (\$1 + 1)) & \$2 in dom f & g1 <= f . \$2 ); A4: now__::_thesis:_for_n_being_Element_of_NAT_ex_r1_being_Real_st_S1[n,r1] let n be Element of NAT ; ::_thesis: ex r1 being Real st S1[n,r1] x0 < x0 + (1 / (n + 1)) by Lm3; then consider r1 being Real such that A5: r1 < x0 + (1 / (n + 1)) and A6: x0 < r1 and A7: r1 in dom f and A8: g1 <= f . r1 by A3; take r1 = r1; ::_thesis: S1[n,r1] thus S1[n,r1] by A5, A6, A7, A8; ::_thesis: verum end; consider s being Real_Sequence such that A9: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A4); A10: rng s c= (dom f) /\ (right_open_halfline x0) by A9, Th6; A11: lim s = x0 by A9, Th6; s is convergent by A9, Th6; then f /* s is divergent_to-infty by A1, A11, A10, Def6; then consider n being Element of NAT such that A12: for k being Element of NAT st n <= k holds (f /* s) . k < g1 by LIMFUNC1:def_5; A13: (f /* s) . n < g1 by A12; rng s c= dom f by A9, Th6; then f . (s . n) < g1 by A13, FUNCT_2:108; hence contradiction by A9; ::_thesis: verum end; assume that A14: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) and A15: for g1 being Real ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds f . r1 < g1 ) ) ; ::_thesis: f is_right_divergent_to-infty_in x0 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 by A15, Lm5; hence f is_right_divergent_to-infty_in x0 by A14, Def6; ::_thesis: verum end; theorem :: LIMFUNC2:13 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_left_divergent_to+infty_in x0 & f2 is_left_divergent_to+infty_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in (dom f1) /\ (dom f2) ) ) holds ( f1 + f2 is_left_divergent_to+infty_in x0 & f1 (#) f2 is_left_divergent_to+infty_in x0 ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_left_divergent_to+infty_in x0 & f2 is_left_divergent_to+infty_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in (dom f1) /\ (dom f2) ) ) holds ( f1 + f2 is_left_divergent_to+infty_in x0 & f1 (#) f2 is_left_divergent_to+infty_in x0 ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_divergent_to+infty_in x0 & f2 is_left_divergent_to+infty_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in (dom f1) /\ (dom f2) ) ) implies ( f1 + f2 is_left_divergent_to+infty_in x0 & f1 (#) f2 is_left_divergent_to+infty_in x0 ) ) assume that A1: f1 is_left_divergent_to+infty_in x0 and A2: f2 is_left_divergent_to+infty_in x0 and A3: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in (dom f1) /\ (dom f2) ) ; ::_thesis: ( f1 + f2 is_left_divergent_to+infty_in x0 & f1 (#) f2 is_left_divergent_to+infty_in x0 ) A4: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(f1_+_f2))_/\_(left_open_halfline_x0)_holds_ (f1_+_f2)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f1 + f2)) /\ (left_open_halfline x0) implies (f1 + f2) /* seq is divergent_to+infty ) assume that A5: seq is convergent and A6: lim seq = x0 and A7: rng seq c= (dom (f1 + f2)) /\ (left_open_halfline x0) ; ::_thesis: (f1 + f2) /* seq is divergent_to+infty rng seq c= (dom f2) /\ (left_open_halfline x0) by A7, Lm4; then A8: f2 /* seq is divergent_to+infty by A2, A5, A6, Def2; rng seq c= (dom f1) /\ (left_open_halfline x0) by A7, Lm4; then f1 /* seq is divergent_to+infty by A1, A5, A6, Def2; then A9: (f1 /* seq) + (f2 /* seq) is divergent_to+infty by A8, LIMFUNC1:8; A10: dom (f1 + f2) = (dom f1) /\ (dom f2) by A7, Lm4; rng seq c= dom (f1 + f2) by A7, Lm4; hence (f1 + f2) /* seq is divergent_to+infty by A10, A9, RFUNCT_2:8; ::_thesis: verum end; A11: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(f1_(#)_f2))_/\_(left_open_halfline_x0)_holds_ (f1_(#)_f2)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f1 (#) f2)) /\ (left_open_halfline x0) implies (f1 (#) f2) /* seq is divergent_to+infty ) assume that A12: seq is convergent and A13: lim seq = x0 and A14: rng seq c= (dom (f1 (#) f2)) /\ (left_open_halfline x0) ; ::_thesis: (f1 (#) f2) /* seq is divergent_to+infty rng seq c= (dom f2) /\ (left_open_halfline x0) by A14, Lm2; then A15: f2 /* seq is divergent_to+infty by A2, A12, A13, Def2; rng seq c= (dom f1) /\ (left_open_halfline x0) by A14, Lm2; then f1 /* seq is divergent_to+infty by A1, A12, A13, Def2; then A16: (f1 /* seq) (#) (f2 /* seq) is divergent_to+infty by A15, LIMFUNC1:10; A17: dom (f1 (#) f2) = (dom f1) /\ (dom f2) by A14, Lm2; rng seq c= dom (f1 (#) f2) by A14, Lm2; hence (f1 (#) f2) /* seq is divergent_to+infty by A17, A16, RFUNCT_2:8; ::_thesis: verum end; now__::_thesis:_for_r_being_Real_st_r_<_x0_holds_ ex_g_being_Real_st_ (_r_<_g_&_g_<_x0_&_g_in_dom_(f1_+_f2)_) let r be Real; ::_thesis: ( r < x0 implies ex g being Real st ( r < g & g < x0 & g in dom (f1 + f2) ) ) assume r < x0 ; ::_thesis: ex g being Real st ( r < g & g < x0 & g in dom (f1 + f2) ) then consider g being Real such that A18: r < g and A19: g < x0 and A20: g in (dom f1) /\ (dom f2) by A3; take g = g; ::_thesis: ( r < g & g < x0 & g in dom (f1 + f2) ) thus ( r < g & g < x0 & g in dom (f1 + f2) ) by A18, A19, A20, VALUED_1:def_1; ::_thesis: verum end; hence f1 + f2 is_left_divergent_to+infty_in x0 by A4, Def2; ::_thesis: f1 (#) f2 is_left_divergent_to+infty_in x0 now__::_thesis:_for_r_being_Real_st_r_<_x0_holds_ ex_g_being_Real_st_ (_r_<_g_&_g_<_x0_&_g_in_dom_(f1_(#)_f2)_) let r be Real; ::_thesis: ( r < x0 implies ex g being Real st ( r < g & g < x0 & g in dom (f1 (#) f2) ) ) assume r < x0 ; ::_thesis: ex g being Real st ( r < g & g < x0 & g in dom (f1 (#) f2) ) then consider g being Real such that A21: r < g and A22: g < x0 and A23: g in (dom f1) /\ (dom f2) by A3; take g = g; ::_thesis: ( r < g & g < x0 & g in dom (f1 (#) f2) ) thus ( r < g & g < x0 & g in dom (f1 (#) f2) ) by A21, A22, A23, VALUED_1:def_4; ::_thesis: verum end; hence f1 (#) f2 is_left_divergent_to+infty_in x0 by A11, Def2; ::_thesis: verum end; theorem :: LIMFUNC2:14 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_left_divergent_to-infty_in x0 & f2 is_left_divergent_to-infty_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in (dom f1) /\ (dom f2) ) ) holds ( f1 + f2 is_left_divergent_to-infty_in x0 & f1 (#) f2 is_left_divergent_to+infty_in x0 ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_left_divergent_to-infty_in x0 & f2 is_left_divergent_to-infty_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in (dom f1) /\ (dom f2) ) ) holds ( f1 + f2 is_left_divergent_to-infty_in x0 & f1 (#) f2 is_left_divergent_to+infty_in x0 ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_divergent_to-infty_in x0 & f2 is_left_divergent_to-infty_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in (dom f1) /\ (dom f2) ) ) implies ( f1 + f2 is_left_divergent_to-infty_in x0 & f1 (#) f2 is_left_divergent_to+infty_in x0 ) ) assume that A1: f1 is_left_divergent_to-infty_in x0 and A2: f2 is_left_divergent_to-infty_in x0 and A3: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in (dom f1) /\ (dom f2) ) ; ::_thesis: ( f1 + f2 is_left_divergent_to-infty_in x0 & f1 (#) f2 is_left_divergent_to+infty_in x0 ) A4: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(f1_+_f2))_/\_(left_open_halfline_x0)_holds_ (f1_+_f2)_/*_seq_is_divergent_to-infty let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f1 + f2)) /\ (left_open_halfline x0) implies (f1 + f2) /* seq is divergent_to-infty ) assume that A5: seq is convergent and A6: lim seq = x0 and A7: rng seq c= (dom (f1 + f2)) /\ (left_open_halfline x0) ; ::_thesis: (f1 + f2) /* seq is divergent_to-infty rng seq c= (dom f2) /\ (left_open_halfline x0) by A7, Lm4; then A8: f2 /* seq is divergent_to-infty by A2, A5, A6, Def3; rng seq c= (dom f1) /\ (left_open_halfline x0) by A7, Lm4; then f1 /* seq is divergent_to-infty by A1, A5, A6, Def3; then A9: (f1 /* seq) + (f2 /* seq) is divergent_to-infty by A8, LIMFUNC1:11; A10: dom (f1 + f2) = (dom f1) /\ (dom f2) by A7, Lm4; rng seq c= dom (f1 + f2) by A7, Lm4; hence (f1 + f2) /* seq is divergent_to-infty by A10, A9, RFUNCT_2:8; ::_thesis: verum end; A11: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(f1_(#)_f2))_/\_(left_open_halfline_x0)_holds_ (f1_(#)_f2)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f1 (#) f2)) /\ (left_open_halfline x0) implies (f1 (#) f2) /* seq is divergent_to+infty ) assume that A12: seq is convergent and A13: lim seq = x0 and A14: rng seq c= (dom (f1 (#) f2)) /\ (left_open_halfline x0) ; ::_thesis: (f1 (#) f2) /* seq is divergent_to+infty rng seq c= (dom f2) /\ (left_open_halfline x0) by A14, Lm2; then A15: f2 /* seq is divergent_to-infty by A2, A12, A13, Def3; rng seq c= (dom f1) /\ (left_open_halfline x0) by A14, Lm2; then f1 /* seq is divergent_to-infty by A1, A12, A13, Def3; then A16: (f1 /* seq) (#) (f2 /* seq) is divergent_to+infty by A15, LIMFUNC1:24; A17: dom (f1 (#) f2) = (dom f1) /\ (dom f2) by A14, Lm2; rng seq c= dom (f1 (#) f2) by A14, Lm2; hence (f1 (#) f2) /* seq is divergent_to+infty by A17, A16, RFUNCT_2:8; ::_thesis: verum end; now__::_thesis:_for_r_being_Real_st_r_<_x0_holds_ ex_g_being_Real_st_ (_r_<_g_&_g_<_x0_&_g_in_dom_(f1_+_f2)_) let r be Real; ::_thesis: ( r < x0 implies ex g being Real st ( r < g & g < x0 & g in dom (f1 + f2) ) ) assume r < x0 ; ::_thesis: ex g being Real st ( r < g & g < x0 & g in dom (f1 + f2) ) then consider g being Real such that A18: r < g and A19: g < x0 and A20: g in (dom f1) /\ (dom f2) by A3; take g = g; ::_thesis: ( r < g & g < x0 & g in dom (f1 + f2) ) thus ( r < g & g < x0 & g in dom (f1 + f2) ) by A18, A19, A20, VALUED_1:def_1; ::_thesis: verum end; hence f1 + f2 is_left_divergent_to-infty_in x0 by A4, Def3; ::_thesis: f1 (#) f2 is_left_divergent_to+infty_in x0 now__::_thesis:_for_r_being_Real_st_r_<_x0_holds_ ex_g_being_Real_st_ (_r_<_g_&_g_<_x0_&_g_in_dom_(f1_(#)_f2)_) let r be Real; ::_thesis: ( r < x0 implies ex g being Real st ( r < g & g < x0 & g in dom (f1 (#) f2) ) ) assume r < x0 ; ::_thesis: ex g being Real st ( r < g & g < x0 & g in dom (f1 (#) f2) ) then consider g being Real such that A21: r < g and A22: g < x0 and A23: g in (dom f1) /\ (dom f2) by A3; take g = g; ::_thesis: ( r < g & g < x0 & g in dom (f1 (#) f2) ) thus ( r < g & g < x0 & g in dom (f1 (#) f2) ) by A21, A22, A23, VALUED_1:def_4; ::_thesis: verum end; hence f1 (#) f2 is_left_divergent_to+infty_in x0 by A11, Def2; ::_thesis: verum end; theorem :: LIMFUNC2:15 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_right_divergent_to+infty_in x0 & f2 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 f1) /\ (dom f2) ) ) holds ( f1 + f2 is_right_divergent_to+infty_in x0 & f1 (#) f2 is_right_divergent_to+infty_in x0 ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_right_divergent_to+infty_in x0 & f2 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 f1) /\ (dom f2) ) ) holds ( f1 + f2 is_right_divergent_to+infty_in x0 & f1 (#) f2 is_right_divergent_to+infty_in x0 ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_right_divergent_to+infty_in x0 & f2 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 f1) /\ (dom f2) ) ) implies ( f1 + f2 is_right_divergent_to+infty_in x0 & f1 (#) f2 is_right_divergent_to+infty_in x0 ) ) assume that A1: f1 is_right_divergent_to+infty_in x0 and A2: f2 is_right_divergent_to+infty_in x0 and A3: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in (dom f1) /\ (dom f2) ) ; ::_thesis: ( f1 + f2 is_right_divergent_to+infty_in x0 & f1 (#) f2 is_right_divergent_to+infty_in x0 ) A4: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(f1_+_f2))_/\_(right_open_halfline_x0)_holds_ (f1_+_f2)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f1 + f2)) /\ (right_open_halfline x0) implies (f1 + f2) /* seq is divergent_to+infty ) assume that A5: seq is convergent and A6: lim seq = x0 and A7: rng seq c= (dom (f1 + f2)) /\ (right_open_halfline x0) ; ::_thesis: (f1 + f2) /* seq is divergent_to+infty rng seq c= (dom f2) /\ (right_open_halfline x0) by A7, Lm4; then A8: f2 /* seq is divergent_to+infty by A2, A5, A6, Def5; rng seq c= (dom f1) /\ (right_open_halfline x0) by A7, Lm4; then f1 /* seq is divergent_to+infty by A1, A5, A6, Def5; then A9: (f1 /* seq) + (f2 /* seq) is divergent_to+infty by A8, LIMFUNC1:8; A10: dom (f1 + f2) = (dom f1) /\ (dom f2) by A7, Lm4; rng seq c= dom (f1 + f2) by A7, Lm4; hence (f1 + f2) /* seq is divergent_to+infty by A10, A9, RFUNCT_2:8; ::_thesis: verum end; A11: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(f1_(#)_f2))_/\_(right_open_halfline_x0)_holds_ (f1_(#)_f2)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f1 (#) f2)) /\ (right_open_halfline x0) implies (f1 (#) f2) /* seq is divergent_to+infty ) assume that A12: seq is convergent and A13: lim seq = x0 and A14: rng seq c= (dom (f1 (#) f2)) /\ (right_open_halfline x0) ; ::_thesis: (f1 (#) f2) /* seq is divergent_to+infty rng seq c= (dom f2) /\ (right_open_halfline x0) by A14, Lm2; then A15: f2 /* seq is divergent_to+infty by A2, A12, A13, Def5; rng seq c= (dom f1) /\ (right_open_halfline x0) by A14, Lm2; then f1 /* seq is divergent_to+infty by A1, A12, A13, Def5; then A16: (f1 /* seq) (#) (f2 /* seq) is divergent_to+infty by A15, LIMFUNC1:10; A17: dom (f1 (#) f2) = (dom f1) /\ (dom f2) by A14, Lm2; rng seq c= dom (f1 (#) f2) by A14, Lm2; hence (f1 (#) f2) /* seq is divergent_to+infty by A17, A16, RFUNCT_2:8; ::_thesis: verum end; now__::_thesis:_for_r_being_Real_st_x0_<_r_holds_ ex_g_being_Real_st_ (_g_<_r_&_x0_<_g_&_g_in_dom_(f1_+_f2)_) let r be Real; ::_thesis: ( x0 < r implies ex g being Real st ( g < r & x0 < g & g in dom (f1 + f2) ) ) assume x0 < r ; ::_thesis: ex g being Real st ( g < r & x0 < g & g in dom (f1 + f2) ) then consider g being Real such that A18: g < r and A19: x0 < g and A20: g in (dom f1) /\ (dom f2) by A3; take g = g; ::_thesis: ( g < r & x0 < g & g in dom (f1 + f2) ) thus ( g < r & x0 < g & g in dom (f1 + f2) ) by A18, A19, A20, VALUED_1:def_1; ::_thesis: verum end; hence f1 + f2 is_right_divergent_to+infty_in x0 by A4, Def5; ::_thesis: f1 (#) f2 is_right_divergent_to+infty_in x0 now__::_thesis:_for_r_being_Real_st_x0_<_r_holds_ ex_g_being_Real_st_ (_g_<_r_&_x0_<_g_&_g_in_dom_(f1_(#)_f2)_) let r be Real; ::_thesis: ( x0 < r implies ex g being Real st ( g < r & x0 < g & g in dom (f1 (#) f2) ) ) assume x0 < r ; ::_thesis: ex g being Real st ( g < r & x0 < g & g in dom (f1 (#) f2) ) then consider g being Real such that A21: g < r and A22: x0 < g and A23: g in (dom f1) /\ (dom f2) by A3; take g = g; ::_thesis: ( g < r & x0 < g & g in dom (f1 (#) f2) ) thus ( g < r & x0 < g & g in dom (f1 (#) f2) ) by A21, A22, A23, VALUED_1:def_4; ::_thesis: verum end; hence f1 (#) f2 is_right_divergent_to+infty_in x0 by A11, Def5; ::_thesis: verum end; theorem :: LIMFUNC2:16 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_right_divergent_to-infty_in x0 & f2 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 f1) /\ (dom f2) ) ) holds ( f1 + f2 is_right_divergent_to-infty_in x0 & f1 (#) f2 is_right_divergent_to+infty_in x0 ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_right_divergent_to-infty_in x0 & f2 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 f1) /\ (dom f2) ) ) holds ( f1 + f2 is_right_divergent_to-infty_in x0 & f1 (#) f2 is_right_divergent_to+infty_in x0 ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_right_divergent_to-infty_in x0 & f2 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 f1) /\ (dom f2) ) ) implies ( f1 + f2 is_right_divergent_to-infty_in x0 & f1 (#) f2 is_right_divergent_to+infty_in x0 ) ) assume that A1: f1 is_right_divergent_to-infty_in x0 and A2: f2 is_right_divergent_to-infty_in x0 and A3: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in (dom f1) /\ (dom f2) ) ; ::_thesis: ( f1 + f2 is_right_divergent_to-infty_in x0 & f1 (#) f2 is_right_divergent_to+infty_in x0 ) A4: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(f1_+_f2))_/\_(right_open_halfline_x0)_holds_ (f1_+_f2)_/*_seq_is_divergent_to-infty let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f1 + f2)) /\ (right_open_halfline x0) implies (f1 + f2) /* seq is divergent_to-infty ) assume that A5: seq is convergent and A6: lim seq = x0 and A7: rng seq c= (dom (f1 + f2)) /\ (right_open_halfline x0) ; ::_thesis: (f1 + f2) /* seq is divergent_to-infty rng seq c= (dom f2) /\ (right_open_halfline x0) by A7, Lm4; then A8: f2 /* seq is divergent_to-infty by A2, A5, A6, Def6; rng seq c= (dom f1) /\ (right_open_halfline x0) by A7, Lm4; then f1 /* seq is divergent_to-infty by A1, A5, A6, Def6; then A9: (f1 /* seq) + (f2 /* seq) is divergent_to-infty by A8, LIMFUNC1:11; A10: dom (f1 + f2) = (dom f1) /\ (dom f2) by A7, Lm4; rng seq c= dom (f1 + f2) by A7, Lm4; hence (f1 + f2) /* seq is divergent_to-infty by A10, A9, RFUNCT_2:8; ::_thesis: verum end; A11: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(f1_(#)_f2))_/\_(right_open_halfline_x0)_holds_ (f1_(#)_f2)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f1 (#) f2)) /\ (right_open_halfline x0) implies (f1 (#) f2) /* seq is divergent_to+infty ) assume that A12: seq is convergent and A13: lim seq = x0 and A14: rng seq c= (dom (f1 (#) f2)) /\ (right_open_halfline x0) ; ::_thesis: (f1 (#) f2) /* seq is divergent_to+infty rng seq c= (dom f2) /\ (right_open_halfline x0) by A14, Lm2; then A15: f2 /* seq is divergent_to-infty by A2, A12, A13, Def6; rng seq c= (dom f1) /\ (right_open_halfline x0) by A14, Lm2; then f1 /* seq is divergent_to-infty by A1, A12, A13, Def6; then A16: (f1 /* seq) (#) (f2 /* seq) is divergent_to+infty by A15, LIMFUNC1:24; A17: dom (f1 (#) f2) = (dom f1) /\ (dom f2) by A14, Lm2; rng seq c= dom (f1 (#) f2) by A14, Lm2; hence (f1 (#) f2) /* seq is divergent_to+infty by A17, A16, RFUNCT_2:8; ::_thesis: verum end; now__::_thesis:_for_r_being_Real_st_x0_<_r_holds_ ex_g_being_Real_st_ (_g_<_r_&_x0_<_g_&_g_in_dom_(f1_+_f2)_) let r be Real; ::_thesis: ( x0 < r implies ex g being Real st ( g < r & x0 < g & g in dom (f1 + f2) ) ) assume x0 < r ; ::_thesis: ex g being Real st ( g < r & x0 < g & g in dom (f1 + f2) ) then consider g being Real such that A18: g < r and A19: x0 < g and A20: g in (dom f1) /\ (dom f2) by A3; take g = g; ::_thesis: ( g < r & x0 < g & g in dom (f1 + f2) ) thus ( g < r & x0 < g & g in dom (f1 + f2) ) by A18, A19, A20, VALUED_1:def_1; ::_thesis: verum end; hence f1 + f2 is_right_divergent_to-infty_in x0 by A4, Def6; ::_thesis: f1 (#) f2 is_right_divergent_to+infty_in x0 now__::_thesis:_for_r_being_Real_st_x0_<_r_holds_ ex_g_being_Real_st_ (_g_<_r_&_x0_<_g_&_g_in_dom_(f1_(#)_f2)_) let r be Real; ::_thesis: ( x0 < r implies ex g being Real st ( g < r & x0 < g & g in dom (f1 (#) f2) ) ) assume x0 < r ; ::_thesis: ex g being Real st ( g < r & x0 < g & g in dom (f1 (#) f2) ) then consider g being Real such that A21: g < r and A22: x0 < g and A23: g in (dom f1) /\ (dom f2) by A3; take g = g; ::_thesis: ( g < r & x0 < g & g in dom (f1 (#) f2) ) thus ( g < r & x0 < g & g in dom (f1 (#) f2) ) by A21, A22, A23, VALUED_1:def_4; ::_thesis: verum end; hence f1 (#) f2 is_right_divergent_to+infty_in x0 by A11, Def5; ::_thesis: verum end; theorem :: LIMFUNC2:17 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_left_divergent_to+infty_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f1 + f2) ) ) & ex r being Real st ( 0 < r & f2 | ].(x0 - r),x0.[ is bounded_below ) holds f1 + f2 is_left_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_left_divergent_to+infty_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f1 + f2) ) ) & ex r being Real st ( 0 < r & f2 | ].(x0 - r),x0.[ is bounded_below ) holds f1 + f2 is_left_divergent_to+infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_divergent_to+infty_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f1 + f2) ) ) & ex r being Real st ( 0 < r & f2 | ].(x0 - r),x0.[ is bounded_below ) implies f1 + f2 is_left_divergent_to+infty_in x0 ) assume that A1: f1 is_left_divergent_to+infty_in x0 and A2: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f1 + f2) ) ; ::_thesis: ( for r being Real holds ( not 0 < r or not f2 | ].(x0 - r),x0.[ is bounded_below ) or f1 + f2 is_left_divergent_to+infty_in x0 ) given r being Real such that A3: 0 < r and A4: f2 | ].(x0 - r),x0.[ is bounded_below ; ::_thesis: f1 + f2 is_left_divergent_to+infty_in x0 now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(f1_+_f2))_/\_(left_open_halfline_x0)_holds_ (f1_+_f2)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f1 + f2)) /\ (left_open_halfline x0) implies (f1 + f2) /* seq is divergent_to+infty ) assume that A5: seq is convergent and A6: lim seq = x0 and A7: rng seq c= (dom (f1 + f2)) /\ (left_open_halfline x0) ; ::_thesis: (f1 + f2) /* seq is divergent_to+infty x0 - r < x0 by A3, Lm1; then consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds x0 - r < seq . n by A5, A6, Th1; A9: (dom (f1 + f2)) /\ (left_open_halfline x0) c= dom (f1 + f2) by XBOOLE_1:17; rng (seq ^\ k) c= rng seq by VALUED_0:21; then A10: rng (seq ^\ k) c= (dom (f1 + f2)) /\ (left_open_halfline x0) by A7, XBOOLE_1:1; then A11: rng (seq ^\ k) c= dom (f1 + f2) by A9, XBOOLE_1:1; A12: dom (f1 + f2) = (dom f1) /\ (dom f2) by VALUED_1:def_1; then A13: dom (f1 + f2) c= dom f2 by XBOOLE_1:17; then A14: rng (seq ^\ k) c= dom f2 by A11, XBOOLE_1:1; dom (f1 + f2) c= dom f1 by A12, XBOOLE_1:17; then A15: rng (seq ^\ k) c= dom f1 by A11, XBOOLE_1:1; then rng (seq ^\ k) c= (dom f1) /\ (dom f2) by A14, XBOOLE_1:19; then A16: (f1 /* (seq ^\ k)) + (f2 /* (seq ^\ k)) = (f1 + f2) /* (seq ^\ k) by RFUNCT_2:8 .= ((f1 + f2) /* seq) ^\ k by A7, A9, VALUED_0:27, XBOOLE_1:1 ; (dom (f1 + f2)) /\ (left_open_halfline x0) c= left_open_halfline x0 by XBOOLE_1:17; then A17: rng (seq ^\ k) c= left_open_halfline x0 by A10, XBOOLE_1:1; then A18: rng (seq ^\ k) c= (dom f1) /\ (left_open_halfline x0) by A15, XBOOLE_1:19; now__::_thesis:_ex_r2_being_Element_of_REAL_st_ for_n_being_Element_of_NAT_holds_r2_<_(f2_/*_(seq_^\_k))_._n consider r1 being real number such that A19: for g being set st g in ].(x0 - r),x0.[ /\ (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 /* (seq ^\ k)) . n let n be Element of NAT ; ::_thesis: r2 < (f2 /* (seq ^\ k)) . n x0 - r < seq . (n + k) by A8, NAT_1:12; then A20: x0 - r < (seq ^\ k) . n by NAT_1:def_3; A21: (seq ^\ k) . n in rng (seq ^\ k) by VALUED_0:28; then (seq ^\ k) . n in left_open_halfline x0 by A17; then (seq ^\ k) . n in { g1 where g1 is Real : g1 < x0 } by XXREAL_1:229; then ex g being Real st ( g = (seq ^\ k) . n & g < x0 ) ; then (seq ^\ k) . n in { g2 where g2 is Real : ( x0 - r < g2 & g2 < x0 ) } by A20; then (seq ^\ k) . n in ].(x0 - r),x0.[ by RCOMP_1:def_2; then (seq ^\ k) . n in ].(x0 - r),x0.[ /\ (dom f2) by A14, A21, XBOOLE_0:def_4; then r1 - 1 < (f2 . ((seq ^\ k) . n)) - 0 by A19, XREAL_1:15; hence r2 < (f2 /* (seq ^\ k)) . n by A11, A13, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; then A22: f2 /* (seq ^\ k) is bounded_below by SEQ_2:def_4; lim (seq ^\ k) = x0 by A5, A6, SEQ_4:20; then f1 /* (seq ^\ k) is divergent_to+infty by A1, A5, A18, Def2; then (f1 /* (seq ^\ k)) + (f2 /* (seq ^\ k)) is divergent_to+infty by A22, LIMFUNC1:9; hence (f1 + f2) /* seq is divergent_to+infty by A16, LIMFUNC1:7; ::_thesis: verum end; hence f1 + f2 is_left_divergent_to+infty_in x0 by A2, Def2; ::_thesis: verum end; theorem :: LIMFUNC2:18 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_left_divergent_to+infty_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g 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.[ holds r1 <= f2 . g ) ) holds f1 (#) f2 is_left_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_left_divergent_to+infty_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g 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.[ holds r1 <= f2 . g ) ) holds f1 (#) f2 is_left_divergent_to+infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_divergent_to+infty_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g 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.[ holds r1 <= f2 . g ) ) implies f1 (#) f2 is_left_divergent_to+infty_in x0 ) assume that A1: f1 is_left_divergent_to+infty_in x0 and A2: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g 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.[ & not r1 <= f2 . g ) ) or f1 (#) f2 is_left_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.[ holds t <= f2 . g ; ::_thesis: f1 (#) f2 is_left_divergent_to+infty_in x0 now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(f1_(#)_f2))_/\_(left_open_halfline_x0)_holds_ (f1_(#)_f2)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f1 (#) f2)) /\ (left_open_halfline x0) implies (f1 (#) f2) /* seq is divergent_to+infty ) assume that A6: seq is convergent and A7: lim seq = x0 and A8: rng seq c= (dom (f1 (#) f2)) /\ (left_open_halfline x0) ; ::_thesis: (f1 (#) f2) /* seq is divergent_to+infty x0 - r < x0 by A3, Lm1; then consider k being Element of NAT such that A9: for n being Element of NAT st k <= n holds x0 - r < seq . n by A6, A7, Th1; A10: rng seq c= dom (f1 (#) f2) by A8, Lm2; A11: dom (f1 (#) f2) = (dom f1) /\ (dom f2) by A8, Lm2; rng (seq ^\ k) c= rng seq by VALUED_0:21; then A12: rng (seq ^\ k) c= (dom (f1 (#) f2)) /\ (left_open_halfline x0) by A8, XBOOLE_1:1; then A13: rng (seq ^\ k) c= dom f2 by Lm2; A14: rng (seq ^\ k) c= left_open_halfline x0 by A12, Lm2; A15: now__::_thesis:_(_0_<_t_&_(_for_n_being_Element_of_NAT_holds_t_<=_(f2_/*_(seq_^\_k))_._n_)_) thus 0 < t by A4; ::_thesis: for n being Element of NAT holds t <= (f2 /* (seq ^\ k)) . n let n be Element of NAT ; ::_thesis: t <= (f2 /* (seq ^\ k)) . n x0 - r < seq . (n + k) by A9, NAT_1:12; then A16: x0 - r < (seq ^\ k) . n by NAT_1:def_3; A17: (seq ^\ k) . n in rng (seq ^\ k) by VALUED_0:28; then (seq ^\ k) . n in left_open_halfline x0 by A14; then (seq ^\ k) . n in { g1 where g1 is Real : g1 < x0 } by XXREAL_1:229; then ex g being Real st ( g = (seq ^\ k) . n & g < x0 ) ; then (seq ^\ k) . n in { g2 where g2 is Real : ( x0 - r < g2 & g2 < x0 ) } by A16; then (seq ^\ k) . n in ].(x0 - r),x0.[ by RCOMP_1:def_2; then (seq ^\ k) . n in (dom f2) /\ ].(x0 - r),x0.[ by A13, A17, XBOOLE_0:def_4; then t <= f2 . ((seq ^\ k) . n) by A5; hence t <= (f2 /* (seq ^\ k)) . n by A13, FUNCT_2:108; ::_thesis: verum end; A18: rng (seq ^\ k) c= (dom f1) /\ (left_open_halfline x0) by A12, Lm2; lim (seq ^\ k) = x0 by A6, A7, SEQ_4:20; then f1 /* (seq ^\ k) is divergent_to+infty by A1, A6, A18, Def2; then A19: (f1 /* (seq ^\ k)) (#) (f2 /* (seq ^\ k)) is divergent_to+infty by A15, LIMFUNC1:22; rng (seq ^\ k) c= dom (f1 (#) f2) by A12, Lm2; then (f1 /* (seq ^\ k)) (#) (f2 /* (seq ^\ k)) = (f1 (#) f2) /* (seq ^\ k) by A11, RFUNCT_2:8 .= ((f1 (#) f2) /* seq) ^\ k by A10, VALUED_0:27 ; hence (f1 (#) f2) /* seq is divergent_to+infty by A19, LIMFUNC1:7; ::_thesis: verum end; hence f1 (#) f2 is_left_divergent_to+infty_in x0 by A2, Def2; ::_thesis: verum end; theorem :: LIMFUNC2:19 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 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 (f1 + f2) ) ) & ex r being Real st ( 0 < r & f2 | ].x0,(x0 + r).[ is bounded_below ) holds f1 + f2 is_right_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 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 (f1 + f2) ) ) & ex r being Real st ( 0 < r & f2 | ].x0,(x0 + r).[ is bounded_below ) holds f1 + f2 is_right_divergent_to+infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 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 (f1 + f2) ) ) & ex r being Real st ( 0 < r & f2 | ].x0,(x0 + r).[ is bounded_below ) implies f1 + f2 is_right_divergent_to+infty_in x0 ) assume that A1: f1 is_right_divergent_to+infty_in x0 and A2: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f1 + f2) ) ; ::_thesis: ( for r being Real holds ( not 0 < r or not f2 | ].x0,(x0 + r).[ is bounded_below ) or f1 + f2 is_right_divergent_to+infty_in x0 ) given r being Real such that A3: 0 < r and A4: f2 | ].x0,(x0 + r).[ is bounded_below ; ::_thesis: f1 + f2 is_right_divergent_to+infty_in x0 now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(f1_+_f2))_/\_(right_open_halfline_x0)_holds_ (f1_+_f2)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f1 + f2)) /\ (right_open_halfline x0) implies (f1 + f2) /* seq is divergent_to+infty ) assume that A5: seq is convergent and A6: lim seq = x0 and A7: rng seq c= (dom (f1 + f2)) /\ (right_open_halfline x0) ; ::_thesis: (f1 + f2) /* seq is divergent_to+infty x0 < x0 + r by A3, Lm1; then consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds seq . n < x0 + r by A5, A6, Th2; A9: (dom (f1 + f2)) /\ (right_open_halfline x0) c= dom (f1 + f2) by XBOOLE_1:17; rng (seq ^\ k) c= rng seq by VALUED_0:21; then A10: rng (seq ^\ k) c= (dom (f1 + f2)) /\ (right_open_halfline x0) by A7, XBOOLE_1:1; then A11: rng (seq ^\ k) c= dom (f1 + f2) by A9, XBOOLE_1:1; A12: dom (f1 + f2) = (dom f1) /\ (dom f2) by VALUED_1:def_1; then A13: dom (f1 + f2) c= dom f2 by XBOOLE_1:17; then A14: rng (seq ^\ k) c= dom f2 by A11, XBOOLE_1:1; dom (f1 + f2) c= dom f1 by A12, XBOOLE_1:17; then A15: rng (seq ^\ k) c= dom f1 by A11, XBOOLE_1:1; then rng (seq ^\ k) c= (dom f1) /\ (dom f2) by A14, XBOOLE_1:19; then A16: (f1 /* (seq ^\ k)) + (f2 /* (seq ^\ k)) = (f1 + f2) /* (seq ^\ k) by RFUNCT_2:8 .= ((f1 + f2) /* seq) ^\ k by A7, A9, VALUED_0:27, XBOOLE_1:1 ; (dom (f1 + f2)) /\ (right_open_halfline x0) c= right_open_halfline x0 by XBOOLE_1:17; then A17: rng (seq ^\ k) c= right_open_halfline x0 by A10, XBOOLE_1:1; then A18: rng (seq ^\ k) c= (dom f1) /\ (right_open_halfline x0) by A15, XBOOLE_1:19; now__::_thesis:_ex_r2_being_Element_of_REAL_st_ for_n_being_Element_of_NAT_holds_r2_<_(f2_/*_(seq_^\_k))_._n consider r1 being real number such that A19: for g being set st g in ].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 /* (seq ^\ k)) . n let n be Element of NAT ; ::_thesis: r2 < (f2 /* (seq ^\ k)) . n seq . (n + k) < x0 + r by A8, NAT_1:12; then A20: (seq ^\ k) . n < x0 + r by NAT_1:def_3; A21: (seq ^\ k) . n in rng (seq ^\ k) by VALUED_0:28; then (seq ^\ k) . n in right_open_halfline x0 by A17; then (seq ^\ k) . n in { g1 where g1 is Real : x0 < g1 } by XXREAL_1:230; then ex g being Real st ( g = (seq ^\ k) . n & x0 < g ) ; then (seq ^\ k) . n in { g2 where g2 is Real : ( x0 < g2 & g2 < x0 + r ) } by A20; then (seq ^\ k) . n in ].x0,(x0 + r).[ by RCOMP_1:def_2; then (seq ^\ k) . n in ].x0,(x0 + r).[ /\ (dom f2) by A14, A21, XBOOLE_0:def_4; then r1 - 1 < (f2 . ((seq ^\ k) . n)) - 0 by A19, XREAL_1:15; hence r2 < (f2 /* (seq ^\ k)) . n by A11, A13, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; then A22: f2 /* (seq ^\ k) is bounded_below by SEQ_2:def_4; lim (seq ^\ k) = x0 by A5, A6, SEQ_4:20; then f1 /* (seq ^\ k) is divergent_to+infty by A1, A5, A18, Def5; then (f1 /* (seq ^\ k)) + (f2 /* (seq ^\ k)) is divergent_to+infty by A22, LIMFUNC1:9; hence (f1 + f2) /* seq is divergent_to+infty by A16, LIMFUNC1:7; ::_thesis: verum end; hence f1 + f2 is_right_divergent_to+infty_in x0 by A2, Def5; ::_thesis: verum end; theorem :: LIMFUNC2:20 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 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 (f1 (#) f2) ) ) & ex r, r1 being Real st ( 0 < r & 0 < r1 & ( for g being Real st g in (dom f2) /\ ].x0,(x0 + r).[ holds r1 <= f2 . g ) ) holds f1 (#) f2 is_right_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 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 (f1 (#) f2) ) ) & ex r, r1 being Real st ( 0 < r & 0 < r1 & ( for g being Real st g in (dom f2) /\ ].x0,(x0 + r).[ holds r1 <= f2 . g ) ) holds f1 (#) f2 is_right_divergent_to+infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 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 (f1 (#) f2) ) ) & ex r, r1 being Real st ( 0 < r & 0 < r1 & ( for g being Real st g in (dom f2) /\ ].x0,(x0 + r).[ holds r1 <= f2 . g ) ) implies f1 (#) f2 is_right_divergent_to+infty_in x0 ) assume that A1: f1 is_right_divergent_to+infty_in x0 and A2: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g 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,(x0 + r).[ & not r1 <= f2 . g ) ) or f1 (#) f2 is_right_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,(x0 + r).[ holds t <= f2 . g ; ::_thesis: f1 (#) f2 is_right_divergent_to+infty_in x0 now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(f1_(#)_f2))_/\_(right_open_halfline_x0)_holds_ (f1_(#)_f2)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f1 (#) f2)) /\ (right_open_halfline x0) implies (f1 (#) f2) /* seq is divergent_to+infty ) assume that A6: seq is convergent and A7: lim seq = x0 and A8: rng seq c= (dom (f1 (#) f2)) /\ (right_open_halfline x0) ; ::_thesis: (f1 (#) f2) /* seq is divergent_to+infty x0 < x0 + r by A3, Lm1; then consider k being Element of NAT such that A9: for n being Element of NAT st k <= n holds seq . n < x0 + r by A6, A7, Th2; A10: rng seq c= dom (f1 (#) f2) by A8, Lm2; A11: dom (f1 (#) f2) = (dom f1) /\ (dom f2) by A8, Lm2; rng (seq ^\ k) c= rng seq by VALUED_0:21; then A12: rng (seq ^\ k) c= (dom (f1 (#) f2)) /\ (right_open_halfline x0) by A8, XBOOLE_1:1; then A13: rng (seq ^\ k) c= dom f2 by Lm2; A14: rng (seq ^\ k) c= right_open_halfline x0 by A12, Lm2; A15: now__::_thesis:_(_0_<_t_&_(_for_n_being_Element_of_NAT_holds_t_<=_(f2_/*_(seq_^\_k))_._n_)_) thus 0 < t by A4; ::_thesis: for n being Element of NAT holds t <= (f2 /* (seq ^\ k)) . n let n be Element of NAT ; ::_thesis: t <= (f2 /* (seq ^\ k)) . n seq . (n + k) < x0 + r by A9, NAT_1:12; then A16: (seq ^\ k) . n < x0 + r by NAT_1:def_3; A17: (seq ^\ k) . n in rng (seq ^\ k) by VALUED_0:28; then (seq ^\ k) . n in right_open_halfline x0 by A14; then (seq ^\ k) . n in { g1 where g1 is Real : x0 < g1 } by XXREAL_1:230; then ex g being Real st ( g = (seq ^\ k) . n & x0 < g ) ; then (seq ^\ k) . n in { g2 where g2 is Real : ( x0 < g2 & g2 < x0 + r ) } by A16; then (seq ^\ k) . n in ].x0,(x0 + r).[ by RCOMP_1:def_2; then (seq ^\ k) . n in (dom f2) /\ ].x0,(x0 + r).[ by A13, A17, XBOOLE_0:def_4; then t <= f2 . ((seq ^\ k) . n) by A5; hence t <= (f2 /* (seq ^\ k)) . n by A13, FUNCT_2:108; ::_thesis: verum end; A18: rng (seq ^\ k) c= (dom f1) /\ (right_open_halfline x0) by A12, Lm2; lim (seq ^\ k) = x0 by A6, A7, SEQ_4:20; then f1 /* (seq ^\ k) is divergent_to+infty by A1, A6, A18, Def5; then A19: (f1 /* (seq ^\ k)) (#) (f2 /* (seq ^\ k)) is divergent_to+infty by A15, LIMFUNC1:22; rng (seq ^\ k) c= dom (f1 (#) f2) by A12, Lm2; then (f1 /* (seq ^\ k)) (#) (f2 /* (seq ^\ k)) = (f1 (#) f2) /* (seq ^\ k) by A11, RFUNCT_2:8 .= ((f1 (#) f2) /* seq) ^\ k by A10, VALUED_0:27 ; hence (f1 (#) f2) /* seq is divergent_to+infty by A19, LIMFUNC1:7; ::_thesis: verum end; hence f1 (#) f2 is_right_divergent_to+infty_in x0 by A2, Def5; ::_thesis: verum end; theorem :: LIMFUNC2:21 for x0, r being Real for f being PartFunc of REAL,REAL holds ( ( f is_left_divergent_to+infty_in x0 & r > 0 implies r (#) f is_left_divergent_to+infty_in x0 ) & ( f is_left_divergent_to+infty_in x0 & r < 0 implies r (#) f is_left_divergent_to-infty_in x0 ) & ( f is_left_divergent_to-infty_in x0 & r > 0 implies r (#) f is_left_divergent_to-infty_in x0 ) & ( f is_left_divergent_to-infty_in x0 & r < 0 implies r (#) f is_left_divergent_to+infty_in x0 ) ) proof let x0, r be Real; ::_thesis: for f being PartFunc of REAL,REAL holds ( ( f is_left_divergent_to+infty_in x0 & r > 0 implies r (#) f is_left_divergent_to+infty_in x0 ) & ( f is_left_divergent_to+infty_in x0 & r < 0 implies r (#) f is_left_divergent_to-infty_in x0 ) & ( f is_left_divergent_to-infty_in x0 & r > 0 implies r (#) f is_left_divergent_to-infty_in x0 ) & ( f is_left_divergent_to-infty_in x0 & r < 0 implies r (#) f is_left_divergent_to+infty_in x0 ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( ( f is_left_divergent_to+infty_in x0 & r > 0 implies r (#) f is_left_divergent_to+infty_in x0 ) & ( f is_left_divergent_to+infty_in x0 & r < 0 implies r (#) f is_left_divergent_to-infty_in x0 ) & ( f is_left_divergent_to-infty_in x0 & r > 0 implies r (#) f is_left_divergent_to-infty_in x0 ) & ( f is_left_divergent_to-infty_in x0 & r < 0 implies r (#) f is_left_divergent_to+infty_in x0 ) ) A1: (dom f) /\ (left_open_halfline x0) c= dom f by XBOOLE_1:17; thus ( f is_left_divergent_to+infty_in x0 & r > 0 implies r (#) f is_left_divergent_to+infty_in x0 ) ::_thesis: ( ( f is_left_divergent_to+infty_in x0 & r < 0 implies r (#) f is_left_divergent_to-infty_in x0 ) & ( f is_left_divergent_to-infty_in x0 & r > 0 implies r (#) f is_left_divergent_to-infty_in x0 ) & ( f is_left_divergent_to-infty_in x0 & r < 0 implies r (#) f is_left_divergent_to+infty_in x0 ) ) proof assume that A2: f is_left_divergent_to+infty_in x0 and A3: r > 0 ; ::_thesis: r (#) f is_left_divergent_to+infty_in x0 thus for r1 being Real st r1 < x0 holds ex g being Real st ( r1 < g & g < x0 & g in dom (r (#) f) ) :: according to LIMFUNC2:def_2 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom (r (#) f)) /\ (left_open_halfline x0) holds (r (#) f) /* seq is divergent_to+infty proof let r1 be Real; ::_thesis: ( r1 < x0 implies ex g being Real st ( r1 < g & g < x0 & g in dom (r (#) f) ) ) assume r1 < x0 ; ::_thesis: ex g being Real st ( r1 < g & g < x0 & g in dom (r (#) f) ) then consider g being Real such that A4: r1 < g and A5: g < x0 and A6: g in dom f by A2, Def2; take g ; ::_thesis: ( r1 < g & g < x0 & g in dom (r (#) f) ) thus ( r1 < g & g < x0 & g in dom (r (#) f) ) by A4, A5, A6, 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)) /\ (left_open_halfline x0) implies (r (#) f) /* seq is divergent_to+infty ) assume that A7: seq is convergent and A8: lim seq = x0 and A9: rng seq c= (dom (r (#) f)) /\ (left_open_halfline x0) ; ::_thesis: (r (#) f) /* seq is divergent_to+infty A10: (dom f) /\ (left_open_halfline x0) c= dom f by XBOOLE_1:17; A11: rng seq c= (dom f) /\ (left_open_halfline x0) by A9, VALUED_1:def_5; then f /* seq is divergent_to+infty by A2, A7, A8, Def2; then r (#) (f /* seq) is divergent_to+infty by A3, LIMFUNC1:13; hence (r (#) f) /* seq is divergent_to+infty by A11, A10, RFUNCT_2:9, XBOOLE_1:1; ::_thesis: verum end; thus ( f is_left_divergent_to+infty_in x0 & r < 0 implies r (#) f is_left_divergent_to-infty_in x0 ) ::_thesis: ( ( f is_left_divergent_to-infty_in x0 & r > 0 implies r (#) f is_left_divergent_to-infty_in x0 ) & ( f is_left_divergent_to-infty_in x0 & r < 0 implies r (#) f is_left_divergent_to+infty_in x0 ) ) proof assume that A12: f is_left_divergent_to+infty_in x0 and A13: r < 0 ; ::_thesis: r (#) f is_left_divergent_to-infty_in x0 thus for r1 being Real st r1 < x0 holds ex g being Real st ( r1 < g & g < x0 & g in dom (r (#) f) ) :: according to LIMFUNC2:def_3 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom (r (#) f)) /\ (left_open_halfline x0) holds (r (#) f) /* seq is divergent_to-infty proof let r1 be Real; ::_thesis: ( r1 < x0 implies ex g being Real st ( r1 < g & g < x0 & g in dom (r (#) f) ) ) assume r1 < x0 ; ::_thesis: ex g being Real st ( r1 < g & g < x0 & g in dom (r (#) f) ) then consider g being Real such that A14: r1 < g and A15: g < x0 and A16: g in dom f by A12, Def2; take g ; ::_thesis: ( r1 < g & g < x0 & g in dom (r (#) f) ) thus ( r1 < g & g < x0 & g in dom (r (#) f) ) by A14, A15, A16, 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)) /\ (left_open_halfline x0) implies (r (#) f) /* seq is divergent_to-infty ) assume that A17: seq is convergent and A18: lim seq = x0 and A19: rng seq c= (dom (r (#) f)) /\ (left_open_halfline x0) ; ::_thesis: (r (#) f) /* seq is divergent_to-infty A20: (dom f) /\ (left_open_halfline x0) c= dom f by XBOOLE_1:17; A21: rng seq c= (dom f) /\ (left_open_halfline x0) by A19, VALUED_1:def_5; then f /* seq is divergent_to+infty by A12, A17, A18, Def2; then r (#) (f /* seq) is divergent_to-infty by A13, LIMFUNC1:13; hence (r (#) f) /* seq is divergent_to-infty by A21, A20, RFUNCT_2:9, XBOOLE_1:1; ::_thesis: verum end; thus ( f is_left_divergent_to-infty_in x0 & r > 0 implies r (#) f is_left_divergent_to-infty_in x0 ) ::_thesis: ( f is_left_divergent_to-infty_in x0 & r < 0 implies r (#) f is_left_divergent_to+infty_in x0 ) proof assume that A22: f is_left_divergent_to-infty_in x0 and A23: r > 0 ; ::_thesis: r (#) f is_left_divergent_to-infty_in x0 thus for r1 being Real st r1 < x0 holds ex g being Real st ( r1 < g & g < x0 & g in dom (r (#) f) ) :: according to LIMFUNC2:def_3 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom (r (#) f)) /\ (left_open_halfline x0) holds (r (#) f) /* seq is divergent_to-infty proof let r1 be Real; ::_thesis: ( r1 < x0 implies ex g being Real st ( r1 < g & g < x0 & g in dom (r (#) f) ) ) assume r1 < x0 ; ::_thesis: ex g being Real st ( r1 < g & g < x0 & g in dom (r (#) f) ) then consider g being Real such that A24: r1 < g and A25: g < x0 and A26: g in dom f by A22, Def3; take g ; ::_thesis: ( r1 < g & g < x0 & g in dom (r (#) f) ) thus ( r1 < g & g < x0 & g in dom (r (#) f) ) by A24, A25, A26, 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)) /\ (left_open_halfline x0) implies (r (#) f) /* seq is divergent_to-infty ) assume that A27: seq is convergent and A28: lim seq = x0 and A29: rng seq c= (dom (r (#) f)) /\ (left_open_halfline x0) ; ::_thesis: (r (#) f) /* seq is divergent_to-infty A30: (dom f) /\ (left_open_halfline x0) c= dom f by XBOOLE_1:17; A31: rng seq c= (dom f) /\ (left_open_halfline x0) by A29, VALUED_1:def_5; then f /* seq is divergent_to-infty by A22, A27, A28, Def3; then r (#) (f /* seq) is divergent_to-infty by A23, LIMFUNC1:14; hence (r (#) f) /* seq is divergent_to-infty by A31, A30, RFUNCT_2:9, XBOOLE_1:1; ::_thesis: verum end; assume that A32: f is_left_divergent_to-infty_in x0 and A33: r < 0 ; ::_thesis: r (#) f is_left_divergent_to+infty_in x0 thus for r1 being Real st r1 < x0 holds ex g being Real st ( r1 < g & g < x0 & g in dom (r (#) f) ) :: according to LIMFUNC2:def_2 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom (r (#) f)) /\ (left_open_halfline x0) holds (r (#) f) /* seq is divergent_to+infty proof let r1 be Real; ::_thesis: ( r1 < x0 implies ex g being Real st ( r1 < g & g < x0 & g in dom (r (#) f) ) ) assume r1 < x0 ; ::_thesis: ex g being Real st ( r1 < g & g < x0 & g in dom (r (#) f) ) then consider g being Real such that A34: r1 < g and A35: g < x0 and A36: g in dom f by A32, Def3; take g ; ::_thesis: ( r1 < g & g < x0 & g in dom (r (#) f) ) thus ( r1 < g & g < x0 & g in dom (r (#) f) ) by A34, A35, A36, 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)) /\ (left_open_halfline x0) implies (r (#) f) /* seq is divergent_to+infty ) assume that A37: seq is convergent and A38: lim seq = x0 and A39: rng seq c= (dom (r (#) f)) /\ (left_open_halfline x0) ; ::_thesis: (r (#) f) /* seq is divergent_to+infty A40: rng seq c= (dom f) /\ (left_open_halfline x0) by A39, VALUED_1:def_5; then f /* seq is divergent_to-infty by A32, A37, A38, Def3; then r (#) (f /* seq) is divergent_to+infty by A33, LIMFUNC1:14; hence (r (#) f) /* seq is divergent_to+infty by A40, A1, RFUNCT_2:9, XBOOLE_1:1; ::_thesis: verum end; theorem :: LIMFUNC2:22 for x0, r being Real for f being PartFunc of REAL,REAL holds ( ( f is_right_divergent_to+infty_in x0 & r > 0 implies r (#) f is_right_divergent_to+infty_in x0 ) & ( f is_right_divergent_to+infty_in x0 & r < 0 implies r (#) f is_right_divergent_to-infty_in x0 ) & ( f is_right_divergent_to-infty_in x0 & r > 0 implies r (#) f is_right_divergent_to-infty_in x0 ) & ( f is_right_divergent_to-infty_in x0 & r < 0 implies r (#) f is_right_divergent_to+infty_in x0 ) ) proof let x0, r be Real; ::_thesis: for f being PartFunc of REAL,REAL holds ( ( f is_right_divergent_to+infty_in x0 & r > 0 implies r (#) f is_right_divergent_to+infty_in x0 ) & ( f is_right_divergent_to+infty_in x0 & r < 0 implies r (#) f is_right_divergent_to-infty_in x0 ) & ( f is_right_divergent_to-infty_in x0 & r > 0 implies r (#) f is_right_divergent_to-infty_in x0 ) & ( f is_right_divergent_to-infty_in x0 & r < 0 implies r (#) f is_right_divergent_to+infty_in x0 ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( ( f is_right_divergent_to+infty_in x0 & r > 0 implies r (#) f is_right_divergent_to+infty_in x0 ) & ( f is_right_divergent_to+infty_in x0 & r < 0 implies r (#) f is_right_divergent_to-infty_in x0 ) & ( f is_right_divergent_to-infty_in x0 & r > 0 implies r (#) f is_right_divergent_to-infty_in x0 ) & ( f is_right_divergent_to-infty_in x0 & r < 0 implies r (#) f is_right_divergent_to+infty_in x0 ) ) A1: (dom f) /\ (right_open_halfline x0) c= dom f by XBOOLE_1:17; thus ( f is_right_divergent_to+infty_in x0 & r > 0 implies r (#) f is_right_divergent_to+infty_in x0 ) ::_thesis: ( ( f is_right_divergent_to+infty_in x0 & r < 0 implies r (#) f is_right_divergent_to-infty_in x0 ) & ( f is_right_divergent_to-infty_in x0 & r > 0 implies r (#) f is_right_divergent_to-infty_in x0 ) & ( f is_right_divergent_to-infty_in x0 & r < 0 implies r (#) f is_right_divergent_to+infty_in x0 ) ) proof assume that A2: f is_right_divergent_to+infty_in x0 and A3: r > 0 ; ::_thesis: r (#) f is_right_divergent_to+infty_in x0 thus for r1 being Real st x0 < r1 holds ex g being Real st ( g < r1 & x0 < g & g in dom (r (#) f) ) :: according to LIMFUNC2:def_5 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom (r (#) f)) /\ (right_open_halfline x0) holds (r (#) f) /* seq is divergent_to+infty proof let r1 be Real; ::_thesis: ( x0 < r1 implies ex g being Real st ( g < r1 & x0 < g & g in dom (r (#) f) ) ) assume x0 < r1 ; ::_thesis: ex g being Real st ( g < r1 & x0 < g & g in dom (r (#) f) ) then consider g being Real such that A4: g < r1 and A5: x0 < g and A6: g in dom f by A2, Def5; take g ; ::_thesis: ( g < r1 & x0 < g & g in dom (r (#) f) ) thus ( g < r1 & x0 < g & g in dom (r (#) f) ) by A4, A5, A6, 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)) /\ (right_open_halfline x0) implies (r (#) f) /* seq is divergent_to+infty ) assume that A7: seq is convergent and A8: lim seq = x0 and A9: rng seq c= (dom (r (#) f)) /\ (right_open_halfline x0) ; ::_thesis: (r (#) f) /* seq is divergent_to+infty A10: (dom f) /\ (right_open_halfline x0) c= dom f by XBOOLE_1:17; A11: rng seq c= (dom f) /\ (right_open_halfline x0) by A9, VALUED_1:def_5; then f /* seq is divergent_to+infty by A2, A7, A8, Def5; then r (#) (f /* seq) is divergent_to+infty by A3, LIMFUNC1:13; hence (r (#) f) /* seq is divergent_to+infty by A11, A10, RFUNCT_2:9, XBOOLE_1:1; ::_thesis: verum end; thus ( f is_right_divergent_to+infty_in x0 & r < 0 implies r (#) f is_right_divergent_to-infty_in x0 ) ::_thesis: ( ( f is_right_divergent_to-infty_in x0 & r > 0 implies r (#) f is_right_divergent_to-infty_in x0 ) & ( f is_right_divergent_to-infty_in x0 & r < 0 implies r (#) f is_right_divergent_to+infty_in x0 ) ) proof assume that A12: f is_right_divergent_to+infty_in x0 and A13: r < 0 ; ::_thesis: r (#) f is_right_divergent_to-infty_in x0 thus for r1 being Real st x0 < r1 holds ex g being Real st ( g < r1 & x0 < g & g in dom (r (#) f) ) :: according to LIMFUNC2:def_6 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom (r (#) f)) /\ (right_open_halfline x0) holds (r (#) f) /* seq is divergent_to-infty proof let r1 be Real; ::_thesis: ( x0 < r1 implies ex g being Real st ( g < r1 & x0 < g & g in dom (r (#) f) ) ) assume x0 < r1 ; ::_thesis: ex g being Real st ( g < r1 & x0 < g & g in dom (r (#) f) ) then consider g being Real such that A14: g < r1 and A15: x0 < g and A16: g in dom f by A12, Def5; take g ; ::_thesis: ( g < r1 & x0 < g & g in dom (r (#) f) ) thus ( g < r1 & x0 < g & g in dom (r (#) f) ) by A14, A15, A16, 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)) /\ (right_open_halfline x0) implies (r (#) f) /* seq is divergent_to-infty ) assume that A17: seq is convergent and A18: lim seq = x0 and A19: rng seq c= (dom (r (#) f)) /\ (right_open_halfline x0) ; ::_thesis: (r (#) f) /* seq is divergent_to-infty A20: (dom f) /\ (right_open_halfline x0) c= dom f by XBOOLE_1:17; A21: rng seq c= (dom f) /\ (right_open_halfline x0) by A19, VALUED_1:def_5; then f /* seq is divergent_to+infty by A12, A17, A18, Def5; then r (#) (f /* seq) is divergent_to-infty by A13, LIMFUNC1:13; hence (r (#) f) /* seq is divergent_to-infty by A21, A20, RFUNCT_2:9, XBOOLE_1:1; ::_thesis: verum end; thus ( f is_right_divergent_to-infty_in x0 & r > 0 implies r (#) f is_right_divergent_to-infty_in x0 ) ::_thesis: ( f is_right_divergent_to-infty_in x0 & r < 0 implies r (#) f is_right_divergent_to+infty_in x0 ) proof assume that A22: f is_right_divergent_to-infty_in x0 and A23: r > 0 ; ::_thesis: r (#) f is_right_divergent_to-infty_in x0 thus for r1 being Real st x0 < r1 holds ex g being Real st ( g < r1 & x0 < g & g in dom (r (#) f) ) :: according to LIMFUNC2:def_6 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom (r (#) f)) /\ (right_open_halfline x0) holds (r (#) f) /* seq is divergent_to-infty proof let r1 be Real; ::_thesis: ( x0 < r1 implies ex g being Real st ( g < r1 & x0 < g & g in dom (r (#) f) ) ) assume x0 < r1 ; ::_thesis: ex g being Real st ( g < r1 & x0 < g & g in dom (r (#) f) ) then consider g being Real such that A24: g < r1 and A25: x0 < g and A26: g in dom f by A22, Def6; take g ; ::_thesis: ( g < r1 & x0 < g & g in dom (r (#) f) ) thus ( g < r1 & x0 < g & g in dom (r (#) f) ) by A24, A25, A26, 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)) /\ (right_open_halfline x0) implies (r (#) f) /* seq is divergent_to-infty ) assume that A27: seq is convergent and A28: lim seq = x0 and A29: rng seq c= (dom (r (#) f)) /\ (right_open_halfline x0) ; ::_thesis: (r (#) f) /* seq is divergent_to-infty A30: (dom f) /\ (right_open_halfline x0) c= dom f by XBOOLE_1:17; A31: rng seq c= (dom f) /\ (right_open_halfline x0) by A29, VALUED_1:def_5; then f /* seq is divergent_to-infty by A22, A27, A28, Def6; then r (#) (f /* seq) is divergent_to-infty by A23, LIMFUNC1:14; hence (r (#) f) /* seq is divergent_to-infty by A31, A30, RFUNCT_2:9, XBOOLE_1:1; ::_thesis: verum end; assume that A32: f is_right_divergent_to-infty_in x0 and A33: r < 0 ; ::_thesis: r (#) f is_right_divergent_to+infty_in x0 thus for r1 being Real st x0 < r1 holds ex g being Real st ( g < r1 & x0 < g & g in dom (r (#) f) ) :: according to LIMFUNC2:def_5 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom (r (#) f)) /\ (right_open_halfline x0) holds (r (#) f) /* seq is divergent_to+infty proof let r1 be Real; ::_thesis: ( x0 < r1 implies ex g being Real st ( g < r1 & x0 < g & g in dom (r (#) f) ) ) assume x0 < r1 ; ::_thesis: ex g being Real st ( g < r1 & x0 < g & g in dom (r (#) f) ) then consider g being Real such that A34: g < r1 and A35: x0 < g and A36: g in dom f by A32, Def6; take g ; ::_thesis: ( g < r1 & x0 < g & g in dom (r (#) f) ) thus ( g < r1 & x0 < g & g in dom (r (#) f) ) by A34, A35, A36, 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)) /\ (right_open_halfline x0) implies (r (#) f) /* seq is divergent_to+infty ) assume that A37: seq is convergent and A38: lim seq = x0 and A39: rng seq c= (dom (r (#) f)) /\ (right_open_halfline x0) ; ::_thesis: (r (#) f) /* seq is divergent_to+infty A40: rng seq c= (dom f) /\ (right_open_halfline x0) by A39, VALUED_1:def_5; then f /* seq is divergent_to-infty by A32, A37, A38, Def6; then r (#) (f /* seq) is divergent_to+infty by A33, LIMFUNC1:14; hence (r (#) f) /* seq is divergent_to+infty by A40, A1, RFUNCT_2:9, XBOOLE_1:1; ::_thesis: verum end; theorem :: LIMFUNC2:23 for x0 being Real for f being PartFunc of REAL,REAL st ( f is_left_divergent_to+infty_in x0 or f is_left_divergent_to-infty_in x0 ) holds abs f is_left_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st ( f is_left_divergent_to+infty_in x0 or f is_left_divergent_to-infty_in x0 ) holds abs f is_left_divergent_to+infty_in x0 let f be PartFunc of REAL,REAL; ::_thesis: ( ( f is_left_divergent_to+infty_in x0 or f is_left_divergent_to-infty_in x0 ) implies abs f is_left_divergent_to+infty_in x0 ) assume A1: ( f is_left_divergent_to+infty_in x0 or f is_left_divergent_to-infty_in x0 ) ; ::_thesis: abs f is_left_divergent_to+infty_in x0 now__::_thesis:_abs_f_is_left_divergent_to+infty_in_x0 percases ( f is_left_divergent_to+infty_in x0 or f is_left_divergent_to-infty_in x0 ) by A1; supposeA2: f is_left_divergent_to+infty_in x0 ; ::_thesis: abs f is_left_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))_/\_(left_open_halfline_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)) /\ (left_open_halfline 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)) /\ (left_open_halfline x0) ; ::_thesis: (abs f) /* seq is divergent_to+infty A7: rng seq c= (dom f) /\ (left_open_halfline 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; (dom f) /\ (left_open_halfline x0) c= dom f by XBOOLE_1:17; then 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_r_being_Real_st_r_<_x0_holds_ ex_g_being_Real_st_ (_r_<_g_&_g_<_x0_&_g_in_dom_(abs_f)_) let r be Real; ::_thesis: ( r < x0 implies ex g being Real st ( r < g & g < x0 & g in dom (abs f) ) ) assume r < x0 ; ::_thesis: ex g being Real st ( r < g & g < x0 & g in dom (abs f) ) then consider g being Real such that A9: r < g and A10: g < x0 and A11: g in dom f by A2, Def2; take g = g; ::_thesis: ( r < g & g < x0 & g in dom (abs f) ) thus ( r < g & g < x0 & g in dom (abs f) ) by A9, A10, A11, VALUED_1:def_11; ::_thesis: verum end; hence abs f is_left_divergent_to+infty_in x0 by A3, Def2; ::_thesis: verum end; supposeA12: f is_left_divergent_to-infty_in x0 ; ::_thesis: abs f is_left_divergent_to+infty_in x0 A13: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(abs_f))_/\_(left_open_halfline_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)) /\ (left_open_halfline x0) implies (abs f) /* seq is divergent_to+infty ) assume that A14: seq is convergent and A15: lim seq = x0 and A16: rng seq c= (dom (abs f)) /\ (left_open_halfline x0) ; ::_thesis: (abs f) /* seq is divergent_to+infty A17: rng seq c= (dom f) /\ (left_open_halfline x0) by A16, VALUED_1:def_11; then f /* seq is divergent_to-infty by A12, A14, A15, Def3; then A18: abs (f /* seq) is divergent_to+infty by LIMFUNC1:25; (dom f) /\ (left_open_halfline x0) c= dom f by XBOOLE_1:17; then rng seq c= dom f by A17, XBOOLE_1:1; hence (abs f) /* seq is divergent_to+infty by A18, RFUNCT_2:10; ::_thesis: verum end; now__::_thesis:_for_r_being_Real_st_r_<_x0_holds_ ex_g_being_Real_st_ (_r_<_g_&_g_<_x0_&_g_in_dom_(abs_f)_) let r be Real; ::_thesis: ( r < x0 implies ex g being Real st ( r < g & g < x0 & g in dom (abs f) ) ) assume r < x0 ; ::_thesis: ex g being Real st ( r < g & g < x0 & g in dom (abs f) ) then consider g being Real such that A19: r < g and A20: g < x0 and A21: g in dom f by A12, Def3; take g = g; ::_thesis: ( r < g & g < x0 & g in dom (abs f) ) thus ( r < g & g < x0 & g in dom (abs f) ) by A19, A20, A21, VALUED_1:def_11; ::_thesis: verum end; hence abs f is_left_divergent_to+infty_in x0 by A13, Def2; ::_thesis: verum end; end; end; hence abs f is_left_divergent_to+infty_in x0 ; ::_thesis: verum end; theorem :: LIMFUNC2:24 for x0 being Real for f being PartFunc of REAL,REAL st ( f is_right_divergent_to+infty_in x0 or f is_right_divergent_to-infty_in x0 ) holds abs f is_right_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st ( f is_right_divergent_to+infty_in x0 or f is_right_divergent_to-infty_in x0 ) holds abs f is_right_divergent_to+infty_in x0 let f be PartFunc of REAL,REAL; ::_thesis: ( ( f is_right_divergent_to+infty_in x0 or f is_right_divergent_to-infty_in x0 ) implies abs f is_right_divergent_to+infty_in x0 ) assume A1: ( f is_right_divergent_to+infty_in x0 or f is_right_divergent_to-infty_in x0 ) ; ::_thesis: abs f is_right_divergent_to+infty_in x0 now__::_thesis:_abs_f_is_right_divergent_to+infty_in_x0 percases ( f is_right_divergent_to+infty_in x0 or f is_right_divergent_to-infty_in x0 ) by A1; supposeA2: f is_right_divergent_to+infty_in x0 ; ::_thesis: abs f is_right_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))_/\_(right_open_halfline_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)) /\ (right_open_halfline 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)) /\ (right_open_halfline x0) ; ::_thesis: (abs f) /* seq is divergent_to+infty A7: rng seq c= (dom f) /\ (right_open_halfline x0) by A6, VALUED_1:def_11; then f /* seq is divergent_to+infty by A2, A4, A5, Def5; then A8: abs (f /* seq) is divergent_to+infty by LIMFUNC1:25; (dom f) /\ (right_open_halfline x0) c= dom f by XBOOLE_1:17; then 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_r_being_Real_st_x0_<_r_holds_ ex_g_being_Real_st_ (_g_<_r_&_x0_<_g_&_g_in_dom_(abs_f)_) let r be Real; ::_thesis: ( x0 < r implies ex g being Real st ( g < r & x0 < g & g in dom (abs f) ) ) assume x0 < r ; ::_thesis: ex g being Real st ( g < r & x0 < g & g in dom (abs f) ) then consider g being Real such that A9: g < r and A10: x0 < g and A11: g in dom f by A2, Def5; take g = g; ::_thesis: ( g < r & x0 < g & g in dom (abs f) ) thus ( g < r & x0 < g & g in dom (abs f) ) by A9, A10, A11, VALUED_1:def_11; ::_thesis: verum end; hence abs f is_right_divergent_to+infty_in x0 by A3, Def5; ::_thesis: verum end; supposeA12: f is_right_divergent_to-infty_in x0 ; ::_thesis: abs f is_right_divergent_to+infty_in x0 A13: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(abs_f))_/\_(right_open_halfline_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)) /\ (right_open_halfline x0) implies (abs f) /* seq is divergent_to+infty ) assume that A14: seq is convergent and A15: lim seq = x0 and A16: rng seq c= (dom (abs f)) /\ (right_open_halfline x0) ; ::_thesis: (abs f) /* seq is divergent_to+infty A17: rng seq c= (dom f) /\ (right_open_halfline x0) by A16, VALUED_1:def_11; then f /* seq is divergent_to-infty by A12, A14, A15, Def6; then A18: abs (f /* seq) is divergent_to+infty by LIMFUNC1:25; (dom f) /\ (right_open_halfline x0) c= dom f by XBOOLE_1:17; then rng seq c= dom f by A17, XBOOLE_1:1; hence (abs f) /* seq is divergent_to+infty by A18, RFUNCT_2:10; ::_thesis: verum end; now__::_thesis:_for_r_being_Real_st_x0_<_r_holds_ ex_g_being_Real_st_ (_g_<_r_&_x0_<_g_&_g_in_dom_(abs_f)_) let r be Real; ::_thesis: ( x0 < r implies ex g being Real st ( g < r & x0 < g & g in dom (abs f) ) ) assume x0 < r ; ::_thesis: ex g being Real st ( g < r & x0 < g & g in dom (abs f) ) then consider g being Real such that A19: g < r and A20: x0 < g and A21: g in dom f by A12, Def6; take g = g; ::_thesis: ( g < r & x0 < g & g in dom (abs f) ) thus ( g < r & x0 < g & g in dom (abs f) ) by A19, A20, A21, VALUED_1:def_11; ::_thesis: verum end; hence abs f is_right_divergent_to+infty_in x0 by A13, Def5; ::_thesis: verum end; end; end; hence abs f is_right_divergent_to+infty_in x0 ; ::_thesis: verum end; theorem Th25: :: LIMFUNC2:25 for x0 being Real for f being PartFunc of REAL,REAL st ex r being Real st ( f | ].(x0 - r),x0.[ is non-decreasing & not f | ].(x0 - r),x0.[ is bounded_above ) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) holds f is_left_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 & not f | ].(x0 - r),x0.[ is bounded_above ) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) holds f is_left_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 & not f | ].(x0 - r),x0.[ is bounded_above ) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) implies f is_left_divergent_to+infty_in x0 ) given r being Real such that A1: f | ].(x0 - r),x0.[ is non-decreasing and A2: not f | ].(x0 - r),x0.[ is bounded_above ; ::_thesis: ( ex r being Real st ( r < x0 & ( for g being Real holds ( not r < g or not g < x0 or not g in dom f ) ) ) or f is_left_divergent_to+infty_in x0 ) assume for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ; ::_thesis: f is_left_divergent_to+infty_in x0 hence for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ; :: according to LIMFUNC2:def_2 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (left_open_halfline x0) holds f /* seq is divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (left_open_halfline x0) implies f /* seq is divergent_to+infty ) assume that A3: seq is convergent and A4: lim seq = x0 and A5: rng seq c= (dom f) /\ (left_open_halfline x0) ; ::_thesis: f /* seq is divergent_to+infty now__::_thesis:_for_t_being_Real_ex_n_being_Element_of_NAT_st_ for_k_being_Element_of_NAT_st_n_<=_k_holds_ t_<_(f_/*_seq)_._k let t be Real; ::_thesis: ex n being Element of NAT st for k being Element of NAT st n <= k holds t < (f /* seq) . k A6: (dom f) /\ (left_open_halfline x0) c= dom f by XBOOLE_1:17; consider g1 being set such that A7: g1 in ].(x0 - r),x0.[ /\ (dom f) and A8: t < f . g1 by A2, RFUNCT_1:70; reconsider g1 = g1 as Real by A7; g1 in ].(x0 - r),x0.[ by A7, XBOOLE_0:def_4; then g1 in { r1 where r1 is Real : ( x0 - r < r1 & r1 < x0 ) } by RCOMP_1:def_2; then A9: ex r1 being Real st ( r1 = g1 & x0 - r < r1 & r1 < x0 ) ; then consider n being Element of NAT such that A10: for k being Element of NAT st n <= k holds g1 < seq . k by A3, A4, Th1; take n = n; ::_thesis: for k being Element of NAT st n <= k holds t < (f /* seq) . k let k be Element of NAT ; ::_thesis: ( n <= k implies t < (f /* seq) . k ) seq . k in rng seq by VALUED_0:28; then A11: seq . k in (dom f) /\ (left_open_halfline x0) by A5; (dom f) /\ (left_open_halfline x0) c= left_open_halfline x0 by XBOOLE_1:17; then seq . k in left_open_halfline x0 by A11; then seq . k in { r2 where r2 is Real : r2 < x0 } by XXREAL_1:229; then A12: ex r2 being Real st ( r2 = seq . k & r2 < x0 ) ; assume n <= k ; ::_thesis: t < (f /* seq) . k then A13: g1 < seq . k by A10; then x0 - r < seq . k by A9, XXREAL_0:2; then seq . k in { g2 where g2 is Real : ( x0 - r < g2 & g2 < x0 ) } by A12; then seq . k in ].(x0 - r),x0.[ by RCOMP_1:def_2; then seq . k in ].(x0 - r),x0.[ /\ (dom f) by A11, A6, XBOOLE_0:def_4; then f . g1 <= f . (seq . k) by A1, A7, A13, RFUNCT_2:22; then t < f . (seq . k) by A8, XXREAL_0:2; hence t < (f /* seq) . k by A5, A6, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; hence f /* seq is divergent_to+infty by LIMFUNC1:def_4; ::_thesis: verum end; theorem :: LIMFUNC2:26 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 & not f | ].(x0 - r),x0.[ is bounded_above ) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) holds f is_left_divergent_to+infty_in x0 by Th25; theorem Th27: :: LIMFUNC2:27 for x0 being Real for f being PartFunc of REAL,REAL st ex r being Real st ( f | ].(x0 - r),x0.[ is non-increasing & not f | ].(x0 - r),x0.[ is bounded_below ) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) holds f is_left_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 & not f | ].(x0 - r),x0.[ is bounded_below ) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) holds f is_left_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 & not f | ].(x0 - r),x0.[ is bounded_below ) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) implies f is_left_divergent_to-infty_in x0 ) given r being Real such that A1: f | ].(x0 - r),x0.[ is non-increasing and A2: not f | ].(x0 - r),x0.[ is bounded_below ; ::_thesis: ( ex r being Real st ( r < x0 & ( for g being Real holds ( not r < g or not g < x0 or not g in dom f ) ) ) or f is_left_divergent_to-infty_in x0 ) assume for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ; ::_thesis: f is_left_divergent_to-infty_in x0 hence for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ; :: according to LIMFUNC2:def_3 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (left_open_halfline x0) holds f /* seq is divergent_to-infty let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (left_open_halfline x0) implies f /* seq is divergent_to-infty ) assume that A3: seq is convergent and A4: lim seq = x0 and A5: rng seq c= (dom f) /\ (left_open_halfline x0) ; ::_thesis: f /* seq is divergent_to-infty now__::_thesis:_for_t_being_Real_ex_n_being_Element_of_NAT_st_ for_k_being_Element_of_NAT_st_n_<=_k_holds_ (f_/*_seq)_._k_<_t let t be Real; ::_thesis: ex n being Element of NAT st for k being Element of NAT st n <= k holds (f /* seq) . k < t A6: (dom f) /\ (left_open_halfline x0) c= dom f by XBOOLE_1:17; consider g1 being set such that A7: g1 in ].(x0 - r),x0.[ /\ (dom f) and A8: f . g1 < t by A2, RFUNCT_1:71; reconsider g1 = g1 as Real by A7; g1 in ].(x0 - r),x0.[ by A7, XBOOLE_0:def_4; then g1 in { r1 where r1 is Real : ( x0 - r < r1 & r1 < x0 ) } by RCOMP_1:def_2; then A9: ex r1 being Real st ( r1 = g1 & x0 - r < r1 & r1 < x0 ) ; then consider n being Element of NAT such that A10: for k being Element of NAT st n <= k holds g1 < seq . k by A3, A4, Th1; take n = n; ::_thesis: for k being Element of NAT st n <= k holds (f /* seq) . k < t let k be Element of NAT ; ::_thesis: ( n <= k implies (f /* seq) . k < t ) seq . k in rng seq by VALUED_0:28; then A11: seq . k in (dom f) /\ (left_open_halfline x0) by A5; (dom f) /\ (left_open_halfline x0) c= left_open_halfline x0 by XBOOLE_1:17; then seq . k in left_open_halfline x0 by A11; then seq . k in { r2 where r2 is Real : r2 < x0 } by XXREAL_1:229; then A12: ex r2 being Real st ( r2 = seq . k & r2 < x0 ) ; assume n <= k ; ::_thesis: (f /* seq) . k < t then A13: g1 < seq . k by A10; then x0 - r < seq . k by A9, XXREAL_0:2; then seq . k in { g2 where g2 is Real : ( x0 - r < g2 & g2 < x0 ) } by A12; then seq . k in ].(x0 - r),x0.[ by RCOMP_1:def_2; then seq . k in ].(x0 - r),x0.[ /\ (dom f) by A11, A6, XBOOLE_0:def_4; then f . (seq . k) <= f . g1 by A1, A7, A13, RFUNCT_2:23; then f . (seq . k) < t by A8, XXREAL_0:2; hence (f /* seq) . k < t by A5, A6, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; hence f /* seq is divergent_to-infty by LIMFUNC1:def_5; ::_thesis: verum end; theorem :: LIMFUNC2:28 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 & not f | ].(x0 - r),x0.[ is bounded_below ) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) holds f is_left_divergent_to-infty_in x0 by Th27; theorem Th29: :: LIMFUNC2:29 for x0 being Real for f being PartFunc of REAL,REAL st ex r being Real st ( f | ].x0,(x0 + r).[ is non-increasing & not f | ].x0,(x0 + r).[ is bounded_above ) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) holds f is_right_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,(x0 + r).[ is non-increasing & not f | ].x0,(x0 + r).[ is bounded_above ) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) holds f is_right_divergent_to+infty_in x0 let f be PartFunc of REAL,REAL; ::_thesis: ( ex r being Real st ( f | ].x0,(x0 + r).[ is non-increasing & not f | ].x0,(x0 + r).[ is bounded_above ) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) implies f is_right_divergent_to+infty_in x0 ) given r being Real such that A1: f | ].x0,(x0 + r).[ is non-increasing and A2: not f | ].x0,(x0 + r).[ is bounded_above ; ::_thesis: ( ex r being Real st ( x0 < r & ( for g being Real holds ( not g < r or not x0 < g or not g in dom f ) ) ) or f is_right_divergent_to+infty_in x0 ) assume for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ; ::_thesis: f is_right_divergent_to+infty_in x0 hence for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ; :: according to LIMFUNC2:def_5 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (right_open_halfline x0) holds f /* seq is divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (right_open_halfline x0) implies f /* seq is divergent_to+infty ) assume that A3: seq is convergent and A4: lim seq = x0 and A5: rng seq c= (dom f) /\ (right_open_halfline x0) ; ::_thesis: f /* seq is divergent_to+infty now__::_thesis:_for_t_being_Real_ex_n_being_Element_of_NAT_st_ for_k_being_Element_of_NAT_st_n_<=_k_holds_ t_<_(f_/*_seq)_._k let t be Real; ::_thesis: ex n being Element of NAT st for k being Element of NAT st n <= k holds t < (f /* seq) . k A6: (dom f) /\ (right_open_halfline x0) c= dom f by XBOOLE_1:17; consider g1 being set such that A7: g1 in ].x0,(x0 + r).[ /\ (dom f) and A8: t < f . g1 by A2, RFUNCT_1:70; reconsider g1 = g1 as Real by A7; g1 in ].x0,(x0 + r).[ by A7, XBOOLE_0:def_4; then g1 in { r1 where r1 is Real : ( x0 < r1 & r1 < x0 + r ) } by RCOMP_1:def_2; then A9: ex r1 being Real st ( r1 = g1 & x0 < r1 & r1 < x0 + r ) ; then consider n being Element of NAT such that A10: for k being Element of NAT st n <= k holds seq . k < g1 by A3, A4, Th2; take n = n; ::_thesis: for k being Element of NAT st n <= k holds t < (f /* seq) . k let k be Element of NAT ; ::_thesis: ( n <= k implies t < (f /* seq) . k ) seq . k in rng seq by VALUED_0:28; then A11: seq . k in (dom f) /\ (right_open_halfline x0) by A5; (dom f) /\ (right_open_halfline x0) c= right_open_halfline x0 by XBOOLE_1:17; then seq . k in right_open_halfline x0 by A11; then seq . k in { r2 where r2 is Real : x0 < r2 } by XXREAL_1:230; then A12: ex r2 being Real st ( r2 = seq . k & x0 < r2 ) ; assume n <= k ; ::_thesis: t < (f /* seq) . k then A13: seq . k < g1 by A10; then seq . k < x0 + r by A9, XXREAL_0:2; then seq . k in { g2 where g2 is Real : ( x0 < g2 & g2 < x0 + r ) } by A12; then seq . k in ].x0,(x0 + r).[ by RCOMP_1:def_2; then seq . k in ].x0,(x0 + r).[ /\ (dom f) by A11, A6, XBOOLE_0:def_4; then f . g1 <= f . (seq . k) by A1, A7, A13, RFUNCT_2:23; then t < f . (seq . k) by A8, XXREAL_0:2; hence t < (f /* seq) . k by A5, A6, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; hence f /* seq is divergent_to+infty by LIMFUNC1:def_4; ::_thesis: verum end; theorem :: LIMFUNC2:30 for x0 being Real for f being PartFunc of REAL,REAL st ex r being Real st ( 0 < r & f | ].x0,(x0 + r).[ is decreasing & not f | ].x0,(x0 + r).[ is bounded_above ) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) holds f is_right_divergent_to+infty_in x0 by Th29; theorem Th31: :: LIMFUNC2:31 for x0 being Real for f being PartFunc of REAL,REAL st ex r being Real st ( f | ].x0,(x0 + r).[ is non-decreasing & not f | ].x0,(x0 + r).[ is bounded_below ) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) holds f is_right_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,(x0 + r).[ is non-decreasing & not f | ].x0,(x0 + r).[ is bounded_below ) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) holds f is_right_divergent_to-infty_in x0 let f be PartFunc of REAL,REAL; ::_thesis: ( ex r being Real st ( f | ].x0,(x0 + r).[ is non-decreasing & not f | ].x0,(x0 + r).[ is bounded_below ) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) implies f is_right_divergent_to-infty_in x0 ) given r being Real such that A1: f | ].x0,(x0 + r).[ is non-decreasing and A2: not f | ].x0,(x0 + r).[ is bounded_below ; ::_thesis: ( ex r being Real st ( x0 < r & ( for g being Real holds ( not g < r or not x0 < g or not g in dom f ) ) ) or f is_right_divergent_to-infty_in x0 ) assume for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ; ::_thesis: f is_right_divergent_to-infty_in x0 hence for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ; :: according to LIMFUNC2:def_6 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (right_open_halfline x0) holds f /* seq is divergent_to-infty let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (right_open_halfline x0) implies f /* seq is divergent_to-infty ) assume that A3: seq is convergent and A4: lim seq = x0 and A5: rng seq c= (dom f) /\ (right_open_halfline x0) ; ::_thesis: f /* seq is divergent_to-infty now__::_thesis:_for_t_being_Real_ex_n_being_Element_of_NAT_st_ for_k_being_Element_of_NAT_st_n_<=_k_holds_ (f_/*_seq)_._k_<_t let t be Real; ::_thesis: ex n being Element of NAT st for k being Element of NAT st n <= k holds (f /* seq) . k < t A6: (dom f) /\ (right_open_halfline x0) c= dom f by XBOOLE_1:17; consider g1 being set such that A7: g1 in ].x0,(x0 + r).[ /\ (dom f) and A8: f . g1 < t by A2, RFUNCT_1:71; reconsider g1 = g1 as Real by A7; g1 in ].x0,(x0 + r).[ by A7, XBOOLE_0:def_4; then g1 in { r1 where r1 is Real : ( x0 < r1 & r1 < x0 + r ) } by RCOMP_1:def_2; then A9: ex r1 being Real st ( r1 = g1 & x0 < r1 & r1 < x0 + r ) ; then consider n being Element of NAT such that A10: for k being Element of NAT st n <= k holds seq . k < g1 by A3, A4, Th2; take n = n; ::_thesis: for k being Element of NAT st n <= k holds (f /* seq) . k < t let k be Element of NAT ; ::_thesis: ( n <= k implies (f /* seq) . k < t ) seq . k in rng seq by VALUED_0:28; then A11: seq . k in (dom f) /\ (right_open_halfline x0) by A5; (dom f) /\ (right_open_halfline x0) c= right_open_halfline x0 by XBOOLE_1:17; then seq . k in right_open_halfline x0 by A11; then seq . k in { r2 where r2 is Real : x0 < r2 } by XXREAL_1:230; then A12: ex r2 being Real st ( r2 = seq . k & x0 < r2 ) ; assume n <= k ; ::_thesis: (f /* seq) . k < t then A13: seq . k < g1 by A10; then seq . k < x0 + r by A9, XXREAL_0:2; then seq . k in { g2 where g2 is Real : ( x0 < g2 & g2 < x0 + r ) } by A12; then seq . k in ].x0,(x0 + r).[ by RCOMP_1:def_2; then seq . k in ].x0,(x0 + r).[ /\ (dom f) by A11, A6, XBOOLE_0:def_4; then f . (seq . k) <= f . g1 by A1, A7, A13, RFUNCT_2:22; then f . (seq . k) < t by A8, XXREAL_0:2; hence (f /* seq) . k < t by A5, A6, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; hence f /* seq is divergent_to-infty by LIMFUNC1:def_5; ::_thesis: verum end; theorem :: LIMFUNC2:32 for x0 being Real for f being PartFunc of REAL,REAL st ex r being Real st ( 0 < r & f | ].x0,(x0 + r).[ is increasing & not f | ].x0,(x0 + r).[ is bounded_below ) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) holds f is_right_divergent_to-infty_in x0 by Th31; theorem Th33: :: LIMFUNC2:33 for x0 being Real for f1, f being PartFunc of REAL,REAL st f1 is_left_divergent_to+infty_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ex r being Real st ( 0 < r & (dom f) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds f1 . g <= f . g ) ) holds f is_left_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f1, f being PartFunc of REAL,REAL st f1 is_left_divergent_to+infty_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ex r being Real st ( 0 < r & (dom f) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds f1 . g <= f . g ) ) holds f is_left_divergent_to+infty_in x0 let f1, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_divergent_to+infty_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ex r being Real st ( 0 < r & (dom f) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds f1 . g <= f . g ) ) implies f is_left_divergent_to+infty_in x0 ) assume that A1: f1 is_left_divergent_to+infty_in x0 and A2: 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 holds ( not 0 < r or not (dom f) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ or ex g being Real st ( g in (dom f) /\ ].(x0 - r),x0.[ & not f1 . g <= f . g ) ) or f is_left_divergent_to+infty_in x0 ) given r being Real such that A3: 0 < r and A4: (dom f) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ and A5: for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds f1 . g <= f . g ; ::_thesis: f is_left_divergent_to+infty_in x0 thus for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) by A2; :: according to LIMFUNC2:def_2 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (left_open_halfline x0) holds f /* seq is divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (left_open_halfline x0) implies f /* seq is divergent_to+infty ) assume that A6: seq is convergent and A7: lim seq = x0 and A8: rng seq c= (dom f) /\ (left_open_halfline x0) ; ::_thesis: f /* seq is divergent_to+infty x0 - r < x0 by A3, Lm1; then consider k being Element of NAT such that A9: for n being Element of NAT st k <= n holds x0 - r < seq . n by A6, A7, Th1; A10: rng (seq ^\ k) c= rng seq by VALUED_0:21; (dom f) /\ (left_open_halfline x0) c= left_open_halfline x0 by XBOOLE_1:17; then rng seq c= left_open_halfline x0 by A8, XBOOLE_1:1; then A11: rng (seq ^\ k) c= left_open_halfline x0 by A10, XBOOLE_1:1; now__::_thesis:_for_x_being_set_st_x_in_rng_(seq_^\_k)_holds_ x_in_].(x0_-_r),x0.[ let x be set ; ::_thesis: ( x in rng (seq ^\ k) implies x in ].(x0 - r),x0.[ ) assume A12: x in rng (seq ^\ k) ; ::_thesis: x in ].(x0 - r),x0.[ then consider n being Element of NAT such that A13: (seq ^\ k) . n = x by FUNCT_2:113; (seq ^\ k) . n in left_open_halfline x0 by A11, A12, A13; then (seq ^\ k) . n in { g where g is Real : g < x0 } by XXREAL_1:229; then A14: ex r1 being Real st ( r1 = (seq ^\ k) . n & r1 < x0 ) ; x0 - r < seq . (n + k) by A9, NAT_1:12; then x0 - r < (seq ^\ k) . n by NAT_1:def_3; then x in { g1 where g1 is Real : ( x0 - r < g1 & g1 < x0 ) } by A13, A14; hence x in ].(x0 - r),x0.[ by RCOMP_1:def_2; ::_thesis: verum end; then A15: rng (seq ^\ k) c= ].(x0 - r),x0.[ by TARSKI:def_3; A16: (dom f) /\ (left_open_halfline x0) c= dom f by XBOOLE_1:17; then A17: rng seq c= dom f by A8, XBOOLE_1:1; then rng (seq ^\ k) c= dom f by A10, XBOOLE_1:1; then A18: rng (seq ^\ k) c= (dom f) /\ ].(x0 - r),x0.[ by A15, XBOOLE_1:19; then A19: rng (seq ^\ k) c= (dom f1) /\ ].(x0 - r),x0.[ by A4, XBOOLE_1:1; A20: (dom f1) /\ ].(x0 - r),x0.[ c= dom f1 by XBOOLE_1:17; then rng (seq ^\ k) c= dom f1 by A19, XBOOLE_1:1; then A21: rng (seq ^\ k) c= (dom f1) /\ (left_open_halfline x0) by A11, XBOOLE_1:19; A22: now__::_thesis:_for_n_being_Element_of_NAT_holds_(f1_/*_(seq_^\_k))_._n_<=_(f_/*_(seq_^\_k))_._n let n be Element of NAT ; ::_thesis: (f1 /* (seq ^\ k)) . n <= (f /* (seq ^\ k)) . n (seq ^\ k) . n in rng (seq ^\ k) by VALUED_0:28; then f1 . ((seq ^\ k) . n) <= f . ((seq ^\ k) . n) by A5, A18; then (f1 /* (seq ^\ k)) . n <= f . ((seq ^\ k) . n) by A19, A20, FUNCT_2:108, XBOOLE_1:1; hence (f1 /* (seq ^\ k)) . n <= (f /* (seq ^\ k)) . n by A17, A10, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; lim (seq ^\ k) = x0 by A6, A7, SEQ_4:20; then f1 /* (seq ^\ k) is divergent_to+infty by A1, A6, A21, Def2; then f /* (seq ^\ k) is divergent_to+infty by A22, LIMFUNC1:42; then (f /* seq) ^\ k is divergent_to+infty by A8, A16, VALUED_0:27, XBOOLE_1:1; hence f /* seq is divergent_to+infty by LIMFUNC1:7; ::_thesis: verum end; theorem Th34: :: LIMFUNC2:34 for x0 being Real for f1, f being PartFunc of REAL,REAL st f1 is_left_divergent_to-infty_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ex r being Real st ( 0 < r & (dom f) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds f . g <= f1 . g ) ) holds f is_left_divergent_to-infty_in x0 proof let x0 be Real; ::_thesis: for f1, f being PartFunc of REAL,REAL st f1 is_left_divergent_to-infty_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ex r being Real st ( 0 < r & (dom f) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds f . g <= f1 . g ) ) holds f is_left_divergent_to-infty_in x0 let f1, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_divergent_to-infty_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ex r being Real st ( 0 < r & (dom f) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds f . g <= f1 . g ) ) implies f is_left_divergent_to-infty_in x0 ) assume that A1: f1 is_left_divergent_to-infty_in x0 and A2: 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 holds ( not 0 < r or not (dom f) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ or ex g being Real st ( g in (dom f) /\ ].(x0 - r),x0.[ & not f . g <= f1 . g ) ) or f is_left_divergent_to-infty_in x0 ) given r being Real such that A3: 0 < r and A4: (dom f) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ and A5: for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds f . g <= f1 . g ; ::_thesis: f is_left_divergent_to-infty_in x0 thus for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) by A2; :: according to LIMFUNC2:def_3 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (left_open_halfline x0) holds f /* seq is divergent_to-infty let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (left_open_halfline x0) implies f /* seq is divergent_to-infty ) assume that A6: seq is convergent and A7: lim seq = x0 and A8: rng seq c= (dom f) /\ (left_open_halfline x0) ; ::_thesis: f /* seq is divergent_to-infty x0 - r < x0 by A3, Lm1; then consider k being Element of NAT such that A9: for n being Element of NAT st k <= n holds x0 - r < seq . n by A6, A7, Th1; A10: rng (seq ^\ k) c= rng seq by VALUED_0:21; (dom f) /\ (left_open_halfline x0) c= left_open_halfline x0 by XBOOLE_1:17; then rng seq c= left_open_halfline x0 by A8, XBOOLE_1:1; then A11: rng (seq ^\ k) c= left_open_halfline x0 by A10, XBOOLE_1:1; now__::_thesis:_for_x_being_set_st_x_in_rng_(seq_^\_k)_holds_ x_in_].(x0_-_r),x0.[ let x be set ; ::_thesis: ( x in rng (seq ^\ k) implies x in ].(x0 - r),x0.[ ) assume A12: x in rng (seq ^\ k) ; ::_thesis: x in ].(x0 - r),x0.[ then consider n being Element of NAT such that A13: (seq ^\ k) . n = x by FUNCT_2:113; (seq ^\ k) . n in left_open_halfline x0 by A11, A12, A13; then (seq ^\ k) . n in { g where g is Real : g < x0 } by XXREAL_1:229; then A14: ex r1 being Real st ( r1 = (seq ^\ k) . n & r1 < x0 ) ; x0 - r < seq . (n + k) by A9, NAT_1:12; then x0 - r < (seq ^\ k) . n by NAT_1:def_3; then x in { g1 where g1 is Real : ( x0 - r < g1 & g1 < x0 ) } by A13, A14; hence x in ].(x0 - r),x0.[ by RCOMP_1:def_2; ::_thesis: verum end; then A15: rng (seq ^\ k) c= ].(x0 - r),x0.[ by TARSKI:def_3; A16: (dom f) /\ (left_open_halfline x0) c= dom f by XBOOLE_1:17; then A17: rng seq c= dom f by A8, XBOOLE_1:1; then rng (seq ^\ k) c= dom f by A10, XBOOLE_1:1; then A18: rng (seq ^\ k) c= (dom f) /\ ].(x0 - r),x0.[ by A15, XBOOLE_1:19; then A19: rng (seq ^\ k) c= (dom f1) /\ ].(x0 - r),x0.[ by A4, XBOOLE_1:1; A20: (dom f1) /\ ].(x0 - r),x0.[ c= dom f1 by XBOOLE_1:17; then rng (seq ^\ k) c= dom f1 by A19, XBOOLE_1:1; then A21: rng (seq ^\ k) c= (dom f1) /\ (left_open_halfline x0) by A11, XBOOLE_1:19; A22: now__::_thesis:_for_n_being_Element_of_NAT_holds_(f_/*_(seq_^\_k))_._n_<=_(f1_/*_(seq_^\_k))_._n let n be Element of NAT ; ::_thesis: (f /* (seq ^\ k)) . n <= (f1 /* (seq ^\ k)) . n (seq ^\ k) . n in rng (seq ^\ k) by VALUED_0:28; then f . ((seq ^\ k) . n) <= f1 . ((seq ^\ k) . n) by A5, A18; then (f /* (seq ^\ k)) . n <= f1 . ((seq ^\ k) . n) by A17, A10, FUNCT_2:108, XBOOLE_1:1; hence (f /* (seq ^\ k)) . n <= (f1 /* (seq ^\ k)) . n by A19, A20, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; lim (seq ^\ k) = x0 by A6, A7, SEQ_4:20; then f1 /* (seq ^\ k) is divergent_to-infty by A1, A6, A21, Def3; then f /* (seq ^\ k) is divergent_to-infty by A22, LIMFUNC1:43; then (f /* seq) ^\ k is divergent_to-infty by A8, A16, VALUED_0:27, XBOOLE_1:1; hence f /* seq is divergent_to-infty by LIMFUNC1:7; ::_thesis: verum end; theorem Th35: :: LIMFUNC2:35 for x0 being Real for f1, f being PartFunc of REAL,REAL st f1 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 ) ) & ex r being Real st ( 0 < r & (dom f) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds f1 . g <= f . g ) ) holds f is_right_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f1, f being PartFunc of REAL,REAL st f1 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 ) ) & ex r being Real st ( 0 < r & (dom f) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds f1 . g <= f . g ) ) holds f is_right_divergent_to+infty_in x0 let f1, f be PartFunc of REAL,REAL; ::_thesis: ( f1 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 ) ) & ex r being Real st ( 0 < r & (dom f) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds f1 . g <= f . g ) ) implies f is_right_divergent_to+infty_in x0 ) assume that A1: f1 is_right_divergent_to+infty_in x0 and A2: 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 holds ( not 0 < r or not (dom f) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ or ex g being Real st ( g in (dom f) /\ ].x0,(x0 + r).[ & not f1 . g <= f . g ) ) or f is_right_divergent_to+infty_in x0 ) given r being Real such that A3: 0 < r and A4: (dom f) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ and A5: for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds f1 . g <= f . g ; ::_thesis: f is_right_divergent_to+infty_in x0 thus for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) by A2; :: according to LIMFUNC2:def_5 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (right_open_halfline x0) holds f /* seq is divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (right_open_halfline x0) implies f /* seq is divergent_to+infty ) assume that A6: seq is convergent and A7: lim seq = x0 and A8: rng seq c= (dom f) /\ (right_open_halfline x0) ; ::_thesis: f /* seq is divergent_to+infty x0 < x0 + r by A3, Lm1; then consider k being Element of NAT such that A9: for n being Element of NAT st k <= n holds seq . n < x0 + r by A6, A7, Th2; A10: rng (seq ^\ k) c= rng seq by VALUED_0:21; (dom f) /\ (right_open_halfline x0) c= right_open_halfline x0 by XBOOLE_1:17; then rng seq c= right_open_halfline x0 by A8, XBOOLE_1:1; then A11: rng (seq ^\ k) c= right_open_halfline x0 by A10, XBOOLE_1:1; now__::_thesis:_for_x_being_set_st_x_in_rng_(seq_^\_k)_holds_ x_in_].x0,(x0_+_r).[ let x be set ; ::_thesis: ( x in rng (seq ^\ k) implies x in ].x0,(x0 + r).[ ) assume A12: x in rng (seq ^\ k) ; ::_thesis: x in ].x0,(x0 + r).[ then consider n being Element of NAT such that A13: (seq ^\ k) . n = x by FUNCT_2:113; (seq ^\ k) . n in right_open_halfline x0 by A11, A12, A13; then (seq ^\ k) . n in { g where g is Real : x0 < g } by XXREAL_1:230; then A14: ex r1 being Real st ( r1 = (seq ^\ k) . n & x0 < r1 ) ; seq . (n + k) < x0 + r by A9, NAT_1:12; then (seq ^\ k) . n < x0 + r by NAT_1:def_3; then x in { g1 where g1 is Real : ( x0 < g1 & g1 < x0 + r ) } by A13, A14; hence x in ].x0,(x0 + r).[ by RCOMP_1:def_2; ::_thesis: verum end; then A15: rng (seq ^\ k) c= ].x0,(x0 + r).[ by TARSKI:def_3; A16: (dom f) /\ (right_open_halfline x0) c= dom f by XBOOLE_1:17; then A17: rng seq c= dom f by A8, XBOOLE_1:1; then rng (seq ^\ k) c= dom f by A10, XBOOLE_1:1; then A18: rng (seq ^\ k) c= (dom f) /\ ].x0,(x0 + r).[ by A15, XBOOLE_1:19; then A19: rng (seq ^\ k) c= (dom f1) /\ ].x0,(x0 + r).[ by A4, XBOOLE_1:1; A20: (dom f1) /\ ].x0,(x0 + r).[ c= dom f1 by XBOOLE_1:17; then rng (seq ^\ k) c= dom f1 by A19, XBOOLE_1:1; then A21: rng (seq ^\ k) c= (dom f1) /\ (right_open_halfline x0) by A11, XBOOLE_1:19; A22: now__::_thesis:_for_n_being_Element_of_NAT_holds_(f1_/*_(seq_^\_k))_._n_<=_(f_/*_(seq_^\_k))_._n let n be Element of NAT ; ::_thesis: (f1 /* (seq ^\ k)) . n <= (f /* (seq ^\ k)) . n (seq ^\ k) . n in rng (seq ^\ k) by VALUED_0:28; then f1 . ((seq ^\ k) . n) <= f . ((seq ^\ k) . n) by A5, A18; then (f1 /* (seq ^\ k)) . n <= f . ((seq ^\ k) . n) by A19, A20, FUNCT_2:108, XBOOLE_1:1; hence (f1 /* (seq ^\ k)) . n <= (f /* (seq ^\ k)) . n by A17, A10, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; lim (seq ^\ k) = x0 by A6, A7, SEQ_4:20; then f1 /* (seq ^\ k) is divergent_to+infty by A1, A6, A21, Def5; then f /* (seq ^\ k) is divergent_to+infty by A22, LIMFUNC1:42; then (f /* seq) ^\ k is divergent_to+infty by A8, A16, VALUED_0:27, XBOOLE_1:1; hence f /* seq is divergent_to+infty by LIMFUNC1:7; ::_thesis: verum end; theorem Th36: :: LIMFUNC2:36 for x0 being Real for f1, f being PartFunc of REAL,REAL st f1 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 ) ) & ex r being Real st ( 0 < r & (dom f) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds f . g <= f1 . g ) ) holds f is_right_divergent_to-infty_in x0 proof let x0 be Real; ::_thesis: for f1, f being PartFunc of REAL,REAL st f1 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 ) ) & ex r being Real st ( 0 < r & (dom f) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds f . g <= f1 . g ) ) holds f is_right_divergent_to-infty_in x0 let f1, f be PartFunc of REAL,REAL; ::_thesis: ( f1 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 ) ) & ex r being Real st ( 0 < r & (dom f) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds f . g <= f1 . g ) ) implies f is_right_divergent_to-infty_in x0 ) assume that A1: f1 is_right_divergent_to-infty_in x0 and A2: 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 holds ( not 0 < r or not (dom f) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ or ex g being Real st ( g in (dom f) /\ ].x0,(x0 + r).[ & not f . g <= f1 . g ) ) or f is_right_divergent_to-infty_in x0 ) given r being Real such that A3: 0 < r and A4: (dom f) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ and A5: for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds f . g <= f1 . g ; ::_thesis: f is_right_divergent_to-infty_in x0 thus for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) by A2; :: according to LIMFUNC2:def_6 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (right_open_halfline x0) holds f /* seq is divergent_to-infty let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (right_open_halfline x0) implies f /* seq is divergent_to-infty ) assume that A6: seq is convergent and A7: lim seq = x0 and A8: rng seq c= (dom f) /\ (right_open_halfline x0) ; ::_thesis: f /* seq is divergent_to-infty x0 < x0 + r by A3, Lm1; then consider k being Element of NAT such that A9: for n being Element of NAT st k <= n holds seq . n < x0 + r by A6, A7, Th2; A10: rng (seq ^\ k) c= rng seq by VALUED_0:21; (dom f) /\ (right_open_halfline x0) c= right_open_halfline x0 by XBOOLE_1:17; then rng seq c= right_open_halfline x0 by A8, XBOOLE_1:1; then A11: rng (seq ^\ k) c= right_open_halfline x0 by A10, XBOOLE_1:1; now__::_thesis:_for_x_being_set_st_x_in_rng_(seq_^\_k)_holds_ x_in_].x0,(x0_+_r).[ let x be set ; ::_thesis: ( x in rng (seq ^\ k) implies x in ].x0,(x0 + r).[ ) assume A12: x in rng (seq ^\ k) ; ::_thesis: x in ].x0,(x0 + r).[ then consider n being Element of NAT such that A13: (seq ^\ k) . n = x by FUNCT_2:113; (seq ^\ k) . n in right_open_halfline x0 by A11, A12, A13; then (seq ^\ k) . n in { g where g is Real : x0 < g } by XXREAL_1:230; then A14: ex r1 being Real st ( r1 = (seq ^\ k) . n & x0 < r1 ) ; seq . (n + k) < x0 + r by A9, NAT_1:12; then (seq ^\ k) . n < x0 + r by NAT_1:def_3; then x in { g1 where g1 is Real : ( x0 < g1 & g1 < x0 + r ) } by A13, A14; hence x in ].x0,(x0 + r).[ by RCOMP_1:def_2; ::_thesis: verum end; then A15: rng (seq ^\ k) c= ].x0,(x0 + r).[ by TARSKI:def_3; A16: (dom f) /\ (right_open_halfline x0) c= dom f by XBOOLE_1:17; then A17: rng seq c= dom f by A8, XBOOLE_1:1; then rng (seq ^\ k) c= dom f by A10, XBOOLE_1:1; then A18: rng (seq ^\ k) c= (dom f) /\ ].x0,(x0 + r).[ by A15, XBOOLE_1:19; then A19: rng (seq ^\ k) c= (dom f1) /\ ].x0,(x0 + r).[ by A4, XBOOLE_1:1; A20: (dom f1) /\ ].x0,(x0 + r).[ c= dom f1 by XBOOLE_1:17; then rng (seq ^\ k) c= dom f1 by A19, XBOOLE_1:1; then A21: rng (seq ^\ k) c= (dom f1) /\ (right_open_halfline x0) by A11, XBOOLE_1:19; A22: now__::_thesis:_for_n_being_Element_of_NAT_holds_(f_/*_(seq_^\_k))_._n_<=_(f1_/*_(seq_^\_k))_._n let n be Element of NAT ; ::_thesis: (f /* (seq ^\ k)) . n <= (f1 /* (seq ^\ k)) . n (seq ^\ k) . n in rng (seq ^\ k) by VALUED_0:28; then f . ((seq ^\ k) . n) <= f1 . ((seq ^\ k) . n) by A5, A18; then (f /* (seq ^\ k)) . n <= f1 . ((seq ^\ k) . n) by A17, A10, FUNCT_2:108, XBOOLE_1:1; hence (f /* (seq ^\ k)) . n <= (f1 /* (seq ^\ k)) . n by A19, A20, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; lim (seq ^\ k) = x0 by A6, A7, SEQ_4:20; then f1 /* (seq ^\ k) is divergent_to-infty by A1, A6, A21, Def6; then f /* (seq ^\ k) is divergent_to-infty by A22, LIMFUNC1:43; then (f /* seq) ^\ k is divergent_to-infty by A8, A16, VALUED_0:27, XBOOLE_1:1; hence f /* seq is divergent_to-infty by LIMFUNC1:7; ::_thesis: verum end; theorem :: LIMFUNC2:37 for x0 being Real for f1, f being PartFunc of REAL,REAL st f1 is_left_divergent_to+infty_in x0 & ex r being Real st ( 0 < r & ].(x0 - r),x0.[ c= (dom f) /\ (dom f1) & ( for g being Real st g in ].(x0 - r),x0.[ holds f1 . g <= f . g ) ) holds f is_left_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f1, f being PartFunc of REAL,REAL st f1 is_left_divergent_to+infty_in x0 & ex r being Real st ( 0 < r & ].(x0 - r),x0.[ c= (dom f) /\ (dom f1) & ( for g being Real st g in ].(x0 - r),x0.[ holds f1 . g <= f . g ) ) holds f is_left_divergent_to+infty_in x0 let f1, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_divergent_to+infty_in x0 & ex r being Real st ( 0 < r & ].(x0 - r),x0.[ c= (dom f) /\ (dom f1) & ( for g being Real st g in ].(x0 - r),x0.[ holds f1 . g <= f . g ) ) implies f is_left_divergent_to+infty_in x0 ) assume A1: f1 is_left_divergent_to+infty_in x0 ; ::_thesis: ( for r being Real holds ( not 0 < r or not ].(x0 - r),x0.[ c= (dom f) /\ (dom f1) or ex g being Real st ( g in ].(x0 - r),x0.[ & not f1 . g <= f . g ) ) or f is_left_divergent_to+infty_in x0 ) given r being Real such that A2: 0 < r and A3: ].(x0 - r),x0.[ c= (dom f) /\ (dom f1) and A4: for g being Real st g in ].(x0 - r),x0.[ holds f1 . g <= f . g ; ::_thesis: f is_left_divergent_to+infty_in x0 A5: (dom f) /\ (dom f1) c= dom f by XBOOLE_1:17; then A6: ].(x0 - r),x0.[ = (dom f) /\ ].(x0 - r),x0.[ by A3, XBOOLE_1:1, XBOOLE_1:28; (dom f) /\ (dom f1) c= dom f1 by XBOOLE_1:17; then A7: ].(x0 - r),x0.[ = (dom f1) /\ ].(x0 - r),x0.[ by A3, XBOOLE_1:1, XBOOLE_1:28; for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) by A2, A3, A5, Th3, XBOOLE_1:1; hence f is_left_divergent_to+infty_in x0 by A1, A2, A4, A6, A7, Th33; ::_thesis: verum end; theorem :: LIMFUNC2:38 for x0 being Real for f1, f being PartFunc of REAL,REAL st f1 is_left_divergent_to-infty_in x0 & ex r being Real st ( 0 < r & ].(x0 - r),x0.[ c= (dom f) /\ (dom f1) & ( for g being Real st g in ].(x0 - r),x0.[ holds f . g <= f1 . g ) ) holds f is_left_divergent_to-infty_in x0 proof let x0 be Real; ::_thesis: for f1, f being PartFunc of REAL,REAL st f1 is_left_divergent_to-infty_in x0 & ex r being Real st ( 0 < r & ].(x0 - r),x0.[ c= (dom f) /\ (dom f1) & ( for g being Real st g in ].(x0 - r),x0.[ holds f . g <= f1 . g ) ) holds f is_left_divergent_to-infty_in x0 let f1, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_divergent_to-infty_in x0 & ex r being Real st ( 0 < r & ].(x0 - r),x0.[ c= (dom f) /\ (dom f1) & ( for g being Real st g in ].(x0 - r),x0.[ holds f . g <= f1 . g ) ) implies f is_left_divergent_to-infty_in x0 ) assume A1: f1 is_left_divergent_to-infty_in x0 ; ::_thesis: ( for r being Real holds ( not 0 < r or not ].(x0 - r),x0.[ c= (dom f) /\ (dom f1) or ex g being Real st ( g in ].(x0 - r),x0.[ & not f . g <= f1 . g ) ) or f is_left_divergent_to-infty_in x0 ) given r being Real such that A2: 0 < r and A3: ].(x0 - r),x0.[ c= (dom f) /\ (dom f1) and A4: for g being Real st g in ].(x0 - r),x0.[ holds f . g <= f1 . g ; ::_thesis: f is_left_divergent_to-infty_in x0 A5: (dom f) /\ (dom f1) c= dom f by XBOOLE_1:17; then A6: ].(x0 - r),x0.[ = (dom f) /\ ].(x0 - r),x0.[ by A3, XBOOLE_1:1, XBOOLE_1:28; (dom f) /\ (dom f1) c= dom f1 by XBOOLE_1:17; then A7: ].(x0 - r),x0.[ = (dom f1) /\ ].(x0 - r),x0.[ by A3, XBOOLE_1:1, XBOOLE_1:28; for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) by A2, A3, A5, Th3, XBOOLE_1:1; hence f is_left_divergent_to-infty_in x0 by A1, A2, A4, A6, A7, Th34; ::_thesis: verum end; theorem :: LIMFUNC2:39 for x0 being Real for f1, f being PartFunc of REAL,REAL st f1 is_right_divergent_to+infty_in x0 & ex r being Real st ( 0 < r & ].x0,(x0 + r).[ c= (dom f) /\ (dom f1) & ( for g being Real st g in ].x0,(x0 + r).[ holds f1 . g <= f . g ) ) holds f is_right_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f1, f being PartFunc of REAL,REAL st f1 is_right_divergent_to+infty_in x0 & ex r being Real st ( 0 < r & ].x0,(x0 + r).[ c= (dom f) /\ (dom f1) & ( for g being Real st g in ].x0,(x0 + r).[ holds f1 . g <= f . g ) ) holds f is_right_divergent_to+infty_in x0 let f1, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is_right_divergent_to+infty_in x0 & ex r being Real st ( 0 < r & ].x0,(x0 + r).[ c= (dom f) /\ (dom f1) & ( for g being Real st g in ].x0,(x0 + r).[ holds f1 . g <= f . g ) ) implies f is_right_divergent_to+infty_in x0 ) assume A1: f1 is_right_divergent_to+infty_in x0 ; ::_thesis: ( for r being Real holds ( not 0 < r or not ].x0,(x0 + r).[ c= (dom f) /\ (dom f1) or ex g being Real st ( g in ].x0,(x0 + r).[ & not f1 . g <= f . g ) ) or f is_right_divergent_to+infty_in x0 ) given r being Real such that A2: 0 < r and A3: ].x0,(x0 + r).[ c= (dom f) /\ (dom f1) and A4: for g being Real st g in ].x0,(x0 + r).[ holds f1 . g <= f . g ; ::_thesis: f is_right_divergent_to+infty_in x0 A5: (dom f) /\ (dom f1) c= dom f by XBOOLE_1:17; then A6: ].x0,(x0 + r).[ = (dom f) /\ ].x0,(x0 + r).[ by A3, XBOOLE_1:1, XBOOLE_1:28; (dom f) /\ (dom f1) c= dom f1 by XBOOLE_1:17; then A7: ].x0,(x0 + r).[ = (dom f1) /\ ].x0,(x0 + r).[ by A3, XBOOLE_1:1, XBOOLE_1:28; for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) by A2, A3, A5, Th4, XBOOLE_1:1; hence f is_right_divergent_to+infty_in x0 by A1, A2, A4, A6, A7, Th35; ::_thesis: verum end; theorem :: LIMFUNC2:40 for x0 being Real for f1, f being PartFunc of REAL,REAL st f1 is_right_divergent_to-infty_in x0 & ex r being Real st ( 0 < r & ].x0,(x0 + r).[ c= (dom f) /\ (dom f1) & ( for g being Real st g in ].x0,(x0 + r).[ holds f . g <= f1 . g ) ) holds f is_right_divergent_to-infty_in x0 proof let x0 be Real; ::_thesis: for f1, f being PartFunc of REAL,REAL st f1 is_right_divergent_to-infty_in x0 & ex r being Real st ( 0 < r & ].x0,(x0 + r).[ c= (dom f) /\ (dom f1) & ( for g being Real st g in ].x0,(x0 + r).[ holds f . g <= f1 . g ) ) holds f is_right_divergent_to-infty_in x0 let f1, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is_right_divergent_to-infty_in x0 & ex r being Real st ( 0 < r & ].x0,(x0 + r).[ c= (dom f) /\ (dom f1) & ( for g being Real st g in ].x0,(x0 + r).[ holds f . g <= f1 . g ) ) implies f is_right_divergent_to-infty_in x0 ) assume A1: f1 is_right_divergent_to-infty_in x0 ; ::_thesis: ( for r being Real holds ( not 0 < r or not ].x0,(x0 + r).[ c= (dom f) /\ (dom f1) or ex g being Real st ( g in ].x0,(x0 + r).[ & not f . g <= f1 . g ) ) or f is_right_divergent_to-infty_in x0 ) given r being Real such that A2: 0 < r and A3: ].x0,(x0 + r).[ c= (dom f) /\ (dom f1) and A4: for g being Real st g in ].x0,(x0 + r).[ holds f . g <= f1 . g ; ::_thesis: f is_right_divergent_to-infty_in x0 A5: (dom f) /\ (dom f1) c= dom f by XBOOLE_1:17; then A6: ].x0,(x0 + r).[ = (dom f) /\ ].x0,(x0 + r).[ by A3, XBOOLE_1:1, XBOOLE_1:28; (dom f) /\ (dom f1) c= dom f1 by XBOOLE_1:17; then A7: ].x0,(x0 + r).[ = (dom f1) /\ ].x0,(x0 + r).[ by A3, XBOOLE_1:1, XBOOLE_1:28; for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) by A2, A3, A5, Th4, XBOOLE_1:1; hence f is_right_divergent_to-infty_in x0 by A1, A2, A4, A6, A7, Th36; ::_thesis: verum end; definition let f be PartFunc of REAL,REAL; let x0 be Real; assume A1: f is_left_convergent_in x0 ; func lim_left (f,x0) -> Real means :Def7: :: LIMFUNC2:def 7 for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (left_open_halfline 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) /\ (left_open_halfline 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) /\ (left_open_halfline 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) /\ (left_open_halfline 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 ); let g1, g2 be Real; ::_thesis: ( ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (left_open_halfline 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) /\ (left_open_halfline x0) holds ( f /* seq is convergent & lim (f /* seq) = g2 ) ) implies g1 = g2 ) assume that A2: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (left_open_halfline x0) holds ( f /* seq is convergent & lim (f /* seq) = g1 ) and A3: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (left_open_halfline x0) holds ( f /* seq is convergent & lim (f /* seq) = g2 ) ; ::_thesis: g1 = g2 A4: now__::_thesis:_for_n_being_Element_of_NAT_ex_g_being_Real_st_S1[n,g] let n be Element of NAT ; ::_thesis: ex g being Real st S1[n,g] x0 - (1 / (n + 1)) < x0 by Lm3; then consider g being Real such that A5: x0 - (1 / (n + 1)) < g and A6: g < x0 and A7: g in dom f by A1, Def1; take g = g; ::_thesis: S1[n,g] thus S1[n,g] by A5, A6, A7; ::_thesis: verum end; consider s being Real_Sequence such that A8: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A4); A9: rng s c= (dom f) /\ (left_open_halfline x0) by A8, Th5; A10: lim s = x0 by A8, Th5; A11: s is convergent by A8, Th5; then lim (f /* s) = g1 by A10, A9, A2; hence g1 = g2 by A11, A10, A9, A3; ::_thesis: verum end; end; :: deftheorem Def7 defines lim_left LIMFUNC2:def_7_:_ for f being PartFunc of REAL,REAL for x0 being Real st f is_left_convergent_in x0 holds for b3 being Real holds ( b3 = lim_left (f,x0) iff for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (left_open_halfline x0) holds ( f /* seq is convergent & lim (f /* seq) = b3 ) ); definition let f be PartFunc of REAL,REAL; let x0 be Real; assume A1: f is_right_convergent_in x0 ; func lim_right (f,x0) -> Real means :Def8: :: LIMFUNC2:def 8 for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (right_open_halfline 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) /\ (right_open_halfline x0) holds ( f /* seq is convergent & lim (f /* seq) = b1 ) by A1, Def4; uniqueness for b1, b2 being Real st ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (right_open_halfline 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) /\ (right_open_halfline x0) holds ( f /* seq is convergent & lim (f /* seq) = b2 ) ) holds b1 = b2 proof defpred S1[ Element of NAT , real number ] means ( x0 < \$2 & \$2 < x0 + (1 / (\$1 + 1)) & \$2 in dom f ); let g1, g2 be Real; ::_thesis: ( ( for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (right_open_halfline 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) /\ (right_open_halfline x0) holds ( f /* seq is convergent & lim (f /* seq) = g2 ) ) implies g1 = g2 ) assume that A2: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (right_open_halfline x0) holds ( f /* seq is convergent & lim (f /* seq) = g1 ) and A3: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (right_open_halfline x0) holds ( f /* seq is convergent & lim (f /* seq) = g2 ) ; ::_thesis: g1 = g2 A4: now__::_thesis:_for_n_being_Element_of_NAT_ex_g_being_Real_st_S1[n,g] let n be Element of NAT ; ::_thesis: ex g being Real st S1[n,g] x0 < x0 + (1 / (n + 1)) by Lm3; then consider g being Real such that A5: g < x0 + (1 / (n + 1)) and A6: x0 < g and A7: g in dom f by A1, Def4; take g = g; ::_thesis: S1[n,g] thus S1[n,g] by A5, A6, A7; ::_thesis: verum end; consider s being Real_Sequence such that A8: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A4); A9: rng s c= (dom f) /\ (right_open_halfline x0) by A8, Th6; A10: lim s = x0 by A8, Th6; A11: s is convergent by A8, Th6; then lim (f /* s) = g1 by A10, A9, A2; hence g1 = g2 by A11, A10, A9, A3; ::_thesis: verum end; end; :: deftheorem Def8 defines lim_right LIMFUNC2:def_8_:_ for f being PartFunc of REAL,REAL for x0 being Real st f is_right_convergent_in x0 holds for b3 being Real holds ( b3 = lim_right (f,x0) iff for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (right_open_halfline x0) holds ( f /* seq is convergent & lim (f /* seq) = b3 ) ); theorem :: LIMFUNC2:41 for x0, g being Real for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 holds ( lim_left (f,x0) = g iff for g1 being Real st 0 < g1 holds ex r being Real st ( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & 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_left_convergent_in x0 holds ( lim_left (f,x0) = g iff for g1 being Real st 0 < g1 holds ex r being Real st ( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_left_convergent_in x0 implies ( lim_left (f,x0) = g iff for g1 being Real st 0 < g1 holds ex r being Real st ( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ) ) assume A1: f is_left_convergent_in x0 ; ::_thesis: ( lim_left (f,x0) = g iff for g1 being Real st 0 < g1 holds ex r being Real st ( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ) thus ( lim_left (f,x0) = g implies for g1 being Real st 0 < g1 holds ex r being Real st ( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ) ::_thesis: ( ( for g1 being Real st 0 < g1 holds ex r being Real st ( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ) implies lim_left (f,x0) = g ) proof assume that A2: lim_left (f,x0) = g and A3: ex g1 being Real st ( 0 < g1 & ( for r being Real st r < x0 holds ex r1 being Real st ( r < r1 & r1 < x0 & r1 in dom f & abs ((f . r1) - g) >= g1 ) ) ) ; ::_thesis: contradiction consider g1 being Real such that A4: 0 < g1 and A5: for r being Real st r < x0 holds ex r1 being Real st ( r < r1 & r1 < x0 & r1 in dom f & abs ((f . r1) - g) >= g1 ) by A3; defpred S1[ Element of NAT , real number ] means ( x0 - (1 / (\$1 + 1)) < \$2 & \$2 < x0 & \$2 in dom f & abs ((f . \$2) - g) >= g1 ); A6: now__::_thesis:_for_n_being_Element_of_NAT_ex_g2_being_Real_st_S1[n,g2] let n be Element of NAT ; ::_thesis: ex g2 being Real st S1[n,g2] x0 - (1 / (n + 1)) < x0 by Lm3; then consider g2 being Real such that A7: x0 - (1 / (n + 1)) < g2 and A8: g2 < x0 and A9: g2 in dom f and A10: abs ((f . g2) - g) >= g1 by A5; take g2 = g2; ::_thesis: S1[n,g2] thus S1[n,g2] by A7, A8, A9, A10; ::_thesis: verum end; consider s being Real_Sequence such that A11: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A6); A12: rng s c= (dom f) /\ (left_open_halfline x0) by A11, Th5; A13: lim s = x0 by A11, Th5; A14: s is convergent by A11, Th5; then A15: lim (f /* s) = g by A1, A2, A13, A12, Def7; f /* s is convergent by A1, A2, A14, A13, A12, Def7; then consider n being Element of NAT such that A16: for k being Element of NAT st n <= k holds abs (((f /* s) . k) - g) < g1 by A4, A15, SEQ_2:def_7; A17: abs (((f /* s) . n) - g) < g1 by A16; rng s c= dom f by A11, Th5; then abs ((f . (s . n)) - g) < g1 by A17, FUNCT_2:108; hence contradiction by A11; ::_thesis: verum end; assume A18: for g1 being Real st 0 < g1 holds ex r being Real st ( r < x0 & ( for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ; ::_thesis: lim_left (f,x0) = g 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)_=_g_) 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) = g ) ) assume that A19: s is convergent and A20: lim s = x0 and A21: rng s c= (dom f) /\ (left_open_halfline x0) ; ::_thesis: ( f /* s is convergent & lim (f /* s) = g ) A22: (dom f) /\ (left_open_halfline x0) c= dom f by XBOOLE_1:17; A23: 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 A24: 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 r being Real such that A25: r < x0 and A26: for r1 being Real st r < r1 & r1 < x0 & r1 in dom f holds abs ((f . r1) - g) < g1 by A18, A24; consider n being Element of NAT such that A27: for k being Element of NAT st n <= k holds r < s . k by A19, A20, A25, Th1; 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 A28: n <= k ; ::_thesis: abs (((f /* s) . k) - g) < g1 A29: s . k in rng s by VALUED_0:28; then s . k in left_open_halfline x0 by A21, XBOOLE_0:def_4; then s . k in { g2 where g2 is Real : g2 < x0 } by XXREAL_1:229; then A30: ex g2 being Real st ( g2 = s . k & g2 < x0 ) ; s . k in dom f by A21, A29, XBOOLE_0:def_4; then abs ((f . (s . k)) - g) < g1 by A26, A27, A28, A30; hence abs (((f /* s) . k) - g) < g1 by A21, A22, 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 A23, SEQ_2:def_7; ::_thesis: verum end; hence lim_left (f,x0) = g by A1, Def7; ::_thesis: verum end; theorem :: LIMFUNC2:42 for x0, g being Real for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 holds ( lim_right (f,x0) = g iff for g1 being Real st 0 < g1 holds ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & 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_right_convergent_in x0 holds ( lim_right (f,x0) = g iff for g1 being Real st 0 < g1 holds ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_right_convergent_in x0 implies ( lim_right (f,x0) = g iff for g1 being Real st 0 < g1 holds ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ) ) assume A1: f is_right_convergent_in x0 ; ::_thesis: ( lim_right (f,x0) = g iff for g1 being Real st 0 < g1 holds ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ) thus ( lim_right (f,x0) = g implies for g1 being Real st 0 < g1 holds ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ) ::_thesis: ( ( for g1 being Real st 0 < g1 holds ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ) implies lim_right (f,x0) = g ) proof assume that A2: lim_right (f,x0) = g and A3: ex g1 being Real st ( 0 < g1 & ( for r being Real st x0 < r holds ex r1 being Real st ( r1 < r & x0 < r1 & r1 in dom f & abs ((f . r1) - g) >= g1 ) ) ) ; ::_thesis: contradiction consider g1 being Real such that A4: 0 < g1 and A5: for r being Real st x0 < r holds ex r1 being Real st ( r1 < r & x0 < r1 & r1 in dom f & abs ((f . r1) - g) >= g1 ) by A3; defpred S1[ Element of NAT , real number ] means ( x0 < \$2 & \$2 < x0 + (1 / (\$1 + 1)) & \$2 in dom f & g1 <= abs ((f . \$2) - g) ); A6: now__::_thesis:_for_n_being_Element_of_NAT_ex_r1_being_Real_st_S1[n,r1] let n be Element of NAT ; ::_thesis: ex r1 being Real st S1[n,r1] x0 < x0 + (1 / (n + 1)) by Lm3; then consider r1 being Real such that A7: r1 < x0 + (1 / (n + 1)) and A8: x0 < r1 and A9: r1 in dom f and A10: g1 <= abs ((f . r1) - g) by A5; take r1 = r1; ::_thesis: S1[n,r1] thus S1[n,r1] by A7, A8, A9, A10; ::_thesis: verum end; consider s being Real_Sequence such that A11: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A6); A12: rng s c= (dom f) /\ (right_open_halfline x0) by A11, Th6; A13: lim s = x0 by A11, Th6; A14: s is convergent by A11, Th6; then A15: lim (f /* s) = g by A1, A2, A13, A12, Def8; f /* s is convergent by A1, A2, A14, A13, A12, Def8; then consider n being Element of NAT such that A16: for k being Element of NAT st n <= k holds abs (((f /* s) . k) - g) < g1 by A4, A15, SEQ_2:def_7; A17: abs (((f /* s) . n) - g) < g1 by A16; rng s c= dom f by A11, Th6; then abs ((f . (s . n)) - g) < g1 by A17, FUNCT_2:108; hence contradiction by A11; ::_thesis: verum end; assume A18: for g1 being Real st 0 < g1 holds ex r being Real st ( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ; ::_thesis: lim_right (f,x0) = g 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)_=_g_) 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) = g ) ) assume that A19: s is convergent and A20: lim s = x0 and A21: rng s c= (dom f) /\ (right_open_halfline x0) ; ::_thesis: ( f /* s is convergent & lim (f /* s) = g ) A22: (dom f) /\ (right_open_halfline x0) c= dom f by XBOOLE_1:17; A23: 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 A24: 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 r being Real such that A25: x0 < r and A26: for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 by A18, A24; consider n being Element of NAT such that A27: for k being Element of NAT st n <= k holds s . k < r by A19, A20, A25, Th2; 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 A28: n <= k ; ::_thesis: abs (((f /* s) . k) - g) < g1 A29: s . k in rng s by VALUED_0:28; then s . k in right_open_halfline x0 by A21, XBOOLE_0:def_4; then s . k in { g2 where g2 is Real : x0 < g2 } by XXREAL_1:230; then A30: ex g2 being Real st ( g2 = s . k & x0 < g2 ) ; s . k in dom f by A21, A29, XBOOLE_0:def_4; then abs ((f . (s . k)) - g) < g1 by A26, A27, A28, A30; hence abs (((f /* s) . k) - g) < g1 by A21, A22, 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 A23, SEQ_2:def_7; ::_thesis: verum end; hence lim_right (f,x0) = g by A1, Def8; ::_thesis: verum end; theorem Th43: :: LIMFUNC2:43 for x0, r being Real for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 holds ( r (#) f is_left_convergent_in x0 & lim_left ((r (#) f),x0) = r * (lim_left (f,x0)) ) proof let x0, r be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 holds ( r (#) f is_left_convergent_in x0 & lim_left ((r (#) f),x0) = r * (lim_left (f,x0)) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_left_convergent_in x0 implies ( r (#) f is_left_convergent_in x0 & lim_left ((r (#) f),x0) = r * (lim_left (f,x0)) ) ) assume A1: f is_left_convergent_in x0 ; ::_thesis: ( r (#) f is_left_convergent_in x0 & lim_left ((r (#) f),x0) = r * (lim_left (f,x0)) ) A2: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(r_(#)_f))_/\_(left_open_halfline_x0)_holds_ (_(r_(#)_f)_/*_seq_is_convergent_&_lim_((r_(#)_f)_/*_seq)_=_r_*_(lim_left_(f,x0))_) let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (r (#) f)) /\ (left_open_halfline x0) implies ( (r (#) f) /* seq is convergent & lim ((r (#) f) /* seq) = r * (lim_left (f,x0)) ) ) assume that A3: seq is convergent and A4: lim seq = x0 and A5: rng seq c= (dom (r (#) f)) /\ (left_open_halfline x0) ; ::_thesis: ( (r (#) f) /* seq is convergent & lim ((r (#) f) /* seq) = r * (lim_left (f,x0)) ) A6: rng seq c= (dom f) /\ (left_open_halfline x0) by A5, VALUED_1:def_5; A7: (dom f) /\ (left_open_halfline x0) c= dom f by XBOOLE_1:17; then A8: r (#) (f /* seq) = (r (#) f) /* seq by A6, RFUNCT_2:9, XBOOLE_1:1; lim_left (f,x0) = lim_left (f,x0) ; then A9: f /* seq is convergent by A1, A3, A4, A6, Def7; then r (#) (f /* seq) is convergent by SEQ_2:7; hence (r (#) f) /* seq is convergent by A6, A7, RFUNCT_2:9, XBOOLE_1:1; ::_thesis: lim ((r (#) f) /* seq) = r * (lim_left (f,x0)) lim (f /* seq) = lim_left (f,x0) by A1, A3, A4, A6, Def7; hence lim ((r (#) f) /* seq) = r * (lim_left (f,x0)) by A9, A8, SEQ_2:8; ::_thesis: verum end; now__::_thesis:_for_r1_being_Real_st_r1_<_x0_holds_ ex_g_being_Real_st_ (_r1_<_g_&_g_<_x0_&_g_in_dom_(r_(#)_f)_) let r1 be Real; ::_thesis: ( r1 < x0 implies ex g being Real st ( r1 < g & g < x0 & g in dom (r (#) f) ) ) assume r1 < x0 ; ::_thesis: ex g being Real st ( r1 < g & g < x0 & g in dom (r (#) f) ) then consider g being Real such that A10: r1 < g and A11: g < x0 and A12: g in dom f by A1, Def1; take g = g; ::_thesis: ( r1 < g & g < x0 & g in dom (r (#) f) ) thus ( r1 < g & g < x0 & g in dom (r (#) f) ) by A10, A11, A12, VALUED_1:def_5; ::_thesis: verum end; hence r (#) f is_left_convergent_in x0 by A2, Def1; ::_thesis: lim_left ((r (#) f),x0) = r * (lim_left (f,x0)) hence lim_left ((r (#) f),x0) = r * (lim_left (f,x0)) by A2, Def7; ::_thesis: verum end; theorem Th44: :: LIMFUNC2:44 for x0 being Real for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 holds ( - f is_left_convergent_in x0 & lim_left ((- f),x0) = - (lim_left (f,x0)) ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 holds ( - f is_left_convergent_in x0 & lim_left ((- f),x0) = - (lim_left (f,x0)) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_left_convergent_in x0 implies ( - f is_left_convergent_in x0 & lim_left ((- f),x0) = - (lim_left (f,x0)) ) ) assume A1: f is_left_convergent_in x0 ; ::_thesis: ( - f is_left_convergent_in x0 & lim_left ((- f),x0) = - (lim_left (f,x0)) ) (- 1) (#) f = - f ; hence - f is_left_convergent_in x0 by A1, Th43; ::_thesis: lim_left ((- f),x0) = - (lim_left (f,x0)) thus lim_left ((- f),x0) = (- 1) * (lim_left (f,x0)) by A1, Th43 .= - (lim_left (f,x0)) ; ::_thesis: verum end; theorem Th45: :: LIMFUNC2:45 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f1 + f2) ) ) holds ( f1 + f2 is_left_convergent_in x0 & lim_left ((f1 + f2),x0) = (lim_left (f1,x0)) + (lim_left (f2,x0)) ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f1 + f2) ) ) holds ( f1 + f2 is_left_convergent_in x0 & lim_left ((f1 + f2),x0) = (lim_left (f1,x0)) + (lim_left (f2,x0)) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f1 + f2) ) ) implies ( f1 + f2 is_left_convergent_in x0 & lim_left ((f1 + f2),x0) = (lim_left (f1,x0)) + (lim_left (f2,x0)) ) ) assume that A1: f1 is_left_convergent_in x0 and A2: f2 is_left_convergent_in x0 and A3: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f1 + f2) ) ; ::_thesis: ( f1 + f2 is_left_convergent_in x0 & lim_left ((f1 + f2),x0) = (lim_left (f1,x0)) + (lim_left (f2,x0)) ) A4: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(f1_+_f2))_/\_(left_open_halfline_x0)_holds_ (_(f1_+_f2)_/*_seq_is_convergent_&_lim_((f1_+_f2)_/*_seq)_=_(lim_left_(f1,x0))_+_(lim_left_(f2,x0))_) let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f1 + f2)) /\ (left_open_halfline x0) implies ( (f1 + f2) /* seq is convergent & lim ((f1 + f2) /* seq) = (lim_left (f1,x0)) + (lim_left (f2,x0)) ) ) assume that A5: seq is convergent and A6: lim seq = x0 and A7: rng seq c= (dom (f1 + f2)) /\ (left_open_halfline x0) ; ::_thesis: ( (f1 + f2) /* seq is convergent & lim ((f1 + f2) /* seq) = (lim_left (f1,x0)) + (lim_left (f2,x0)) ) A8: dom (f1 + f2) = (dom f1) /\ (dom f2) by A7, Lm4; A9: rng seq c= (dom f1) /\ (left_open_halfline x0) by A7, Lm4; A10: rng seq c= (dom f2) /\ (left_open_halfline x0) by A7, Lm4; then A11: lim (f2 /* seq) = lim_left (f2,x0) by A2, A5, A6, Def7; lim_left (f2,x0) = lim_left (f2,x0) ; then A12: f2 /* seq is convergent by A2, A5, A6, A10, Def7; rng seq c= dom (f1 + f2) by A7, Lm4; then A13: (f1 /* seq) + (f2 /* seq) = (f1 + f2) /* seq by A8, RFUNCT_2:8; lim_left (f1,x0) = lim_left (f1,x0) ; then A14: f1 /* seq is convergent by A1, A5, A6, A9, Def7; hence (f1 + f2) /* seq is convergent by A12, A13, SEQ_2:5; ::_thesis: lim ((f1 + f2) /* seq) = (lim_left (f1,x0)) + (lim_left (f2,x0)) lim (f1 /* seq) = lim_left (f1,x0) by A1, A5, A6, A9, Def7; hence lim ((f1 + f2) /* seq) = (lim_left (f1,x0)) + (lim_left (f2,x0)) by A14, A12, A11, A13, SEQ_2:6; ::_thesis: verum end; hence f1 + f2 is_left_convergent_in x0 by A3, Def1; ::_thesis: lim_left ((f1 + f2),x0) = (lim_left (f1,x0)) + (lim_left (f2,x0)) hence lim_left ((f1 + f2),x0) = (lim_left (f1,x0)) + (lim_left (f2,x0)) by A4, Def7; ::_thesis: verum end; theorem :: LIMFUNC2:46 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f1 - f2) ) ) holds ( f1 - f2 is_left_convergent_in x0 & lim_left ((f1 - f2),x0) = (lim_left (f1,x0)) - (lim_left (f2,x0)) ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f1 - f2) ) ) holds ( f1 - f2 is_left_convergent_in x0 & lim_left ((f1 - f2),x0) = (lim_left (f1,x0)) - (lim_left (f2,x0)) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f1 - f2) ) ) implies ( f1 - f2 is_left_convergent_in x0 & lim_left ((f1 - f2),x0) = (lim_left (f1,x0)) - (lim_left (f2,x0)) ) ) assume that A1: f1 is_left_convergent_in x0 and A2: f2 is_left_convergent_in x0 and A3: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f1 - f2) ) ; ::_thesis: ( f1 - f2 is_left_convergent_in x0 & lim_left ((f1 - f2),x0) = (lim_left (f1,x0)) - (lim_left (f2,x0)) ) A4: - f2 is_left_convergent_in x0 by A2, Th44; hence f1 - f2 is_left_convergent_in x0 by A1, A3, Th45; ::_thesis: lim_left ((f1 - f2),x0) = (lim_left (f1,x0)) - (lim_left (f2,x0)) thus lim_left ((f1 - f2),x0) = (lim_left (f1,x0)) + (lim_left ((- f2),x0)) by A1, A3, A4, Th45 .= (lim_left (f1,x0)) + (- (lim_left (f2,x0))) by A2, Th44 .= (lim_left (f1,x0)) - (lim_left (f2,x0)) ; ::_thesis: verum end; theorem :: LIMFUNC2:47 for x0 being Real for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 & f " {0} = {} & lim_left (f,x0) <> 0 holds ( f ^ is_left_convergent_in x0 & lim_left ((f ^),x0) = (lim_left (f,x0)) " ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 & f " {0} = {} & lim_left (f,x0) <> 0 holds ( f ^ is_left_convergent_in x0 & lim_left ((f ^),x0) = (lim_left (f,x0)) " ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_left_convergent_in x0 & f " {0} = {} & lim_left (f,x0) <> 0 implies ( f ^ is_left_convergent_in x0 & lim_left ((f ^),x0) = (lim_left (f,x0)) " ) ) assume that A1: f is_left_convergent_in x0 and A2: f " {0} = {} and A3: lim_left (f,x0) <> 0 ; ::_thesis: ( f ^ is_left_convergent_in x0 & lim_left ((f ^),x0) = (lim_left (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_^))_/\_(left_open_halfline_x0)_holds_ (_(f_^)_/*_seq_is_convergent_&_lim_((f_^)_/*_seq)_=_(lim_left_(f,x0))_"_) let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f ^)) /\ (left_open_halfline x0) implies ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = (lim_left (f,x0)) " ) ) assume that A6: seq is convergent and A7: lim seq = x0 and A8: rng seq c= (dom (f ^)) /\ (left_open_halfline x0) ; ::_thesis: ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = (lim_left (f,x0)) " ) A9: lim (f /* seq) = lim_left (f,x0) by A1, A4, A6, A7, A8, Def7; A10: (dom f) /\ (left_open_halfline x0) c= dom f by XBOOLE_1:17; then A11: rng seq c= dom f by A4, A8, XBOOLE_1:1; A12: f /* seq is convergent by A1, A3, A4, A6, A7, A8, Def7; A13: (f /* seq) " = (f ^) /* seq by A4, A8, A10, RFUNCT_2:12, XBOOLE_1:1; hence (f ^) /* seq is convergent by A3, A4, A12, A9, A11, RFUNCT_2:11, SEQ_2:21; ::_thesis: lim ((f ^) /* seq) = (lim_left (f,x0)) " thus lim ((f ^) /* seq) = (lim_left (f,x0)) " by A3, A4, A12, A9, A11, A13, RFUNCT_2:11, SEQ_2:22; ::_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 A1, A4, Def1; hence f ^ is_left_convergent_in x0 by A5, Def1; ::_thesis: lim_left ((f ^),x0) = (lim_left (f,x0)) " hence lim_left ((f ^),x0) = (lim_left (f,x0)) " by A5, Def7; ::_thesis: verum end; theorem :: LIMFUNC2:48 for x0 being Real for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 holds ( abs f is_left_convergent_in x0 & lim_left ((abs f),x0) = abs (lim_left (f,x0)) ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 holds ( abs f is_left_convergent_in x0 & lim_left ((abs f),x0) = abs (lim_left (f,x0)) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_left_convergent_in x0 implies ( abs f is_left_convergent_in x0 & lim_left ((abs f),x0) = abs (lim_left (f,x0)) ) ) assume A1: f is_left_convergent_in x0 ; ::_thesis: ( abs f is_left_convergent_in x0 & lim_left ((abs f),x0) = abs (lim_left (f,x0)) ) A2: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(abs_f))_/\_(left_open_halfline_x0)_holds_ (_(abs_f)_/*_seq_is_convergent_&_lim_((abs_f)_/*_seq)_=_abs_(lim_left_(f,x0))_) let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (abs f)) /\ (left_open_halfline x0) implies ( (abs f) /* seq is convergent & lim ((abs f) /* seq) = abs (lim_left (f,x0)) ) ) assume that A3: seq is convergent and A4: lim seq = x0 and A5: rng seq c= (dom (abs f)) /\ (left_open_halfline x0) ; ::_thesis: ( (abs f) /* seq is convergent & lim ((abs f) /* seq) = abs (lim_left (f,x0)) ) A6: rng seq c= (dom f) /\ (left_open_halfline x0) by A5, VALUED_1:def_11; (dom f) /\ (left_open_halfline x0) c= dom f by XBOOLE_1:17; then rng seq c= dom f by A6, XBOOLE_1:1; then A7: abs (f /* seq) = (abs f) /* seq by RFUNCT_2:10; lim_left (f,x0) = lim_left (f,x0) ; then A8: f /* seq is convergent by A1, A3, A4, A6, Def7; hence (abs f) /* seq is convergent by A7; ::_thesis: lim ((abs f) /* seq) = abs (lim_left (f,x0)) lim (f /* seq) = lim_left (f,x0) by A1, A3, A4, A6, Def7; hence lim ((abs f) /* seq) = abs (lim_left (f,x0)) by A8, A7, SEQ_4:14; ::_thesis: verum end; now__::_thesis:_for_r_being_Real_st_r_<_x0_holds_ ex_g_being_Real_st_ (_r_<_g_&_g_<_x0_&_g_in_dom_(abs_f)_) let r be Real; ::_thesis: ( r < x0 implies ex g being Real st ( r < g & g < x0 & g in dom (abs f) ) ) assume r < x0 ; ::_thesis: ex g being Real st ( r < g & g < x0 & g in dom (abs f) ) then consider g being Real such that A9: r < g and A10: g < x0 and A11: g in dom f by A1, Def1; take g = g; ::_thesis: ( r < g & g < x0 & g in dom (abs f) ) thus ( r < g & g < x0 & g in dom (abs f) ) by A9, A10, A11, VALUED_1:def_11; ::_thesis: verum end; hence abs f is_left_convergent_in x0 by A2, Def1; ::_thesis: lim_left ((abs f),x0) = abs (lim_left (f,x0)) hence lim_left ((abs f),x0) = abs (lim_left (f,x0)) by A2, Def7; ::_thesis: verum end; theorem Th49: :: LIMFUNC2:49 for x0 being Real for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 & lim_left (f,x0) <> 0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f & f . g <> 0 ) ) holds ( f ^ is_left_convergent_in x0 & lim_left ((f ^),x0) = (lim_left (f,x0)) " ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 & lim_left (f,x0) <> 0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f & f . g <> 0 ) ) holds ( f ^ is_left_convergent_in x0 & lim_left ((f ^),x0) = (lim_left (f,x0)) " ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_left_convergent_in x0 & lim_left (f,x0) <> 0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f & f . g <> 0 ) ) implies ( f ^ is_left_convergent_in x0 & lim_left ((f ^),x0) = (lim_left (f,x0)) " ) ) assume that A1: f is_left_convergent_in x0 and A2: lim_left (f,x0) <> 0 and A3: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f & f . g <> 0 ) ; ::_thesis: ( f ^ is_left_convergent_in x0 & lim_left ((f ^),x0) = (lim_left (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_^))_/\_(left_open_halfline_x0)_holds_ (_(f_^)_/*_seq_is_convergent_&_lim_((f_^)_/*_seq)_=_(lim_left_(f,x0))_"_) A6: dom (f ^) c= dom f by A4, XBOOLE_1:36; let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f ^)) /\ (left_open_halfline x0) implies ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = (lim_left (f,x0)) " ) ) assume that A7: seq is convergent and A8: lim seq = x0 and A9: rng seq c= (dom (f ^)) /\ (left_open_halfline x0) ; ::_thesis: ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = (lim_left (f,x0)) " ) A10: (dom (f ^)) /\ (left_open_halfline x0) c= dom (f ^) by XBOOLE_1:17; then A11: f /* seq is non-zero by A9, RFUNCT_2:11, XBOOLE_1:1; rng seq c= dom (f ^) by A9, A10, XBOOLE_1:1; then A12: rng seq c= dom f by A6, XBOOLE_1:1; (dom (f ^)) /\ (left_open_halfline x0) c= left_open_halfline x0 by XBOOLE_1:17; then rng seq c= left_open_halfline x0 by A9, XBOOLE_1:1; then A13: rng seq c= (dom f) /\ (left_open_halfline x0) by A12, XBOOLE_1:19; then A14: lim (f /* seq) = lim_left (f,x0) by A1, A7, A8, Def7; A15: (f /* seq) " = (f ^) /* seq by A9, A10, RFUNCT_2:12, XBOOLE_1:1; A16: f /* seq is convergent by A1, A2, A7, A8, A13, Def7; hence (f ^) /* seq is convergent by A2, A14, A11, A15, SEQ_2:21; ::_thesis: lim ((f ^) /* seq) = (lim_left (f,x0)) " thus lim ((f ^) /* seq) = (lim_left (f,x0)) " by A2, A16, A14, A11, A15, SEQ_2:22; ::_thesis: verum end; now__::_thesis:_for_r_being_Real_st_r_<_x0_holds_ ex_g_being_Real_st_ (_r_<_g_&_g_<_x0_&_g_in_dom_(f_^)_) 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 g being Real such that A17: r < g and A18: g < x0 and A19: g in dom f and A20: f . g <> 0 by A3; take g = g; ::_thesis: ( r < g & g < x0 & g in dom (f ^) ) not f . g in {0} by A20, TARSKI:def_1; then not g in f " {0} by FUNCT_1:def_7; hence ( r < g & g < x0 & g in dom (f ^) ) by A4, A17, A18, A19, XBOOLE_0:def_5; ::_thesis: verum end; hence f ^ is_left_convergent_in x0 by A5, Def1; ::_thesis: lim_left ((f ^),x0) = (lim_left (f,x0)) " hence lim_left ((f ^),x0) = (lim_left (f,x0)) " by A5, Def7; ::_thesis: verum end; theorem Th50: :: LIMFUNC2:50 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f1 (#) f2) ) ) holds ( f1 (#) f2 is_left_convergent_in x0 & lim_left ((f1 (#) f2),x0) = (lim_left (f1,x0)) * (lim_left (f2,x0)) ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f1 (#) f2) ) ) holds ( f1 (#) f2 is_left_convergent_in x0 & lim_left ((f1 (#) f2),x0) = (lim_left (f1,x0)) * (lim_left (f2,x0)) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f1 (#) f2) ) ) implies ( f1 (#) f2 is_left_convergent_in x0 & lim_left ((f1 (#) f2),x0) = (lim_left (f1,x0)) * (lim_left (f2,x0)) ) ) assume that A1: f1 is_left_convergent_in x0 and A2: f2 is_left_convergent_in x0 and A3: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f1 (#) f2) ) ; ::_thesis: ( f1 (#) f2 is_left_convergent_in x0 & lim_left ((f1 (#) f2),x0) = (lim_left (f1,x0)) * (lim_left (f2,x0)) ) A4: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(f1_(#)_f2))_/\_(left_open_halfline_x0)_holds_ (_(f1_(#)_f2)_/*_seq_is_convergent_&_lim_((f1_(#)_f2)_/*_seq)_=_(lim_left_(f1,x0))_*_(lim_left_(f2,x0))_) let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f1 (#) f2)) /\ (left_open_halfline x0) implies ( (f1 (#) f2) /* seq is convergent & lim ((f1 (#) f2) /* seq) = (lim_left (f1,x0)) * (lim_left (f2,x0)) ) ) assume that A5: seq is convergent and A6: lim seq = x0 and A7: rng seq c= (dom (f1 (#) f2)) /\ (left_open_halfline x0) ; ::_thesis: ( (f1 (#) f2) /* seq is convergent & lim ((f1 (#) f2) /* seq) = (lim_left (f1,x0)) * (lim_left (f2,x0)) ) A8: dom (f1 (#) f2) = (dom f1) /\ (dom f2) by A7, Lm2; A9: rng seq c= (dom f1) /\ (left_open_halfline x0) by A7, Lm2; A10: rng seq c= (dom f2) /\ (left_open_halfline x0) by A7, Lm2; then A11: lim (f2 /* seq) = lim_left (f2,x0) by A2, A5, A6, Def7; lim_left (f2,x0) = lim_left (f2,x0) ; then A12: f2 /* seq is convergent by A2, A5, A6, A10, Def7; rng seq c= dom (f1 (#) f2) by A7, Lm2; then A13: (f1 /* seq) (#) (f2 /* seq) = (f1 (#) f2) /* seq by A8, RFUNCT_2:8; lim_left (f1,x0) = lim_left (f1,x0) ; then A14: f1 /* seq is convergent by A1, A5, A6, A9, Def7; hence (f1 (#) f2) /* seq is convergent by A12, A13, SEQ_2:14; ::_thesis: lim ((f1 (#) f2) /* seq) = (lim_left (f1,x0)) * (lim_left (f2,x0)) lim (f1 /* seq) = lim_left (f1,x0) by A1, A5, A6, A9, Def7; hence lim ((f1 (#) f2) /* seq) = (lim_left (f1,x0)) * (lim_left (f2,x0)) by A14, A12, A11, A13, SEQ_2:15; ::_thesis: verum end; hence f1 (#) f2 is_left_convergent_in x0 by A3, Def1; ::_thesis: lim_left ((f1 (#) f2),x0) = (lim_left (f1,x0)) * (lim_left (f2,x0)) hence lim_left ((f1 (#) f2),x0) = (lim_left (f1,x0)) * (lim_left (f2,x0)) by A4, Def7; ::_thesis: verum end; theorem :: LIMFUNC2:51 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & lim_left (f2,x0) <> 0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f1 / f2) ) ) holds ( f1 / f2 is_left_convergent_in x0 & lim_left ((f1 / f2),x0) = (lim_left (f1,x0)) / (lim_left (f2,x0)) ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & lim_left (f2,x0) <> 0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f1 / f2) ) ) holds ( f1 / f2 is_left_convergent_in x0 & lim_left ((f1 / f2),x0) = (lim_left (f1,x0)) / (lim_left (f2,x0)) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & lim_left (f2,x0) <> 0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f1 / f2) ) ) implies ( f1 / f2 is_left_convergent_in x0 & lim_left ((f1 / f2),x0) = (lim_left (f1,x0)) / (lim_left (f2,x0)) ) ) assume that A1: f1 is_left_convergent_in x0 and A2: f2 is_left_convergent_in x0 and A3: lim_left (f2,x0) <> 0 and A4: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f1 / f2) ) ; ::_thesis: ( f1 / f2 is_left_convergent_in x0 & lim_left ((f1 / f2),x0) = (lim_left (f1,x0)) / (lim_left (f2,x0)) ) A5: now__::_thesis:_for_r_being_Real_st_r_<_x0_holds_ ex_g_being_Real_st_ (_r_<_g_&_g_<_x0_&_g_in_dom_f2_&_f2_._g_<>_0_) let r be Real; ::_thesis: ( r < x0 implies ex g being Real st ( r < g & g < x0 & g in dom f2 & f2 . g <> 0 ) ) assume r < x0 ; ::_thesis: ex g being Real st ( r < g & g < x0 & g in dom f2 & f2 . g <> 0 ) then consider g being Real such that A6: r < g and A7: g < x0 and A8: g in dom (f1 / f2) by A4; take g = g; ::_thesis: ( r < g & g < x0 & g in dom f2 & f2 . g <> 0 ) thus ( r < g & g < x0 ) by A6, A7; ::_thesis: ( g in dom f2 & f2 . g <> 0 ) dom (f1 / f2) = (dom f1) /\ ((dom f2) \ (f2 " {0})) by RFUNCT_1:def_1; then A9: g in (dom f2) \ (f2 " {0}) by A8, XBOOLE_0:def_4; then A10: not g in f2 " {0} by XBOOLE_0:def_5; g in dom f2 by A9, XBOOLE_0:def_5; then not f2 . g in {0} by A10, FUNCT_1:def_7; hence ( g in dom f2 & f2 . g <> 0 ) by A9, TARSKI:def_1, XBOOLE_0:def_5; ::_thesis: verum end; then A11: f2 ^ is_left_convergent_in x0 by A2, A3, Th49; A12: f1 / f2 = f1 (#) (f2 ^) by RFUNCT_1:31; hence f1 / f2 is_left_convergent_in x0 by A1, A4, A11, Th50; ::_thesis: lim_left ((f1 / f2),x0) = (lim_left (f1,x0)) / (lim_left (f2,x0)) lim_left ((f2 ^),x0) = (lim_left (f2,x0)) " by A2, A3, A5, Th49; hence lim_left ((f1 / f2),x0) = (lim_left (f1,x0)) * ((lim_left (f2,x0)) ") by A1, A4, A12, A11, Th50 .= (lim_left (f1,x0)) / (lim_left (f2,x0)) by XCMPLX_0:def_9 ; ::_thesis: verum end; theorem Th52: :: LIMFUNC2:52 for x0, r being Real for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 holds ( r (#) f is_right_convergent_in x0 & lim_right ((r (#) f),x0) = r * (lim_right (f,x0)) ) proof let x0, r be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 holds ( r (#) f is_right_convergent_in x0 & lim_right ((r (#) f),x0) = r * (lim_right (f,x0)) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_right_convergent_in x0 implies ( r (#) f is_right_convergent_in x0 & lim_right ((r (#) f),x0) = r * (lim_right (f,x0)) ) ) assume A1: f is_right_convergent_in x0 ; ::_thesis: ( r (#) f is_right_convergent_in x0 & lim_right ((r (#) f),x0) = r * (lim_right (f,x0)) ) A2: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(r_(#)_f))_/\_(right_open_halfline_x0)_holds_ (_(r_(#)_f)_/*_seq_is_convergent_&_lim_((r_(#)_f)_/*_seq)_=_r_*_(lim_right_(f,x0))_) let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (r (#) f)) /\ (right_open_halfline x0) implies ( (r (#) f) /* seq is convergent & lim ((r (#) f) /* seq) = r * (lim_right (f,x0)) ) ) assume that A3: seq is convergent and A4: lim seq = x0 and A5: rng seq c= (dom (r (#) f)) /\ (right_open_halfline x0) ; ::_thesis: ( (r (#) f) /* seq is convergent & lim ((r (#) f) /* seq) = r * (lim_right (f,x0)) ) A6: rng seq c= (dom f) /\ (right_open_halfline x0) by A5, VALUED_1:def_5; A7: (dom f) /\ (right_open_halfline x0) c= dom f by XBOOLE_1:17; then A8: r (#) (f /* seq) = (r (#) f) /* seq by A6, RFUNCT_2:9, XBOOLE_1:1; lim_right (f,x0) = lim_right (f,x0) ; then A9: f /* seq is convergent by A1, A3, A4, A6, Def8; then r (#) (f /* seq) is convergent by SEQ_2:7; hence (r (#) f) /* seq is convergent by A6, A7, RFUNCT_2:9, XBOOLE_1:1; ::_thesis: lim ((r (#) f) /* seq) = r * (lim_right (f,x0)) lim (f /* seq) = lim_right (f,x0) by A1, A3, A4, A6, Def8; hence lim ((r (#) f) /* seq) = r * (lim_right (f,x0)) by A9, A8, SEQ_2:8; ::_thesis: verum end; now__::_thesis:_for_r1_being_Real_st_x0_<_r1_holds_ ex_g_being_Real_st_ (_g_<_r1_&_x0_<_g_&_g_in_dom_(r_(#)_f)_) let r1 be Real; ::_thesis: ( x0 < r1 implies ex g being Real st ( g < r1 & x0 < g & g in dom (r (#) f) ) ) assume x0 < r1 ; ::_thesis: ex g being Real st ( g < r1 & x0 < g & g in dom (r (#) f) ) then consider g being Real such that A10: g < r1 and A11: x0 < g and A12: g in dom f by A1, Def4; take g = g; ::_thesis: ( g < r1 & x0 < g & g in dom (r (#) f) ) thus ( g < r1 & x0 < g & g in dom (r (#) f) ) by A10, A11, A12, VALUED_1:def_5; ::_thesis: verum end; hence r (#) f is_right_convergent_in x0 by A2, Def4; ::_thesis: lim_right ((r (#) f),x0) = r * (lim_right (f,x0)) hence lim_right ((r (#) f),x0) = r * (lim_right (f,x0)) by A2, Def8; ::_thesis: verum end; theorem Th53: :: LIMFUNC2:53 for x0 being Real for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 holds ( - f is_right_convergent_in x0 & lim_right ((- f),x0) = - (lim_right (f,x0)) ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 holds ( - f is_right_convergent_in x0 & lim_right ((- f),x0) = - (lim_right (f,x0)) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_right_convergent_in x0 implies ( - f is_right_convergent_in x0 & lim_right ((- f),x0) = - (lim_right (f,x0)) ) ) assume A1: f is_right_convergent_in x0 ; ::_thesis: ( - f is_right_convergent_in x0 & lim_right ((- f),x0) = - (lim_right (f,x0)) ) (- 1) (#) f = - f ; hence - f is_right_convergent_in x0 by A1, Th52; ::_thesis: lim_right ((- f),x0) = - (lim_right (f,x0)) thus lim_right ((- f),x0) = (- 1) * (lim_right (f,x0)) by A1, Th52 .= - (lim_right (f,x0)) ; ::_thesis: verum end; theorem Th54: :: LIMFUNC2:54 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f1 + f2) ) ) holds ( f1 + f2 is_right_convergent_in x0 & lim_right ((f1 + f2),x0) = (lim_right (f1,x0)) + (lim_right (f2,x0)) ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f1 + f2) ) ) holds ( f1 + f2 is_right_convergent_in x0 & lim_right ((f1 + f2),x0) = (lim_right (f1,x0)) + (lim_right (f2,x0)) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f1 + f2) ) ) implies ( f1 + f2 is_right_convergent_in x0 & lim_right ((f1 + f2),x0) = (lim_right (f1,x0)) + (lim_right (f2,x0)) ) ) assume that A1: f1 is_right_convergent_in x0 and A2: f2 is_right_convergent_in x0 and A3: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f1 + f2) ) ; ::_thesis: ( f1 + f2 is_right_convergent_in x0 & lim_right ((f1 + f2),x0) = (lim_right (f1,x0)) + (lim_right (f2,x0)) ) A4: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(f1_+_f2))_/\_(right_open_halfline_x0)_holds_ (_(f1_+_f2)_/*_seq_is_convergent_&_lim_((f1_+_f2)_/*_seq)_=_(lim_right_(f1,x0))_+_(lim_right_(f2,x0))_) let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f1 + f2)) /\ (right_open_halfline x0) implies ( (f1 + f2) /* seq is convergent & lim ((f1 + f2) /* seq) = (lim_right (f1,x0)) + (lim_right (f2,x0)) ) ) assume that A5: seq is convergent and A6: lim seq = x0 and A7: rng seq c= (dom (f1 + f2)) /\ (right_open_halfline x0) ; ::_thesis: ( (f1 + f2) /* seq is convergent & lim ((f1 + f2) /* seq) = (lim_right (f1,x0)) + (lim_right (f2,x0)) ) A8: dom (f1 + f2) = (dom f1) /\ (dom f2) by A7, Lm4; A9: rng seq c= (dom f1) /\ (right_open_halfline x0) by A7, Lm4; A10: rng seq c= (dom f2) /\ (right_open_halfline x0) by A7, Lm4; then A11: lim (f2 /* seq) = lim_right (f2,x0) by A2, A5, A6, Def8; lim_right (f2,x0) = lim_right (f2,x0) ; then A12: f2 /* seq is convergent by A2, A5, A6, A10, Def8; rng seq c= dom (f1 + f2) by A7, Lm4; then A13: (f1 /* seq) + (f2 /* seq) = (f1 + f2) /* seq by A8, RFUNCT_2:8; lim_right (f1,x0) = lim_right (f1,x0) ; then A14: f1 /* seq is convergent by A1, A5, A6, A9, Def8; hence (f1 + f2) /* seq is convergent by A12, A13, SEQ_2:5; ::_thesis: lim ((f1 + f2) /* seq) = (lim_right (f1,x0)) + (lim_right (f2,x0)) lim (f1 /* seq) = lim_right (f1,x0) by A1, A5, A6, A9, Def8; hence lim ((f1 + f2) /* seq) = (lim_right (f1,x0)) + (lim_right (f2,x0)) by A14, A12, A11, A13, SEQ_2:6; ::_thesis: verum end; hence f1 + f2 is_right_convergent_in x0 by A3, Def4; ::_thesis: lim_right ((f1 + f2),x0) = (lim_right (f1,x0)) + (lim_right (f2,x0)) hence lim_right ((f1 + f2),x0) = (lim_right (f1,x0)) + (lim_right (f2,x0)) by A4, Def8; ::_thesis: verum end; theorem :: LIMFUNC2:55 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f1 - f2) ) ) holds ( f1 - f2 is_right_convergent_in x0 & lim_right ((f1 - f2),x0) = (lim_right (f1,x0)) - (lim_right (f2,x0)) ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f1 - f2) ) ) holds ( f1 - f2 is_right_convergent_in x0 & lim_right ((f1 - f2),x0) = (lim_right (f1,x0)) - (lim_right (f2,x0)) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f1 - f2) ) ) implies ( f1 - f2 is_right_convergent_in x0 & lim_right ((f1 - f2),x0) = (lim_right (f1,x0)) - (lim_right (f2,x0)) ) ) assume that A1: f1 is_right_convergent_in x0 and A2: f2 is_right_convergent_in x0 and A3: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f1 - f2) ) ; ::_thesis: ( f1 - f2 is_right_convergent_in x0 & lim_right ((f1 - f2),x0) = (lim_right (f1,x0)) - (lim_right (f2,x0)) ) A4: - f2 is_right_convergent_in x0 by A2, Th53; hence f1 - f2 is_right_convergent_in x0 by A1, A3, Th54; ::_thesis: lim_right ((f1 - f2),x0) = (lim_right (f1,x0)) - (lim_right (f2,x0)) thus lim_right ((f1 - f2),x0) = (lim_right (f1,x0)) + (lim_right ((- f2),x0)) by A1, A3, A4, Th54 .= (lim_right (f1,x0)) + (- (lim_right (f2,x0))) by A2, Th53 .= (lim_right (f1,x0)) - (lim_right (f2,x0)) ; ::_thesis: verum end; theorem :: LIMFUNC2:56 for x0 being Real for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 & f " {0} = {} & lim_right (f,x0) <> 0 holds ( f ^ is_right_convergent_in x0 & lim_right ((f ^),x0) = (lim_right (f,x0)) " ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 & f " {0} = {} & lim_right (f,x0) <> 0 holds ( f ^ is_right_convergent_in x0 & lim_right ((f ^),x0) = (lim_right (f,x0)) " ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_right_convergent_in x0 & f " {0} = {} & lim_right (f,x0) <> 0 implies ( f ^ is_right_convergent_in x0 & lim_right ((f ^),x0) = (lim_right (f,x0)) " ) ) assume that A1: f is_right_convergent_in x0 and A2: f " {0} = {} and A3: lim_right (f,x0) <> 0 ; ::_thesis: ( f ^ is_right_convergent_in x0 & lim_right ((f ^),x0) = (lim_right (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_^))_/\_(right_open_halfline_x0)_holds_ (_(f_^)_/*_seq_is_convergent_&_lim_((f_^)_/*_seq)_=_(lim_right_(f,x0))_"_) let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f ^)) /\ (right_open_halfline x0) implies ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = (lim_right (f,x0)) " ) ) assume that A6: seq is convergent and A7: lim seq = x0 and A8: rng seq c= (dom (f ^)) /\ (right_open_halfline x0) ; ::_thesis: ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = (lim_right (f,x0)) " ) A9: lim (f /* seq) = lim_right (f,x0) by A1, A4, A6, A7, A8, Def8; A10: (dom f) /\ (right_open_halfline x0) c= dom f by XBOOLE_1:17; then A11: rng seq c= dom f by A4, A8, XBOOLE_1:1; A12: f /* seq is convergent by A1, A3, A4, A6, A7, A8, Def8; A13: (f /* seq) " = (f ^) /* seq by A4, A8, A10, RFUNCT_2:12, XBOOLE_1:1; hence (f ^) /* seq is convergent by A3, A4, A12, A9, A11, RFUNCT_2:11, SEQ_2:21; ::_thesis: lim ((f ^) /* seq) = (lim_right (f,x0)) " thus lim ((f ^) /* seq) = (lim_right (f,x0)) " by A3, A4, A12, A9, A11, A13, RFUNCT_2:11, SEQ_2:22; ::_thesis: verum end; for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f ^) ) by A1, A4, Def4; hence f ^ is_right_convergent_in x0 by A5, Def4; ::_thesis: lim_right ((f ^),x0) = (lim_right (f,x0)) " hence lim_right ((f ^),x0) = (lim_right (f,x0)) " by A5, Def8; ::_thesis: verum end; theorem :: LIMFUNC2:57 for x0 being Real for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 holds ( abs f is_right_convergent_in x0 & lim_right ((abs f),x0) = abs (lim_right (f,x0)) ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 holds ( abs f is_right_convergent_in x0 & lim_right ((abs f),x0) = abs (lim_right (f,x0)) ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_right_convergent_in x0 implies ( abs f is_right_convergent_in x0 & lim_right ((abs f),x0) = abs (lim_right (f,x0)) ) ) assume A1: f is_right_convergent_in x0 ; ::_thesis: ( abs f is_right_convergent_in x0 & lim_right ((abs f),x0) = abs (lim_right (f,x0)) ) A2: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(abs_f))_/\_(right_open_halfline_x0)_holds_ (_(abs_f)_/*_seq_is_convergent_&_lim_((abs_f)_/*_seq)_=_abs_(lim_right_(f,x0))_) let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (abs f)) /\ (right_open_halfline x0) implies ( (abs f) /* seq is convergent & lim ((abs f) /* seq) = abs (lim_right (f,x0)) ) ) assume that A3: seq is convergent and A4: lim seq = x0 and A5: rng seq c= (dom (abs f)) /\ (right_open_halfline x0) ; ::_thesis: ( (abs f) /* seq is convergent & lim ((abs f) /* seq) = abs (lim_right (f,x0)) ) A6: rng seq c= (dom f) /\ (right_open_halfline x0) by A5, VALUED_1:def_11; (dom f) /\ (right_open_halfline x0) c= dom f by XBOOLE_1:17; then rng seq c= dom f by A6, XBOOLE_1:1; then A7: abs (f /* seq) = (abs f) /* seq by RFUNCT_2:10; lim_right (f,x0) = lim_right (f,x0) ; then A8: f /* seq is convergent by A1, A3, A4, A6, Def8; hence (abs f) /* seq is convergent by A7; ::_thesis: lim ((abs f) /* seq) = abs (lim_right (f,x0)) lim (f /* seq) = lim_right (f,x0) by A1, A3, A4, A6, Def8; hence lim ((abs f) /* seq) = abs (lim_right (f,x0)) by A8, A7, SEQ_4:14; ::_thesis: verum end; now__::_thesis:_for_r_being_Real_st_x0_<_r_holds_ ex_g_being_Real_st_ (_g_<_r_&_x0_<_g_&_g_in_dom_(abs_f)_) let r be Real; ::_thesis: ( x0 < r implies ex g being Real st ( g < r & x0 < g & g in dom (abs f) ) ) assume x0 < r ; ::_thesis: ex g being Real st ( g < r & x0 < g & g in dom (abs f) ) then consider g being Real such that A9: g < r and A10: x0 < g and A11: g in dom f by A1, Def4; take g = g; ::_thesis: ( g < r & x0 < g & g in dom (abs f) ) thus ( g < r & x0 < g & g in dom (abs f) ) by A9, A10, A11, VALUED_1:def_11; ::_thesis: verum end; hence abs f is_right_convergent_in x0 by A2, Def4; ::_thesis: lim_right ((abs f),x0) = abs (lim_right (f,x0)) hence lim_right ((abs f),x0) = abs (lim_right (f,x0)) by A2, Def8; ::_thesis: verum end; theorem Th58: :: LIMFUNC2:58 for x0 being Real for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 & lim_right (f,x0) <> 0 & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f & f . g <> 0 ) ) holds ( f ^ is_right_convergent_in x0 & lim_right ((f ^),x0) = (lim_right (f,x0)) " ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 & lim_right (f,x0) <> 0 & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f & f . g <> 0 ) ) holds ( f ^ is_right_convergent_in x0 & lim_right ((f ^),x0) = (lim_right (f,x0)) " ) let f be PartFunc of REAL,REAL; ::_thesis: ( f is_right_convergent_in x0 & lim_right (f,x0) <> 0 & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f & f . g <> 0 ) ) implies ( f ^ is_right_convergent_in x0 & lim_right ((f ^),x0) = (lim_right (f,x0)) " ) ) assume that A1: f is_right_convergent_in x0 and A2: lim_right (f,x0) <> 0 and A3: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f & f . g <> 0 ) ; ::_thesis: ( f ^ is_right_convergent_in x0 & lim_right ((f ^),x0) = (lim_right (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_^))_/\_(right_open_halfline_x0)_holds_ (_(f_^)_/*_seq_is_convergent_&_lim_((f_^)_/*_seq)_=_(lim_right_(f,x0))_"_) A6: dom (f ^) c= dom f by A4, XBOOLE_1:36; let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f ^)) /\ (right_open_halfline x0) implies ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = (lim_right (f,x0)) " ) ) assume that A7: seq is convergent and A8: lim seq = x0 and A9: rng seq c= (dom (f ^)) /\ (right_open_halfline x0) ; ::_thesis: ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = (lim_right (f,x0)) " ) A10: (dom (f ^)) /\ (right_open_halfline x0) c= dom (f ^) by XBOOLE_1:17; then A11: f /* seq is non-zero by A9, RFUNCT_2:11, XBOOLE_1:1; rng seq c= dom (f ^) by A9, A10, XBOOLE_1:1; then A12: rng seq c= dom f by A6, XBOOLE_1:1; (dom (f ^)) /\ (right_open_halfline x0) c= right_open_halfline x0 by XBOOLE_1:17; then rng seq c= right_open_halfline x0 by A9, XBOOLE_1:1; then A13: rng seq c= (dom f) /\ (right_open_halfline x0) by A12, XBOOLE_1:19; then A14: lim (f /* seq) = lim_right (f,x0) by A1, A7, A8, Def8; A15: (f /* seq) " = (f ^) /* seq by A9, A10, RFUNCT_2:12, XBOOLE_1:1; A16: f /* seq is convergent by A1, A2, A7, A8, A13, Def8; hence (f ^) /* seq is convergent by A2, A14, A11, A15, SEQ_2:21; ::_thesis: lim ((f ^) /* seq) = (lim_right (f,x0)) " thus lim ((f ^) /* seq) = (lim_right (f,x0)) " by A2, A16, A14, A11, A15, SEQ_2:22; ::_thesis: verum end; now__::_thesis:_for_r_being_Real_st_x0_<_r_holds_ ex_g_being_Real_st_ (_g_<_r_&_x0_<_g_&_g_in_dom_(f_^)_) 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 g being Real such that A17: g < r and A18: x0 < g and A19: g in dom f and A20: f . g <> 0 by A3; take g = g; ::_thesis: ( g < r & x0 < g & g in dom (f ^) ) not f . g in {0} by A20, TARSKI:def_1; then not g in f " {0} by FUNCT_1:def_7; hence ( g < r & x0 < g & g in dom (f ^) ) by A4, A17, A18, A19, XBOOLE_0:def_5; ::_thesis: verum end; hence f ^ is_right_convergent_in x0 by A5, Def4; ::_thesis: lim_right ((f ^),x0) = (lim_right (f,x0)) " hence lim_right ((f ^),x0) = (lim_right (f,x0)) " by A5, Def8; ::_thesis: verum end; theorem Th59: :: LIMFUNC2:59 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f1 (#) f2) ) ) holds ( f1 (#) f2 is_right_convergent_in x0 & lim_right ((f1 (#) f2),x0) = (lim_right (f1,x0)) * (lim_right (f2,x0)) ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f1 (#) f2) ) ) holds ( f1 (#) f2 is_right_convergent_in x0 & lim_right ((f1 (#) f2),x0) = (lim_right (f1,x0)) * (lim_right (f2,x0)) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f1 (#) f2) ) ) implies ( f1 (#) f2 is_right_convergent_in x0 & lim_right ((f1 (#) f2),x0) = (lim_right (f1,x0)) * (lim_right (f2,x0)) ) ) assume that A1: f1 is_right_convergent_in x0 and A2: f2 is_right_convergent_in x0 and A3: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f1 (#) f2) ) ; ::_thesis: ( f1 (#) f2 is_right_convergent_in x0 & lim_right ((f1 (#) f2),x0) = (lim_right (f1,x0)) * (lim_right (f2,x0)) ) A4: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(f1_(#)_f2))_/\_(right_open_halfline_x0)_holds_ (_(f1_(#)_f2)_/*_seq_is_convergent_&_lim_((f1_(#)_f2)_/*_seq)_=_(lim_right_(f1,x0))_*_(lim_right_(f2,x0))_) let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f1 (#) f2)) /\ (right_open_halfline x0) implies ( (f1 (#) f2) /* seq is convergent & lim ((f1 (#) f2) /* seq) = (lim_right (f1,x0)) * (lim_right (f2,x0)) ) ) assume that A5: seq is convergent and A6: lim seq = x0 and A7: rng seq c= (dom (f1 (#) f2)) /\ (right_open_halfline x0) ; ::_thesis: ( (f1 (#) f2) /* seq is convergent & lim ((f1 (#) f2) /* seq) = (lim_right (f1,x0)) * (lim_right (f2,x0)) ) A8: dom (f1 (#) f2) = (dom f1) /\ (dom f2) by A7, Lm2; A9: rng seq c= (dom f1) /\ (right_open_halfline x0) by A7, Lm2; A10: rng seq c= (dom f2) /\ (right_open_halfline x0) by A7, Lm2; then A11: lim (f2 /* seq) = lim_right (f2,x0) by A2, A5, A6, Def8; lim_right (f2,x0) = lim_right (f2,x0) ; then A12: f2 /* seq is convergent by A2, A5, A6, A10, Def8; rng seq c= dom (f1 (#) f2) by A7, Lm2; then A13: (f1 /* seq) (#) (f2 /* seq) = (f1 (#) f2) /* seq by A8, RFUNCT_2:8; lim_right (f1,x0) = lim_right (f1,x0) ; then A14: f1 /* seq is convergent by A1, A5, A6, A9, Def8; hence (f1 (#) f2) /* seq is convergent by A12, A13, SEQ_2:14; ::_thesis: lim ((f1 (#) f2) /* seq) = (lim_right (f1,x0)) * (lim_right (f2,x0)) lim (f1 /* seq) = lim_right (f1,x0) by A1, A5, A6, A9, Def8; hence lim ((f1 (#) f2) /* seq) = (lim_right (f1,x0)) * (lim_right (f2,x0)) by A14, A12, A11, A13, SEQ_2:15; ::_thesis: verum end; hence f1 (#) f2 is_right_convergent_in x0 by A3, Def4; ::_thesis: lim_right ((f1 (#) f2),x0) = (lim_right (f1,x0)) * (lim_right (f2,x0)) hence lim_right ((f1 (#) f2),x0) = (lim_right (f1,x0)) * (lim_right (f2,x0)) by A4, Def8; ::_thesis: verum end; theorem :: LIMFUNC2:60 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & lim_right (f2,x0) <> 0 & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f1 / f2) ) ) holds ( f1 / f2 is_right_convergent_in x0 & lim_right ((f1 / f2),x0) = (lim_right (f1,x0)) / (lim_right (f2,x0)) ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & lim_right (f2,x0) <> 0 & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f1 / f2) ) ) holds ( f1 / f2 is_right_convergent_in x0 & lim_right ((f1 / f2),x0) = (lim_right (f1,x0)) / (lim_right (f2,x0)) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & lim_right (f2,x0) <> 0 & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f1 / f2) ) ) implies ( f1 / f2 is_right_convergent_in x0 & lim_right ((f1 / f2),x0) = (lim_right (f1,x0)) / (lim_right (f2,x0)) ) ) assume that A1: f1 is_right_convergent_in x0 and A2: f2 is_right_convergent_in x0 and A3: lim_right (f2,x0) <> 0 and A4: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f1 / f2) ) ; ::_thesis: ( f1 / f2 is_right_convergent_in x0 & lim_right ((f1 / f2),x0) = (lim_right (f1,x0)) / (lim_right (f2,x0)) ) A5: now__::_thesis:_for_r_being_Real_st_x0_<_r_holds_ ex_g_being_Real_st_ (_g_<_r_&_x0_<_g_&_g_in_dom_f2_&_f2_._g_<>_0_) let r be Real; ::_thesis: ( x0 < r implies ex g being Real st ( g < r & x0 < g & g in dom f2 & f2 . g <> 0 ) ) assume x0 < r ; ::_thesis: ex g being Real st ( g < r & x0 < g & g in dom f2 & f2 . g <> 0 ) then consider g being Real such that A6: g < r and A7: x0 < g and A8: g in dom (f1 / f2) by A4; take g = g; ::_thesis: ( g < r & x0 < g & g in dom f2 & f2 . g <> 0 ) thus ( g < r & x0 < g ) by A6, A7; ::_thesis: ( g in dom f2 & f2 . g <> 0 ) dom (f1 / f2) = (dom f1) /\ ((dom f2) \ (f2 " {0})) by RFUNCT_1:def_1; then A9: g in (dom f2) \ (f2 " {0}) by A8, XBOOLE_0:def_4; then A10: not g in f2 " {0} by XBOOLE_0:def_5; g in dom f2 by A9, XBOOLE_0:def_5; then not f2 . g in {0} by A10, FUNCT_1:def_7; hence ( g in dom f2 & f2 . g <> 0 ) by A9, TARSKI:def_1, XBOOLE_0:def_5; ::_thesis: verum end; then A11: f2 ^ is_right_convergent_in x0 by A2, A3, Th58; A12: f1 / f2 = f1 (#) (f2 ^) by RFUNCT_1:31; hence f1 / f2 is_right_convergent_in x0 by A1, A4, A11, Th59; ::_thesis: lim_right ((f1 / f2),x0) = (lim_right (f1,x0)) / (lim_right (f2,x0)) lim_right ((f2 ^),x0) = (lim_right (f2,x0)) " by A2, A3, A5, Th58; hence lim_right ((f1 / f2),x0) = (lim_right (f1,x0)) * ((lim_right (f2,x0)) ") by A1, A4, A12, A11, Th59 .= (lim_right (f1,x0)) / (lim_right (f2,x0)) by XCMPLX_0:def_9 ; ::_thesis: verum end; theorem :: LIMFUNC2:61 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & lim_left (f1,x0) = 0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f1 (#) f2) ) ) & ex r being Real st ( 0 < r & f2 | ].(x0 - r),x0.[ is bounded ) holds ( f1 (#) f2 is_left_convergent_in x0 & lim_left ((f1 (#) f2),x0) = 0 ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & lim_left (f1,x0) = 0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f1 (#) f2) ) ) & ex r being Real st ( 0 < r & f2 | ].(x0 - r),x0.[ is bounded ) holds ( f1 (#) f2 is_left_convergent_in x0 & lim_left ((f1 (#) f2),x0) = 0 ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_convergent_in x0 & lim_left (f1,x0) = 0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f1 (#) f2) ) ) & ex r being Real st ( 0 < r & f2 | ].(x0 - r),x0.[ is bounded ) implies ( f1 (#) f2 is_left_convergent_in x0 & lim_left ((f1 (#) f2),x0) = 0 ) ) assume that A1: f1 is_left_convergent_in x0 and A2: lim_left (f1,x0) = 0 and A3: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f1 (#) f2) ) ; ::_thesis: ( for r being Real holds ( not 0 < r or not f2 | ].(x0 - r),x0.[ is bounded ) or ( f1 (#) f2 is_left_convergent_in x0 & lim_left ((f1 (#) f2),x0) = 0 ) ) given r being Real such that A4: 0 < r and A5: f2 | ].(x0 - r),x0.[ is bounded ; ::_thesis: ( f1 (#) f2 is_left_convergent_in x0 & lim_left ((f1 (#) f2),x0) = 0 ) consider g being real number such that A6: for r1 being set st r1 in ].(x0 - r),x0.[ /\ (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))_/\_(left_open_halfline_x0)_holds_ (_(f1_(#)_f2)_/*_s_is_convergent_&_lim_((f1_(#)_f2)_/*_s)_=_0_) set L = left_open_halfline x0; let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f1 (#) f2)) /\ (left_open_halfline 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)) /\ (left_open_halfline x0) ; ::_thesis: ( (f1 (#) f2) /* s is convergent & lim ((f1 (#) f2) /* s) = 0 ) x0 - r < x0 by A4, Lm1; then consider k being Element of NAT such that A11: for n being Element of NAT st k <= n holds x0 - r < s . n by A8, A9, Th1; A12: rng (s ^\ k) c= rng s by VALUED_0:21; A13: 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 A13, A12, XBOOLE_1:1; then A14: (f1 /* (s ^\ k)) (#) (f2 /* (s ^\ k)) = (f1 (#) f2) /* (s ^\ k) by RFUNCT_2:8 .= ((f1 (#) f2) /* s) ^\ k by A13, VALUED_0:27 ; rng s c= (dom f1) /\ (left_open_halfline x0) by A10, Lm2; then A15: rng (s ^\ k) c= (dom f1) /\ (left_open_halfline x0) by A12, XBOOLE_1:1; A16: lim (s ^\ k) = x0 by A8, A9, SEQ_4:20; then A17: f1 /* (s ^\ k) is convergent by A1, A2, A8, A15, Def7; rng s c= left_open_halfline x0 by A10, Lm2; then A18: rng (s ^\ k) c= left_open_halfline x0 by A12, XBOOLE_1:1; A19: rng s c= dom f2 by A10, Lm2; then A20: 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 x0 - r < s . (n + k) by A11, NAT_1:12; then A21: x0 - r < (s ^\ k) . n by NAT_1:def_3; A22: (s ^\ k) . n in rng (s ^\ k) by VALUED_0:28; then (s ^\ k) . n in left_open_halfline x0 by A18; then (s ^\ k) . n in { g1 where g1 is Real : g1 < x0 } by XXREAL_1:229; then ex g1 being Real st ( g1 = (s ^\ k) . n & g1 < x0 ) ; then (s ^\ k) . n in { g2 where g2 is Real : ( x0 - r < g2 & g2 < x0 ) } by A21; then (s ^\ k) . n in ].(x0 - r),x0.[ by RCOMP_1:def_2; then (s ^\ k) . n in ].(x0 - r),x0.[ /\ (dom f2) by A20, A22, XBOOLE_0:def_4; then abs (f2 . ((s ^\ k) . n)) <= g by A6; then A23: abs ((f2 /* (s ^\ k)) . n) <= g by A19, 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 A23, XXREAL_0:2; ::_thesis: verum end; then A24: f2 /* (s ^\ k) is bounded by SEQ_2:3; A25: lim (f1 /* (s ^\ k)) = 0 by A1, A2, A8, A16, A15, Def7; then A26: (f1 /* (s ^\ k)) (#) (f2 /* (s ^\ k)) is convergent by A17, A24, SEQ_2:25; hence (f1 (#) f2) /* s is convergent by A14, SEQ_4:21; ::_thesis: lim ((f1 (#) f2) /* s) = 0 lim ((f1 /* (s ^\ k)) (#) (f2 /* (s ^\ k))) = 0 by A17, A25, A24, SEQ_2:26; hence lim ((f1 (#) f2) /* s) = 0 by A26, A14, SEQ_4:22; ::_thesis: verum end; hence f1 (#) f2 is_left_convergent_in x0 by A3, Def1; ::_thesis: lim_left ((f1 (#) f2),x0) = 0 hence lim_left ((f1 (#) f2),x0) = 0 by A7, Def7; ::_thesis: verum end; theorem :: LIMFUNC2:62 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & lim_right (f1,x0) = 0 & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f1 (#) f2) ) ) & ex r being Real st ( 0 < r & f2 | ].x0,(x0 + r).[ is bounded ) holds ( f1 (#) f2 is_right_convergent_in x0 & lim_right ((f1 (#) f2),x0) = 0 ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & lim_right (f1,x0) = 0 & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f1 (#) f2) ) ) & ex r being Real st ( 0 < r & f2 | ].x0,(x0 + r).[ is bounded ) holds ( f1 (#) f2 is_right_convergent_in x0 & lim_right ((f1 (#) f2),x0) = 0 ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_right_convergent_in x0 & lim_right (f1,x0) = 0 & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f1 (#) f2) ) ) & ex r being Real st ( 0 < r & f2 | ].x0,(x0 + r).[ is bounded ) implies ( f1 (#) f2 is_right_convergent_in x0 & lim_right ((f1 (#) f2),x0) = 0 ) ) assume that A1: f1 is_right_convergent_in x0 and A2: lim_right (f1,x0) = 0 and A3: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f1 (#) f2) ) ; ::_thesis: ( for r being Real holds ( not 0 < r or not f2 | ].x0,(x0 + r).[ is bounded ) or ( f1 (#) f2 is_right_convergent_in x0 & lim_right ((f1 (#) f2),x0) = 0 ) ) given r being Real such that A4: 0 < r and A5: f2 | ].x0,(x0 + r).[ is bounded ; ::_thesis: ( f1 (#) f2 is_right_convergent_in x0 & lim_right ((f1 (#) f2),x0) = 0 ) consider g being real number such that A6: for r1 being set st r1 in ].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))_/\_(right_open_halfline_x0)_holds_ (_(f1_(#)_f2)_/*_s_is_convergent_&_lim_((f1_(#)_f2)_/*_s)_=_0_) set L = right_open_halfline x0; let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f1 (#) f2)) /\ (right_open_halfline 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)) /\ (right_open_halfline x0) ; ::_thesis: ( (f1 (#) f2) /* s is convergent & lim ((f1 (#) f2) /* s) = 0 ) x0 < x0 + r by A4, Lm1; then consider k being Element of NAT such that A11: for n being Element of NAT st k <= n holds s . n < x0 + r by A8, A9, Th2; A12: rng (s ^\ k) c= rng s by VALUED_0:21; A13: 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 A13, A12, XBOOLE_1:1; then A14: (f1 /* (s ^\ k)) (#) (f2 /* (s ^\ k)) = (f1 (#) f2) /* (s ^\ k) by RFUNCT_2:8 .= ((f1 (#) f2) /* s) ^\ k by A13, VALUED_0:27 ; rng s c= (dom f1) /\ (right_open_halfline x0) by A10, Lm2; then A15: rng (s ^\ k) c= (dom f1) /\ (right_open_halfline x0) by A12, XBOOLE_1:1; A16: lim (s ^\ k) = x0 by A8, A9, SEQ_4:20; then A17: f1 /* (s ^\ k) is convergent by A1, A2, A8, A15, Def8; rng s c= right_open_halfline x0 by A10, Lm2; then A18: rng (s ^\ k) c= right_open_halfline x0 by A12, XBOOLE_1:1; A19: rng s c= dom f2 by A10, Lm2; then A20: 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 s . (n + k) < x0 + r by A11, NAT_1:12; then A21: (s ^\ k) . n < x0 + r by NAT_1:def_3; A22: (s ^\ k) . n in rng (s ^\ k) by VALUED_0:28; then (s ^\ k) . n in right_open_halfline x0 by A18; then (s ^\ k) . n in { g1 where g1 is Real : x0 < g1 } by XXREAL_1:230; then ex g1 being Real st ( g1 = (s ^\ k) . n & x0 < g1 ) ; then (s ^\ k) . n in { g2 where g2 is Real : ( x0 < g2 & g2 < x0 + r ) } by A21; then (s ^\ k) . n in ].x0,(x0 + r).[ by RCOMP_1:def_2; then (s ^\ k) . n in ].x0,(x0 + r).[ /\ (dom f2) by A20, A22, XBOOLE_0:def_4; then abs (f2 . ((s ^\ k) . n)) <= g by A6; then A23: abs ((f2 /* (s ^\ k)) . n) <= g by A19, 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 A23, XXREAL_0:2; ::_thesis: verum end; then A24: f2 /* (s ^\ k) is bounded by SEQ_2:3; A25: lim (f1 /* (s ^\ k)) = 0 by A1, A2, A8, A16, A15, Def8; then A26: (f1 /* (s ^\ k)) (#) (f2 /* (s ^\ k)) is convergent by A17, A24, SEQ_2:25; hence (f1 (#) f2) /* s is convergent by A14, SEQ_4:21; ::_thesis: lim ((f1 (#) f2) /* s) = 0 lim ((f1 /* (s ^\ k)) (#) (f2 /* (s ^\ k))) = 0 by A17, A25, A24, SEQ_2:26; hence lim ((f1 (#) f2) /* s) = 0 by A26, A14, SEQ_4:22; ::_thesis: verum end; hence f1 (#) f2 is_right_convergent_in x0 by A3, Def4; ::_thesis: lim_right ((f1 (#) f2),x0) = 0 hence lim_right ((f1 (#) f2),x0) = 0 by A7, Def8; ::_thesis: verum end; theorem Th63: :: LIMFUNC2:63 for x0 being Real for f1, f2, f being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & lim_left (f1,x0) = lim_left (f2,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds ( f1 . g <= f . g & f . g <= f2 . g ) ) & ( ( (dom f1) /\ ].(x0 - r),x0.[ c= (dom f2) /\ ].(x0 - r),x0.[ & (dom f) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ ) or ( (dom f2) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ & (dom f) /\ ].(x0 - r),x0.[ c= (dom f2) /\ ].(x0 - r),x0.[ ) ) ) holds ( f is_left_convergent_in x0 & lim_left (f,x0) = lim_left (f1,x0) ) proof let x0 be Real; ::_thesis: for f1, f2, f being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & lim_left (f1,x0) = lim_left (f2,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds ( f1 . g <= f . g & f . g <= f2 . g ) ) & ( ( (dom f1) /\ ].(x0 - r),x0.[ c= (dom f2) /\ ].(x0 - r),x0.[ & (dom f) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ ) or ( (dom f2) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ & (dom f) /\ ].(x0 - r),x0.[ c= (dom f2) /\ ].(x0 - r),x0.[ ) ) ) holds ( f is_left_convergent_in x0 & lim_left (f,x0) = lim_left (f1,x0) ) let f1, f2, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & lim_left (f1,x0) = lim_left (f2,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds ( f1 . g <= f . g & f . g <= f2 . g ) ) & ( ( (dom f1) /\ ].(x0 - r),x0.[ c= (dom f2) /\ ].(x0 - r),x0.[ & (dom f) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ ) or ( (dom f2) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ & (dom f) /\ ].(x0 - r),x0.[ c= (dom f2) /\ ].(x0 - r),x0.[ ) ) ) implies ( f is_left_convergent_in x0 & lim_left (f,x0) = lim_left (f1,x0) ) ) assume that A1: f1 is_left_convergent_in x0 and A2: f2 is_left_convergent_in x0 and A3: lim_left (f1,x0) = lim_left (f2,x0) and A4: 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 holds ( not 0 < r or ex g being Real st ( g in (dom f) /\ ].(x0 - r),x0.[ & not ( f1 . g <= f . g & f . g <= f2 . g ) ) or ( not ( (dom f1) /\ ].(x0 - r),x0.[ c= (dom f2) /\ ].(x0 - r),x0.[ & (dom f) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ ) & not ( (dom f2) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ & (dom f) /\ ].(x0 - r),x0.[ c= (dom f2) /\ ].(x0 - r),x0.[ ) ) ) or ( f is_left_convergent_in x0 & lim_left (f,x0) = lim_left (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.[ holds ( f1 . g <= f . g & f . g <= f2 . g ) and A7: ( ( (dom f1) /\ ].(x0 - r1),x0.[ c= (dom f2) /\ ].(x0 - r1),x0.[ & (dom f) /\ ].(x0 - r1),x0.[ c= (dom f1) /\ ].(x0 - r1),x0.[ ) or ( (dom f2) /\ ].(x0 - r1),x0.[ c= (dom f1) /\ ].(x0 - r1),x0.[ & (dom f) /\ ].(x0 - r1),x0.[ c= (dom f2) /\ ].(x0 - r1),x0.[ ) ) ; ::_thesis: ( f is_left_convergent_in x0 & lim_left (f,x0) = lim_left (f1,x0) ) now__::_thesis:_(_f_is_left_convergent_in_x0_&_f_is_left_convergent_in_x0_&_lim_left_(f,x0)_=_lim_left_(f1,x0)_) percases ( ( (dom f1) /\ ].(x0 - r1),x0.[ c= (dom f2) /\ ].(x0 - r1),x0.[ & (dom f) /\ ].(x0 - r1),x0.[ c= (dom f1) /\ ].(x0 - r1),x0.[ ) or ( (dom f2) /\ ].(x0 - r1),x0.[ c= (dom f1) /\ ].(x0 - r1),x0.[ & (dom f) /\ ].(x0 - r1),x0.[ c= (dom f2) /\ ].(x0 - r1),x0.[ ) ) by A7; supposeA8: ( (dom f1) /\ ].(x0 - r1),x0.[ c= (dom f2) /\ ].(x0 - r1),x0.[ & (dom f) /\ ].(x0 - r1),x0.[ c= (dom f1) /\ ].(x0 - r1),x0.[ ) ; ::_thesis: ( f is_left_convergent_in x0 & f is_left_convergent_in x0 & lim_left (f,x0) = lim_left (f1,x0) ) A9: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_f)_/\_(left_open_halfline_x0)_holds_ (_f_/*_seq_is_convergent_&_lim_(f_/*_seq)_=_lim_left_(f1,x0)_) let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (left_open_halfline x0) implies ( f /* seq is convergent & lim (f /* seq) = lim_left (f1,x0) ) ) assume that A10: seq is convergent and A11: lim seq = x0 and A12: rng seq c= (dom f) /\ (left_open_halfline x0) ; ::_thesis: ( f /* seq is convergent & lim (f /* seq) = lim_left (f1,x0) ) x0 - r1 < lim seq by A5, A11, Lm1; then consider k being Element of NAT such that A13: for n being Element of NAT st k <= n holds x0 - r1 < seq . n by A10, Th1; A14: rng (seq ^\ k) c= rng seq by VALUED_0:21; (dom f) /\ (left_open_halfline x0) c= left_open_halfline x0 by XBOOLE_1:17; then rng seq c= left_open_halfline x0 by A12, XBOOLE_1:1; then A15: rng (seq ^\ k) c= left_open_halfline x0 by A14, XBOOLE_1:1; now__::_thesis:_for_x_being_set_st_x_in_rng_(seq_^\_k)_holds_ x_in_].(x0_-_r1),x0.[ let x be set ; ::_thesis: ( x in rng (seq ^\ k) implies x in ].(x0 - r1),x0.[ ) assume A16: x in rng (seq ^\ k) ; ::_thesis: x in ].(x0 - r1),x0.[ then consider n being Element of NAT such that A17: x = (seq ^\ k) . n by FUNCT_2:113; (seq ^\ k) . n in left_open_halfline x0 by A15, A16, A17; then (seq ^\ k) . n in { g where g is Real : g < x0 } by XXREAL_1:229; then A18: ex g being Real st ( g = (seq ^\ k) . n & g < x0 ) ; x0 - r1 < seq . (n + k) by A13, NAT_1:12; then x0 - r1 < (seq ^\ k) . n by NAT_1:def_3; then x in { g1 where g1 is Real : ( x0 - r1 < g1 & g1 < x0 ) } by A17, A18; hence x in ].(x0 - r1),x0.[ by RCOMP_1:def_2; ::_thesis: verum end; then A19: rng (seq ^\ k) c= ].(x0 - r1),x0.[ by TARSKI:def_3; ].(x0 - r1),x0.[ c= left_open_halfline x0 by XXREAL_1:263; then A20: rng (seq ^\ k) c= left_open_halfline x0 by A19, XBOOLE_1:1; A21: (dom f) /\ (left_open_halfline x0) c= dom f by XBOOLE_1:17; then A22: rng seq c= dom f by A12, XBOOLE_1:1; then rng (seq ^\ k) c= dom f by A14, XBOOLE_1:1; then A23: rng (seq ^\ k) c= (dom f) /\ ].(x0 - r1),x0.[ by A19, XBOOLE_1:19; then A24: rng (seq ^\ k) c= (dom f1) /\ ].(x0 - r1),x0.[ by A8, XBOOLE_1:1; then A25: rng (seq ^\ k) c= (dom f2) /\ ].(x0 - r1),x0.[ by A8, XBOOLE_1:1; A26: lim (seq ^\ k) = x0 by A10, A11, SEQ_4:20; A27: (dom f2) /\ ].(x0 - r1),x0.[ c= dom f2 by XBOOLE_1:17; then rng (seq ^\ k) c= dom f2 by A25, XBOOLE_1:1; then A28: rng (seq ^\ k) c= (dom f2) /\ (left_open_halfline x0) by A20, XBOOLE_1:19; then A29: lim (f2 /* (seq ^\ k)) = lim_left (f2,x0) by A2, A10, A26, Def7; A30: (dom f1) /\ ].(x0 - r1),x0.[ c= dom f1 by XBOOLE_1:17; then rng (seq ^\ k) c= dom f1 by A24, XBOOLE_1:1; then A31: rng (seq ^\ k) c= (dom f1) /\ (left_open_halfline x0) by A20, XBOOLE_1:19; then A32: lim (f1 /* (seq ^\ k)) = lim_left (f1,x0) by A1, A10, A26, Def7; A33: now__::_thesis:_for_n_being_Element_of_NAT_holds_ (_(f1_/*_(seq_^\_k))_._n_<=_(f_/*_(seq_^\_k))_._n_&_(f_/*_(seq_^\_k))_._n_<=_(f2_/*_(seq_^\_k))_._n_) let n be Element of NAT ; ::_thesis: ( (f1 /* (seq ^\ k)) . n <= (f /* (seq ^\ k)) . n & (f /* (seq ^\ k)) . n <= (f2 /* (seq ^\ k)) . n ) A34: (seq ^\ k) . n in rng (seq ^\ k) by VALUED_0:28; then f . ((seq ^\ k) . n) <= f2 . ((seq ^\ k) . n) by A6, A23; then A35: (f /* (seq ^\ k)) . n <= f2 . ((seq ^\ k) . n) by A14, A22, FUNCT_2:108, XBOOLE_1:1; f1 . ((seq ^\ k) . n) <= f . ((seq ^\ k) . n) by A6, A23, A34; then f1 . ((seq ^\ k) . n) <= (f /* (seq ^\ k)) . n by A14, A22, FUNCT_2:108, XBOOLE_1:1; hence ( (f1 /* (seq ^\ k)) . n <= (f /* (seq ^\ k)) . n & (f /* (seq ^\ k)) . n <= (f2 /* (seq ^\ k)) . n ) by A30, A27, A24, A25, A35, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; A36: f2 /* (seq ^\ k) is convergent by A2, A3, A10, A26, A28, Def7; A37: f1 /* (seq ^\ k) is convergent by A1, A3, A10, A26, A31, Def7; then f /* (seq ^\ k) is convergent by A3, A32, A36, A29, A33, SEQ_2:19; then A38: (f /* seq) ^\ k is convergent by A12, A21, VALUED_0:27, XBOOLE_1:1; hence f /* seq is convergent by SEQ_4:21; ::_thesis: lim (f /* seq) = lim_left (f1,x0) lim (f /* (seq ^\ k)) = lim_left (f1,x0) by A3, A37, A32, A36, A29, A33, SEQ_2:20; then lim ((f /* seq) ^\ k) = lim_left (f1,x0) by A12, A21, VALUED_0:27, XBOOLE_1:1; hence lim (f /* seq) = lim_left (f1,x0) by A38, SEQ_4:22; ::_thesis: verum end; hence f is_left_convergent_in x0 by A4, Def1; ::_thesis: ( f is_left_convergent_in x0 & lim_left (f,x0) = lim_left (f1,x0) ) hence ( f is_left_convergent_in x0 & lim_left (f,x0) = lim_left (f1,x0) ) by A9, Def7; ::_thesis: verum end; supposeA39: ( (dom f2) /\ ].(x0 - r1),x0.[ c= (dom f1) /\ ].(x0 - r1),x0.[ & (dom f) /\ ].(x0 - r1),x0.[ c= (dom f2) /\ ].(x0 - r1),x0.[ ) ; ::_thesis: ( f is_left_convergent_in x0 & f is_left_convergent_in x0 & lim_left (f,x0) = lim_left (f1,x0) ) A40: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_f)_/\_(left_open_halfline_x0)_holds_ (_f_/*_seq_is_convergent_&_lim_(f_/*_seq)_=_lim_left_(f1,x0)_) let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (left_open_halfline x0) implies ( f /* seq is convergent & lim (f /* seq) = lim_left (f1,x0) ) ) assume that A41: seq is convergent and A42: lim seq = x0 and A43: rng seq c= (dom f) /\ (left_open_halfline x0) ; ::_thesis: ( f /* seq is convergent & lim (f /* seq) = lim_left (f1,x0) ) x0 - r1 < lim seq by A5, A42, Lm1; then consider k being Element of NAT such that A44: for n being Element of NAT st k <= n holds x0 - r1 < seq . n by A41, Th1; A45: rng (seq ^\ k) c= rng seq by VALUED_0:21; (dom f) /\ (left_open_halfline x0) c= left_open_halfline x0 by XBOOLE_1:17; then rng seq c= left_open_halfline x0 by A43, XBOOLE_1:1; then A46: rng (seq ^\ k) c= left_open_halfline x0 by A45, XBOOLE_1:1; now__::_thesis:_for_x_being_set_st_x_in_rng_(seq_^\_k)_holds_ x_in_].(x0_-_r1),x0.[ let x be set ; ::_thesis: ( x in rng (seq ^\ k) implies x in ].(x0 - r1),x0.[ ) assume A47: x in rng (seq ^\ k) ; ::_thesis: x in ].(x0 - r1),x0.[ then consider n being Element of NAT such that A48: x = (seq ^\ k) . n by FUNCT_2:113; (seq ^\ k) . n in left_open_halfline x0 by A46, A47, A48; then (seq ^\ k) . n in { g where g is Real : g < x0 } by XXREAL_1:229; then A49: ex g being Real st ( g = (seq ^\ k) . n & g < x0 ) ; x0 - r1 < seq . (n + k) by A44, NAT_1:12; then x0 - r1 < (seq ^\ k) . n by NAT_1:def_3; then x in { g1 where g1 is Real : ( x0 - r1 < g1 & g1 < x0 ) } by A48, A49; hence x in ].(x0 - r1),x0.[ by RCOMP_1:def_2; ::_thesis: verum end; then A50: rng (seq ^\ k) c= ].(x0 - r1),x0.[ by TARSKI:def_3; ].(x0 - r1),x0.[ c= left_open_halfline x0 by XXREAL_1:263; then A51: rng (seq ^\ k) c= left_open_halfline x0 by A50, XBOOLE_1:1; A52: (dom f) /\ (left_open_halfline x0) c= dom f by XBOOLE_1:17; then A53: rng seq c= dom f by A43, XBOOLE_1:1; then rng (seq ^\ k) c= dom f by A45, XBOOLE_1:1; then A54: rng (seq ^\ k) c= (dom f) /\ ].(x0 - r1),x0.[ by A50, XBOOLE_1:19; then A55: rng (seq ^\ k) c= (dom f2) /\ ].(x0 - r1),x0.[ by A39, XBOOLE_1:1; then A56: rng (seq ^\ k) c= (dom f1) /\ ].(x0 - r1),x0.[ by A39, XBOOLE_1:1; A57: lim (seq ^\ k) = x0 by A41, A42, SEQ_4:20; A58: (dom f2) /\ ].(x0 - r1),x0.[ c= dom f2 by XBOOLE_1:17; then rng (seq ^\ k) c= dom f2 by A55, XBOOLE_1:1; then A59: rng (seq ^\ k) c= (dom f2) /\ (left_open_halfline x0) by A51, XBOOLE_1:19; then A60: lim (f2 /* (seq ^\ k)) = lim_left (f2,x0) by A2, A41, A57, Def7; A61: (dom f1) /\ ].(x0 - r1),x0.[ c= dom f1 by XBOOLE_1:17; then rng (seq ^\ k) c= dom f1 by A56, XBOOLE_1:1; then A62: rng (seq ^\ k) c= (dom f1) /\ (left_open_halfline x0) by A51, XBOOLE_1:19; then A63: lim (f1 /* (seq ^\ k)) = lim_left (f1,x0) by A1, A41, A57, Def7; A64: now__::_thesis:_for_n_being_Element_of_NAT_holds_ (_(f1_/*_(seq_^\_k))_._n_<=_(f_/*_(seq_^\_k))_._n_&_(f_/*_(seq_^\_k))_._n_<=_(f2_/*_(seq_^\_k))_._n_) let n be Element of NAT ; ::_thesis: ( (f1 /* (seq ^\ k)) . n <= (f /* (seq ^\ k)) . n & (f /* (seq ^\ k)) . n <= (f2 /* (seq ^\ k)) . n ) A65: (seq ^\ k) . n in rng (seq ^\ k) by VALUED_0:28; then f . ((seq ^\ k) . n) <= f2 . ((seq ^\ k) . n) by A6, A54; then A66: (f /* (seq ^\ k)) . n <= f2 . ((seq ^\ k) . n) by A45, A53, FUNCT_2:108, XBOOLE_1:1; f1 . ((seq ^\ k) . n) <= f . ((seq ^\ k) . n) by A6, A54, A65; then f1 . ((seq ^\ k) . n) <= (f /* (seq ^\ k)) . n by A45, A53, FUNCT_2:108, XBOOLE_1:1; hence ( (f1 /* (seq ^\ k)) . n <= (f /* (seq ^\ k)) . n & (f /* (seq ^\ k)) . n <= (f2 /* (seq ^\ k)) . n ) by A61, A58, A55, A56, A66, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; A67: f2 /* (seq ^\ k) is convergent by A2, A3, A41, A57, A59, Def7; A68: f1 /* (seq ^\ k) is convergent by A1, A3, A41, A57, A62, Def7; then f /* (seq ^\ k) is convergent by A3, A63, A67, A60, A64, SEQ_2:19; then A69: (f /* seq) ^\ k is convergent by A43, A52, VALUED_0:27, XBOOLE_1:1; hence f /* seq is convergent by SEQ_4:21; ::_thesis: lim (f /* seq) = lim_left (f1,x0) lim (f /* (seq ^\ k)) = lim_left (f1,x0) by A3, A68, A63, A67, A60, A64, SEQ_2:20; then lim ((f /* seq) ^\ k) = lim_left (f1,x0) by A43, A52, VALUED_0:27, XBOOLE_1:1; hence lim (f /* seq) = lim_left (f1,x0) by A69, SEQ_4:22; ::_thesis: verum end; hence f is_left_convergent_in x0 by A4, Def1; ::_thesis: ( f is_left_convergent_in x0 & lim_left (f,x0) = lim_left (f1,x0) ) hence ( f is_left_convergent_in x0 & lim_left (f,x0) = lim_left (f1,x0) ) by A40, Def7; ::_thesis: verum end; end; end; hence ( f is_left_convergent_in x0 & lim_left (f,x0) = lim_left (f1,x0) ) ; ::_thesis: verum end; theorem :: LIMFUNC2:64 for x0 being Real for f1, f2, f being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & lim_left (f1,x0) = lim_left (f2,x0) & ex r being Real st ( 0 < r & ].(x0 - r),x0.[ c= ((dom f1) /\ (dom f2)) /\ (dom f) & ( for g being Real st g in ].(x0 - r),x0.[ holds ( f1 . g <= f . g & f . g <= f2 . g ) ) ) holds ( f is_left_convergent_in x0 & lim_left (f,x0) = lim_left (f1,x0) ) proof let x0 be Real; ::_thesis: for f1, f2, f being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & lim_left (f1,x0) = lim_left (f2,x0) & ex r being Real st ( 0 < r & ].(x0 - r),x0.[ c= ((dom f1) /\ (dom f2)) /\ (dom f) & ( for g being Real st g in ].(x0 - r),x0.[ holds ( f1 . g <= f . g & f . g <= f2 . g ) ) ) holds ( f is_left_convergent_in x0 & lim_left (f,x0) = lim_left (f1,x0) ) let f1, f2, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & lim_left (f1,x0) = lim_left (f2,x0) & ex r being Real st ( 0 < r & ].(x0 - r),x0.[ c= ((dom f1) /\ (dom f2)) /\ (dom f) & ( for g being Real st g in ].(x0 - r),x0.[ holds ( f1 . g <= f . g & f . g <= f2 . g ) ) ) implies ( f is_left_convergent_in x0 & lim_left (f,x0) = lim_left (f1,x0) ) ) assume that A1: f1 is_left_convergent_in x0 and A2: f2 is_left_convergent_in x0 and A3: lim_left (f1,x0) = lim_left (f2,x0) ; ::_thesis: ( for r being Real holds ( not 0 < r or not ].(x0 - r),x0.[ c= ((dom f1) /\ (dom f2)) /\ (dom f) or ex g being Real st ( g in ].(x0 - r),x0.[ & not ( f1 . g <= f . g & f . g <= f2 . g ) ) ) or ( f is_left_convergent_in x0 & lim_left (f,x0) = lim_left (f1,x0) ) ) given r being Real such that A4: 0 < r and A5: ].(x0 - r),x0.[ c= ((dom f1) /\ (dom f2)) /\ (dom f) and A6: for g being Real st g in ].(x0 - r),x0.[ holds ( f1 . g <= f . g & f . g <= f2 . g ) ; ::_thesis: ( f is_left_convergent_in x0 & lim_left (f,x0) = lim_left (f1,x0) ) ((dom f1) /\ (dom f2)) /\ (dom f) c= (dom f1) /\ (dom f2) by XBOOLE_1:17; then A7: ].(x0 - r),x0.[ c= (dom f1) /\ (dom f2) by A5, XBOOLE_1:1; A8: ((dom f1) /\ (dom f2)) /\ (dom f) c= dom f by XBOOLE_1:17; then A9: ].(x0 - r),x0.[ c= dom f by A5, XBOOLE_1:1; A10: now__::_thesis:_for_r1_being_Real_st_r1_<_x0_holds_ ex_g_being_Real_st_ (_r1_<_g_&_g_<_x0_&_g_in_dom_f_) let r1 be Real; ::_thesis: ( r1 < x0 implies ex g being Real st ( r1 < g & g < x0 & g in dom f ) ) assume A11: r1 < x0 ; ::_thesis: ex g being Real st ( r1 < g & g < x0 & g in dom f ) now__::_thesis:_ex_g_being_Real_st_ (_r1_<_g_&_g_<_x0_&_g_in_dom_f_) percases ( r1 <= x0 - r or x0 - r <= r1 ) ; supposeA12: r1 <= x0 - r ; ::_thesis: ex g being Real st ( r1 < g & g < x0 & g in dom f ) now__::_thesis:_ex_g_being_Real_st_ (_r1_<_g_&_g_<_x0_&_g_in_dom_f_) x0 - r < x0 by A4, Lm1; then consider g being real number such that A13: x0 - r < g and A14: g < x0 by XREAL_1:5; reconsider g = g as Real by XREAL_0:def_1; take g = g; ::_thesis: ( r1 < g & g < x0 & g in dom f ) thus ( r1 < g & g < x0 ) by A12, A13, A14, XXREAL_0:2; ::_thesis: g in dom f g in { g2 where g2 is Real : ( x0 - r < g2 & g2 < x0 ) } by A13, A14; then g in ].(x0 - r),x0.[ by RCOMP_1:def_2; hence g in dom f by A9; ::_thesis: verum end; hence ex g being Real st ( r1 < g & g < x0 & g in dom f ) ; ::_thesis: verum end; supposeA15: x0 - r <= r1 ; ::_thesis: ex g being Real st ( r1 < g & g < x0 & g in dom f ) now__::_thesis:_ex_g_being_Real_st_ (_r1_<_g_&_g_<_x0_&_g_in_dom_f_) consider g being real number such that A16: r1 < g and A17: g < x0 by A11, XREAL_1:5; reconsider g = g as Real by XREAL_0:def_1; take g = g; ::_thesis: ( r1 < g & g < x0 & g in dom f ) thus ( r1 < g & g < x0 ) by A16, A17; ::_thesis: g in dom f x0 - r < g by A15, A16, XXREAL_0:2; then g in { g2 where g2 is Real : ( x0 - r < g2 & g2 < x0 ) } by A17; then g in ].(x0 - r),x0.[ by RCOMP_1:def_2; hence g in dom f by A9; ::_thesis: verum end; hence ex g being Real st ( r1 < g & g < x0 & g in dom f ) ; ::_thesis: verum end; end; end; hence ex g being Real st ( r1 < g & g < x0 & g in dom f ) ; ::_thesis: verum end; (dom f1) /\ (dom f2) c= dom f2 by XBOOLE_1:17; then A18: (dom f2) /\ ].(x0 - r),x0.[ = ].(x0 - r),x0.[ by A7, XBOOLE_1:1, XBOOLE_1:28; (dom f1) /\ (dom f2) c= dom f1 by XBOOLE_1:17; then A19: (dom f1) /\ ].(x0 - r),x0.[ = ].(x0 - r),x0.[ by A7, XBOOLE_1:1, XBOOLE_1:28; (dom f) /\ ].(x0 - r),x0.[ = ].(x0 - r),x0.[ by A5, A8, XBOOLE_1:1, XBOOLE_1:28; hence ( f is_left_convergent_in x0 & lim_left (f,x0) = lim_left (f1,x0) ) by A1, A2, A3, A4, A6, A19, A18, A10, Th63; ::_thesis: verum end; theorem Th65: :: LIMFUNC2:65 for x0 being Real for f1, f2, f being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & lim_right (f1,x0) = lim_right (f2,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds ( f1 . g <= f . g & f . g <= f2 . g ) ) & ( ( (dom f1) /\ ].x0,(x0 + r).[ c= (dom f2) /\ ].x0,(x0 + r).[ & (dom f) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ ) or ( (dom f2) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ & (dom f) /\ ].x0,(x0 + r).[ c= (dom f2) /\ ].x0,(x0 + r).[ ) ) ) holds ( f is_right_convergent_in x0 & lim_right (f,x0) = lim_right (f1,x0) ) proof let x0 be Real; ::_thesis: for f1, f2, f being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & lim_right (f1,x0) = lim_right (f2,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds ( f1 . g <= f . g & f . g <= f2 . g ) ) & ( ( (dom f1) /\ ].x0,(x0 + r).[ c= (dom f2) /\ ].x0,(x0 + r).[ & (dom f) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ ) or ( (dom f2) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ & (dom f) /\ ].x0,(x0 + r).[ c= (dom f2) /\ ].x0,(x0 + r).[ ) ) ) holds ( f is_right_convergent_in x0 & lim_right (f,x0) = lim_right (f1,x0) ) let f1, f2, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & lim_right (f1,x0) = lim_right (f2,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds ( f1 . g <= f . g & f . g <= f2 . g ) ) & ( ( (dom f1) /\ ].x0,(x0 + r).[ c= (dom f2) /\ ].x0,(x0 + r).[ & (dom f) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ ) or ( (dom f2) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ & (dom f) /\ ].x0,(x0 + r).[ c= (dom f2) /\ ].x0,(x0 + r).[ ) ) ) implies ( f is_right_convergent_in x0 & lim_right (f,x0) = lim_right (f1,x0) ) ) assume that A1: f1 is_right_convergent_in x0 and A2: f2 is_right_convergent_in x0 and A3: lim_right (f1,x0) = lim_right (f2,x0) and A4: 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 holds ( not 0 < r or ex g being Real st ( g in (dom f) /\ ].x0,(x0 + r).[ & not ( f1 . g <= f . g & f . g <= f2 . g ) ) or ( not ( (dom f1) /\ ].x0,(x0 + r).[ c= (dom f2) /\ ].x0,(x0 + r).[ & (dom f) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ ) & not ( (dom f2) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ & (dom f) /\ ].x0,(x0 + r).[ c= (dom f2) /\ ].x0,(x0 + r).[ ) ) ) or ( f is_right_convergent_in x0 & lim_right (f,x0) = lim_right (f1,x0) ) ) given r1 being Real such that A5: 0 < r1 and A6: for g being Real st g in (dom f) /\ ].x0,(x0 + r1).[ holds ( f1 . g <= f . g & f . g <= f2 . g ) and A7: ( ( (dom f1) /\ ].x0,(x0 + r1).[ c= (dom f2) /\ ].x0,(x0 + r1).[ & (dom f) /\ ].x0,(x0 + r1).[ c= (dom f1) /\ ].x0,(x0 + r1).[ ) or ( (dom f2) /\ ].x0,(x0 + r1).[ c= (dom f1) /\ ].x0,(x0 + r1).[ & (dom f) /\ ].x0,(x0 + r1).[ c= (dom f2) /\ ].x0,(x0 + r1).[ ) ) ; ::_thesis: ( f is_right_convergent_in x0 & lim_right (f,x0) = lim_right (f1,x0) ) now__::_thesis:_(_f_is_right_convergent_in_x0_&_f_is_right_convergent_in_x0_&_lim_right_(f,x0)_=_lim_right_(f1,x0)_) percases ( ( (dom f1) /\ ].x0,(x0 + r1).[ c= (dom f2) /\ ].x0,(x0 + r1).[ & (dom f) /\ ].x0,(x0 + r1).[ c= (dom f1) /\ ].x0,(x0 + r1).[ ) or ( (dom f2) /\ ].x0,(x0 + r1).[ c= (dom f1) /\ ].x0,(x0 + r1).[ & (dom f) /\ ].x0,(x0 + r1).[ c= (dom f2) /\ ].x0,(x0 + r1).[ ) ) by A7; supposeA8: ( (dom f1) /\ ].x0,(x0 + r1).[ c= (dom f2) /\ ].x0,(x0 + r1).[ & (dom f) /\ ].x0,(x0 + r1).[ c= (dom f1) /\ ].x0,(x0 + r1).[ ) ; ::_thesis: ( f is_right_convergent_in x0 & f is_right_convergent_in x0 & lim_right (f,x0) = lim_right (f1,x0) ) A9: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_f)_/\_(right_open_halfline_x0)_holds_ (_f_/*_seq_is_convergent_&_lim_(f_/*_seq)_=_lim_right_(f1,x0)_) let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (right_open_halfline x0) implies ( f /* seq is convergent & lim (f /* seq) = lim_right (f1,x0) ) ) assume that A10: seq is convergent and A11: lim seq = x0 and A12: rng seq c= (dom f) /\ (right_open_halfline x0) ; ::_thesis: ( f /* seq is convergent & lim (f /* seq) = lim_right (f1,x0) ) x0 < (lim seq) + r1 by A5, A11, Lm1; then consider k being Element of NAT such that A13: for n being Element of NAT st k <= n holds seq . n < x0 + r1 by A10, A11, Th2; A14: rng (seq ^\ k) c= rng seq by VALUED_0:21; (dom f) /\ (right_open_halfline x0) c= right_open_halfline x0 by XBOOLE_1:17; then rng seq c= right_open_halfline x0 by A12, XBOOLE_1:1; then A15: rng (seq ^\ k) c= right_open_halfline x0 by A14, XBOOLE_1:1; now__::_thesis:_for_x_being_set_st_x_in_rng_(seq_^\_k)_holds_ x_in_].x0,(x0_+_r1).[ let x be set ; ::_thesis: ( x in rng (seq ^\ k) implies x in ].x0,(x0 + r1).[ ) assume A16: x in rng (seq ^\ k) ; ::_thesis: x in ].x0,(x0 + r1).[ then consider n being Element of NAT such that A17: x = (seq ^\ k) . n by FUNCT_2:113; (seq ^\ k) . n in right_open_halfline x0 by A15, A16, A17; then (seq ^\ k) . n in { g where g is Real : x0 < g } by XXREAL_1:230; then A18: ex g being Real st ( g = (seq ^\ k) . n & x0 < g ) ; seq . (n + k) < x0 + r1 by A13, NAT_1:12; then (seq ^\ k) . n < x0 + r1 by NAT_1:def_3; then x in { g1 where g1 is Real : ( x0 < g1 & g1 < x0 + r1 ) } by A17, A18; hence x in ].x0,(x0 + r1).[ by RCOMP_1:def_2; ::_thesis: verum end; then A19: rng (seq ^\ k) c= ].x0,(x0 + r1).[ by TARSKI:def_3; ].x0,(x0 + r1).[ c= right_open_halfline x0 by XXREAL_1:247; then A20: rng (seq ^\ k) c= right_open_halfline x0 by A19, XBOOLE_1:1; A21: (dom f) /\ (right_open_halfline x0) c= dom f by XBOOLE_1:17; then A22: rng seq c= dom f by A12, XBOOLE_1:1; then rng (seq ^\ k) c= dom f by A14, XBOOLE_1:1; then A23: rng (seq ^\ k) c= (dom f) /\ ].x0,(x0 + r1).[ by A19, XBOOLE_1:19; then A24: rng (seq ^\ k) c= (dom f1) /\ ].x0,(x0 + r1).[ by A8, XBOOLE_1:1; then A25: rng (seq ^\ k) c= (dom f2) /\ ].x0,(x0 + r1).[ by A8, XBOOLE_1:1; A26: lim (seq ^\ k) = x0 by A10, A11, SEQ_4:20; A27: (dom f2) /\ ].x0,(x0 + r1).[ c= dom f2 by XBOOLE_1:17; then rng (seq ^\ k) c= dom f2 by A25, XBOOLE_1:1; then A28: rng (seq ^\ k) c= (dom f2) /\ (right_open_halfline x0) by A20, XBOOLE_1:19; then A29: lim (f2 /* (seq ^\ k)) = lim_right (f2,x0) by A2, A10, A26, Def8; A30: (dom f1) /\ ].x0,(x0 + r1).[ c= dom f1 by XBOOLE_1:17; then rng (seq ^\ k) c= dom f1 by A24, XBOOLE_1:1; then A31: rng (seq ^\ k) c= (dom f1) /\ (right_open_halfline x0) by A20, XBOOLE_1:19; then A32: lim (f1 /* (seq ^\ k)) = lim_right (f1,x0) by A1, A10, A26, Def8; A33: now__::_thesis:_for_n_being_Element_of_NAT_holds_ (_(f1_/*_(seq_^\_k))_._n_<=_(f_/*_(seq_^\_k))_._n_&_(f_/*_(seq_^\_k))_._n_<=_(f2_/*_(seq_^\_k))_._n_) let n be Element of NAT ; ::_thesis: ( (f1 /* (seq ^\ k)) . n <= (f /* (seq ^\ k)) . n & (f /* (seq ^\ k)) . n <= (f2 /* (seq ^\ k)) . n ) A34: (seq ^\ k) . n in rng (seq ^\ k) by VALUED_0:28; then f . ((seq ^\ k) . n) <= f2 . ((seq ^\ k) . n) by A6, A23; then A35: (f /* (seq ^\ k)) . n <= f2 . ((seq ^\ k) . n) by A14, A22, FUNCT_2:108, XBOOLE_1:1; f1 . ((seq ^\ k) . n) <= f . ((seq ^\ k) . n) by A6, A23, A34; then f1 . ((seq ^\ k) . n) <= (f /* (seq ^\ k)) . n by A14, A22, FUNCT_2:108, XBOOLE_1:1; hence ( (f1 /* (seq ^\ k)) . n <= (f /* (seq ^\ k)) . n & (f /* (seq ^\ k)) . n <= (f2 /* (seq ^\ k)) . n ) by A30, A27, A24, A25, A35, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; A36: f2 /* (seq ^\ k) is convergent by A2, A3, A10, A26, A28, Def8; A37: f1 /* (seq ^\ k) is convergent by A1, A3, A10, A26, A31, Def8; then f /* (seq ^\ k) is convergent by A3, A32, A36, A29, A33, SEQ_2:19; then A38: (f /* seq) ^\ k is convergent by A12, A21, VALUED_0:27, XBOOLE_1:1; hence f /* seq is convergent by SEQ_4:21; ::_thesis: lim (f /* seq) = lim_right (f1,x0) lim (f /* (seq ^\ k)) = lim_right (f1,x0) by A3, A37, A32, A36, A29, A33, SEQ_2:20; then lim ((f /* seq) ^\ k) = lim_right (f1,x0) by A12, A21, VALUED_0:27, XBOOLE_1:1; hence lim (f /* seq) = lim_right (f1,x0) by A38, SEQ_4:22; ::_thesis: verum end; hence f is_right_convergent_in x0 by A4, Def4; ::_thesis: ( f is_right_convergent_in x0 & lim_right (f,x0) = lim_right (f1,x0) ) hence ( f is_right_convergent_in x0 & lim_right (f,x0) = lim_right (f1,x0) ) by A9, Def8; ::_thesis: verum end; supposeA39: ( (dom f2) /\ ].x0,(x0 + r1).[ c= (dom f1) /\ ].x0,(x0 + r1).[ & (dom f) /\ ].x0,(x0 + r1).[ c= (dom f2) /\ ].x0,(x0 + r1).[ ) ; ::_thesis: ( f is_right_convergent_in x0 & f is_right_convergent_in x0 & lim_right (f,x0) = lim_right (f1,x0) ) A40: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_f)_/\_(right_open_halfline_x0)_holds_ (_f_/*_seq_is_convergent_&_lim_(f_/*_seq)_=_lim_right_(f1,x0)_) let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (right_open_halfline x0) implies ( f /* seq is convergent & lim (f /* seq) = lim_right (f1,x0) ) ) assume that A41: seq is convergent and A42: lim seq = x0 and A43: rng seq c= (dom f) /\ (right_open_halfline x0) ; ::_thesis: ( f /* seq is convergent & lim (f /* seq) = lim_right (f1,x0) ) x0 < (lim seq) + r1 by A5, A42, Lm1; then consider k being Element of NAT such that A44: for n being Element of NAT st k <= n holds seq . n < x0 + r1 by A41, A42, Th2; A45: rng (seq ^\ k) c= rng seq by VALUED_0:21; (dom f) /\ (right_open_halfline x0) c= right_open_halfline x0 by XBOOLE_1:17; then rng seq c= right_open_halfline x0 by A43, XBOOLE_1:1; then A46: rng (seq ^\ k) c= right_open_halfline x0 by A45, XBOOLE_1:1; now__::_thesis:_for_x_being_set_st_x_in_rng_(seq_^\_k)_holds_ x_in_].x0,(x0_+_r1).[ let x be set ; ::_thesis: ( x in rng (seq ^\ k) implies x in ].x0,(x0 + r1).[ ) assume A47: x in rng (seq ^\ k) ; ::_thesis: x in ].x0,(x0 + r1).[ then consider n being Element of NAT such that A48: x = (seq ^\ k) . n by FUNCT_2:113; (seq ^\ k) . n in right_open_halfline x0 by A46, A47, A48; then (seq ^\ k) . n in { g where g is Real : x0 < g } by XXREAL_1:230; then A49: ex g being Real st ( g = (seq ^\ k) . n & x0 < g ) ; seq . (n + k) < x0 + r1 by A44, NAT_1:12; then (seq ^\ k) . n < x0 + r1 by NAT_1:def_3; then x in { g1 where g1 is Real : ( x0 < g1 & g1 < x0 + r1 ) } by A48, A49; hence x in ].x0,(x0 + r1).[ by RCOMP_1:def_2; ::_thesis: verum end; then A50: rng (seq ^\ k) c= ].x0,(x0 + r1).[ by TARSKI:def_3; ].x0,(x0 + r1).[ c= right_open_halfline x0 by XXREAL_1:247; then A51: rng (seq ^\ k) c= right_open_halfline x0 by A50, XBOOLE_1:1; A52: (dom f) /\ (right_open_halfline x0) c= dom f by XBOOLE_1:17; then A53: rng seq c= dom f by A43, XBOOLE_1:1; then rng (seq ^\ k) c= dom f by A45, XBOOLE_1:1; then A54: rng (seq ^\ k) c= (dom f) /\ ].x0,(x0 + r1).[ by A50, XBOOLE_1:19; then A55: rng (seq ^\ k) c= (dom f2) /\ ].x0,(x0 + r1).[ by A39, XBOOLE_1:1; then A56: rng (seq ^\ k) c= (dom f1) /\ ].x0,(x0 + r1).[ by A39, XBOOLE_1:1; A57: lim (seq ^\ k) = x0 by A41, A42, SEQ_4:20; A58: (dom f2) /\ ].x0,(x0 + r1).[ c= dom f2 by XBOOLE_1:17; then rng (seq ^\ k) c= dom f2 by A55, XBOOLE_1:1; then A59: rng (seq ^\ k) c= (dom f2) /\ (right_open_halfline x0) by A51, XBOOLE_1:19; then A60: lim (f2 /* (seq ^\ k)) = lim_right (f2,x0) by A2, A41, A57, Def8; A61: (dom f1) /\ ].x0,(x0 + r1).[ c= dom f1 by XBOOLE_1:17; then rng (seq ^\ k) c= dom f1 by A56, XBOOLE_1:1; then A62: rng (seq ^\ k) c= (dom f1) /\ (right_open_halfline x0) by A51, XBOOLE_1:19; then A63: lim (f1 /* (seq ^\ k)) = lim_right (f1,x0) by A1, A41, A57, Def8; A64: now__::_thesis:_for_n_being_Element_of_NAT_holds_ (_(f1_/*_(seq_^\_k))_._n_<=_(f_/*_(seq_^\_k))_._n_&_(f_/*_(seq_^\_k))_._n_<=_(f2_/*_(seq_^\_k))_._n_) let n be Element of NAT ; ::_thesis: ( (f1 /* (seq ^\ k)) . n <= (f /* (seq ^\ k)) . n & (f /* (seq ^\ k)) . n <= (f2 /* (seq ^\ k)) . n ) A65: (seq ^\ k) . n in rng (seq ^\ k) by VALUED_0:28; then f . ((seq ^\ k) . n) <= f2 . ((seq ^\ k) . n) by A6, A54; then A66: (f /* (seq ^\ k)) . n <= f2 . ((seq ^\ k) . n) by A45, A53, FUNCT_2:108, XBOOLE_1:1; f1 . ((seq ^\ k) . n) <= f . ((seq ^\ k) . n) by A6, A54, A65; then f1 . ((seq ^\ k) . n) <= (f /* (seq ^\ k)) . n by A45, A53, FUNCT_2:108, XBOOLE_1:1; hence ( (f1 /* (seq ^\ k)) . n <= (f /* (seq ^\ k)) . n & (f /* (seq ^\ k)) . n <= (f2 /* (seq ^\ k)) . n ) by A61, A58, A55, A56, A66, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; A67: f2 /* (seq ^\ k) is convergent by A2, A3, A41, A57, A59, Def8; A68: f1 /* (seq ^\ k) is convergent by A1, A3, A41, A57, A62, Def8; then f /* (seq ^\ k) is convergent by A3, A63, A67, A60, A64, SEQ_2:19; then A69: (f /* seq) ^\ k is convergent by A43, A52, VALUED_0:27, XBOOLE_1:1; hence f /* seq is convergent by SEQ_4:21; ::_thesis: lim (f /* seq) = lim_right (f1,x0) lim (f /* (seq ^\ k)) = lim_right (f1,x0) by A3, A68, A63, A67, A60, A64, SEQ_2:20; then lim ((f /* seq) ^\ k) = lim_right (f1,x0) by A43, A52, VALUED_0:27, XBOOLE_1:1; hence lim (f /* seq) = lim_right (f1,x0) by A69, SEQ_4:22; ::_thesis: verum end; hence f is_right_convergent_in x0 by A4, Def4; ::_thesis: ( f is_right_convergent_in x0 & lim_right (f,x0) = lim_right (f1,x0) ) hence ( f is_right_convergent_in x0 & lim_right (f,x0) = lim_right (f1,x0) ) by A40, Def8; ::_thesis: verum end; end; end; hence ( f is_right_convergent_in x0 & lim_right (f,x0) = lim_right (f1,x0) ) ; ::_thesis: verum end; theorem :: LIMFUNC2:66 for x0 being Real for f1, f2, f being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & lim_right (f1,x0) = lim_right (f2,x0) & ex r being Real st ( 0 < r & ].x0,(x0 + r).[ c= ((dom f1) /\ (dom f2)) /\ (dom f) & ( for g being Real st g in ].x0,(x0 + r).[ holds ( f1 . g <= f . g & f . g <= f2 . g ) ) ) holds ( f is_right_convergent_in x0 & lim_right (f,x0) = lim_right (f1,x0) ) proof let x0 be Real; ::_thesis: for f1, f2, f being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & lim_right (f1,x0) = lim_right (f2,x0) & ex r being Real st ( 0 < r & ].x0,(x0 + r).[ c= ((dom f1) /\ (dom f2)) /\ (dom f) & ( for g being Real st g in ].x0,(x0 + r).[ holds ( f1 . g <= f . g & f . g <= f2 . g ) ) ) holds ( f is_right_convergent_in x0 & lim_right (f,x0) = lim_right (f1,x0) ) let f1, f2, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & lim_right (f1,x0) = lim_right (f2,x0) & ex r being Real st ( 0 < r & ].x0,(x0 + r).[ c= ((dom f1) /\ (dom f2)) /\ (dom f) & ( for g being Real st g in ].x0,(x0 + r).[ holds ( f1 . g <= f . g & f . g <= f2 . g ) ) ) implies ( f is_right_convergent_in x0 & lim_right (f,x0) = lim_right (f1,x0) ) ) assume that A1: f1 is_right_convergent_in x0 and A2: f2 is_right_convergent_in x0 and A3: lim_right (f1,x0) = lim_right (f2,x0) ; ::_thesis: ( for r being Real holds ( not 0 < r or not ].x0,(x0 + r).[ c= ((dom f1) /\ (dom f2)) /\ (dom f) or ex g being Real st ( g in ].x0,(x0 + r).[ & not ( f1 . g <= f . g & f . g <= f2 . g ) ) ) or ( f is_right_convergent_in x0 & lim_right (f,x0) = lim_right (f1,x0) ) ) given r being Real such that A4: 0 < r and A5: ].x0,(x0 + r).[ c= ((dom f1) /\ (dom f2)) /\ (dom f) and A6: for g being Real st g in ].x0,(x0 + r).[ holds ( f1 . g <= f . g & f . g <= f2 . g ) ; ::_thesis: ( f is_right_convergent_in x0 & lim_right (f,x0) = lim_right (f1,x0) ) ((dom f1) /\ (dom f2)) /\ (dom f) c= (dom f1) /\ (dom f2) by XBOOLE_1:17; then A7: ].x0,(x0 + r).[ c= (dom f1) /\ (dom f2) by A5, XBOOLE_1:1; A8: ((dom f1) /\ (dom f2)) /\ (dom f) c= dom f by XBOOLE_1:17; then A9: ].x0,(x0 + r).[ c= dom f by A5, XBOOLE_1:1; A10: now__::_thesis:_for_r1_being_Real_st_x0_<_r1_holds_ ex_g_being_Real_st_ (_g_<_r1_&_x0_<_g_&_g_in_dom_f_) let r1 be Real; ::_thesis: ( x0 < r1 implies ex g being Real st ( g < r1 & x0 < g & g in dom f ) ) assume A11: x0 < r1 ; ::_thesis: ex g being Real st ( g < r1 & x0 < g & g in dom f ) now__::_thesis:_ex_g_being_Real_st_ (_g_<_r1_&_x0_<_g_&_g_in_dom_f_) percases ( r1 <= x0 + r or x0 + r <= r1 ) ; supposeA12: r1 <= x0 + r ; ::_thesis: ex g being Real st ( g < r1 & x0 < g & g in dom f ) now__::_thesis:_ex_g_being_Real_st_ (_g_<_r1_&_x0_<_g_&_g_in_dom_f_) consider g being real number such that A13: x0 < g and A14: g < r1 by A11, XREAL_1:5; reconsider g = g as Real by XREAL_0:def_1; take g = g; ::_thesis: ( g < r1 & x0 < g & g in dom f ) thus ( g < r1 & x0 < g ) by A13, A14; ::_thesis: g in dom f g < x0 + r by A12, A14, XXREAL_0:2; then g in { g2 where g2 is Real : ( x0 < g2 & g2 < x0 + r ) } by A13; then g in ].x0,(x0 + r).[ by RCOMP_1:def_2; hence g in dom f by A9; ::_thesis: verum end; hence ex g being Real st ( g < r1 & x0 < g & g in dom f ) ; ::_thesis: verum end; supposeA15: x0 + r <= r1 ; ::_thesis: ex g being Real st ( g < r1 & x0 < g & g in dom f ) now__::_thesis:_ex_g_being_Real_st_ (_g_<_r1_&_x0_<_g_&_g_in_dom_f_) x0 + 0 < x0 + r by A4, XREAL_1:8; then consider g being real number such that A16: x0 < g and A17: g < x0 + r by XREAL_1:5; reconsider g = g as Real by XREAL_0:def_1; take g = g; ::_thesis: ( g < r1 & x0 < g & g in dom f ) thus ( g < r1 & x0 < g ) by A15, A16, A17, XXREAL_0:2; ::_thesis: g in dom f g in { g2 where g2 is Real : ( x0 < g2 & g2 < x0 + r ) } by A16, A17; then g in ].x0,(x0 + r).[ by RCOMP_1:def_2; hence g in dom f by A9; ::_thesis: verum end; hence ex g being Real st ( g < r1 & x0 < g & g in dom f ) ; ::_thesis: verum end; end; end; hence ex g being Real st ( g < r1 & x0 < g & g in dom f ) ; ::_thesis: verum end; (dom f1) /\ (dom f2) c= dom f2 by XBOOLE_1:17; then A18: (dom f2) /\ ].x0,(x0 + r).[ = ].x0,(x0 + r).[ by A7, XBOOLE_1:1, XBOOLE_1:28; (dom f1) /\ (dom f2) c= dom f1 by XBOOLE_1:17; then A19: (dom f1) /\ ].x0,(x0 + r).[ = ].x0,(x0 + r).[ by A7, XBOOLE_1:1, XBOOLE_1:28; (dom f) /\ ].x0,(x0 + r).[ = ].x0,(x0 + r).[ by A5, A8, XBOOLE_1:1, XBOOLE_1:28; hence ( f is_right_convergent_in x0 & lim_right (f,x0) = lim_right (f1,x0) ) by A1, A2, A3, A4, A6, A19, A18, A10, Th65; ::_thesis: verum end; theorem :: LIMFUNC2:67 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & ex r being Real st ( 0 < r & ( ( (dom f1) /\ ].(x0 - r),x0.[ c= (dom f2) /\ ].(x0 - r),x0.[ & ( for g being Real st g in (dom f1) /\ ].(x0 - r),x0.[ holds f1 . g <= f2 . g ) ) or ( (dom f2) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ & ( for g being Real st g in (dom f2) /\ ].(x0 - r),x0.[ holds f1 . g <= f2 . g ) ) ) ) holds lim_left (f1,x0) <= lim_left (f2,x0) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & ex r being Real st ( 0 < r & ( ( (dom f1) /\ ].(x0 - r),x0.[ c= (dom f2) /\ ].(x0 - r),x0.[ & ( for g being Real st g in (dom f1) /\ ].(x0 - r),x0.[ holds f1 . g <= f2 . g ) ) or ( (dom f2) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ & ( for g being Real st g in (dom f2) /\ ].(x0 - r),x0.[ holds f1 . g <= f2 . g ) ) ) ) holds lim_left (f1,x0) <= lim_left (f2,x0) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & ex r being Real st ( 0 < r & ( ( (dom f1) /\ ].(x0 - r),x0.[ c= (dom f2) /\ ].(x0 - r),x0.[ & ( for g being Real st g in (dom f1) /\ ].(x0 - r),x0.[ holds f1 . g <= f2 . g ) ) or ( (dom f2) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ & ( for g being Real st g in (dom f2) /\ ].(x0 - r),x0.[ holds f1 . g <= f2 . g ) ) ) ) implies lim_left (f1,x0) <= lim_left (f2,x0) ) assume that A1: f1 is_left_convergent_in x0 and A2: f2 is_left_convergent_in x0 ; ::_thesis: ( for r being Real holds ( not 0 < r or ( not ( (dom f1) /\ ].(x0 - r),x0.[ c= (dom f2) /\ ].(x0 - r),x0.[ & ( for g being Real st g in (dom f1) /\ ].(x0 - r),x0.[ holds f1 . g <= f2 . g ) ) & not ( (dom f2) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ & ( for g being Real st g in (dom f2) /\ ].(x0 - r),x0.[ holds f1 . g <= f2 . g ) ) ) ) or lim_left (f1,x0) <= lim_left (f2,x0) ) A3: lim_left (f2,x0) = lim_left (f2,x0) ; given r being Real such that A4: 0 < r and A5: ( ( (dom f1) /\ ].(x0 - r),x0.[ c= (dom f2) /\ ].(x0 - r),x0.[ & ( for g being Real st g in (dom f1) /\ ].(x0 - r),x0.[ holds f1 . g <= f2 . g ) ) or ( (dom f2) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ & ( for g being Real st g in (dom f2) /\ ].(x0 - r),x0.[ holds f1 . g <= f2 . g ) ) ) ; ::_thesis: lim_left (f1,x0) <= lim_left (f2,x0) A6: lim_left (f1,x0) = lim_left (f1,x0) ; now__::_thesis:_lim_left_(f1,x0)_<=_lim_left_(f2,x0) percases ( ( (dom f1) /\ ].(x0 - r),x0.[ c= (dom f2) /\ ].(x0 - r),x0.[ & ( for g being Real st g in (dom f1) /\ ].(x0 - r),x0.[ holds f1 . g <= f2 . g ) ) or ( (dom f2) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ & ( for g being Real st g in (dom f2) /\ ].(x0 - r),x0.[ holds f1 . g <= f2 . g ) ) ) by A5; supposeA7: ( (dom f1) /\ ].(x0 - r),x0.[ c= (dom f2) /\ ].(x0 - r),x0.[ & ( for g being Real st g in (dom f1) /\ ].(x0 - r),x0.[ holds f1 . g <= f2 . g ) ) ; ::_thesis: lim_left (f1,x0) <= lim_left (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_g_being_Real_st_S1[n,g] let n be Element of NAT ; ::_thesis: ex g being Real st S1[n,g] x0 - (1 / (n + 1)) < x0 by Lm3; then consider g being Real such that A9: x0 - (1 / (n + 1)) < g and A10: g < x0 and A11: g in dom f1 by A1, Def1; take g = g; ::_thesis: S1[n,g] thus S1[n,g] by A9, A10, A11; ::_thesis: verum end; consider s being Real_Sequence such that A12: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A8); A13: lim s = x0 by A12, Th5; A14: rng s c= (dom f1) /\ (left_open_halfline x0) by A12, Th5; A15: ].(x0 - r),x0.[ c= left_open_halfline x0 by XXREAL_1:263; A16: s is convergent by A12, Th5; 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, A13, Th1; A18: lim (s ^\ k) = x0 by A16, A13, SEQ_4:20; now__::_thesis:_for_x_being_set_st_x_in_rng_(s_^\_k)_holds_ x_in_(dom_f1)_/\_].(x0_-_r),x0.[ let x be set ; ::_thesis: ( x in rng (s ^\ k) implies x in (dom f1) /\ ].(x0 - r),x0.[ ) assume x in rng (s ^\ k) ; ::_thesis: x in (dom f1) /\ ].(x0 - r),x0.[ then consider n being Element of NAT such that A19: (s ^\ k) . n = x by FUNCT_2:113; s . (n + k) < x0 by A12; then A20: (s ^\ k) . n < x0 by NAT_1:def_3; s . (n + k) in dom f1 by A12; then A21: (s ^\ k) . n in dom f1 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 A20; then (s ^\ k) . n in ].(x0 - r),x0.[ by RCOMP_1:def_2; hence x in (dom f1) /\ ].(x0 - r),x0.[ by A19, A21, XBOOLE_0:def_4; ::_thesis: verum end; then A22: rng (s ^\ k) c= (dom f1) /\ ].(x0 - r),x0.[ by TARSKI:def_3; then A23: rng (s ^\ k) c= (dom f2) /\ ].(x0 - r),x0.[ by A7, XBOOLE_1:1; (dom f2) /\ ].(x0 - r),x0.[ c= ].(x0 - r),x0.[ by XBOOLE_1:17; then rng (s ^\ k) c= ].(x0 - r),x0.[ by A23, XBOOLE_1:1; then A24: rng (s ^\ k) c= left_open_halfline x0 by A15, XBOOLE_1:1; A25: (dom f2) /\ ].(x0 - r),x0.[ c= dom f2 by XBOOLE_1:17; then rng (s ^\ k) c= dom f2 by A23, XBOOLE_1:1; then A26: rng (s ^\ k) c= (dom f2) /\ (left_open_halfline x0) by A24, XBOOLE_1:19; then A27: lim (f2 /* (s ^\ k)) = lim_left (f2,x0) by A2, A16, A18, Def7; rng (s ^\ k) c= rng s by VALUED_0:21; then A28: rng (s ^\ k) c= (dom f1) /\ (left_open_halfline x0) by A14, XBOOLE_1:1; then A29: lim (f1 /* (s ^\ k)) = lim_left (f1,x0) by A1, A16, A18, Def7; A30: (dom f1) /\ ].(x0 - r),x0.[ c= dom f1 by XBOOLE_1:17; A31: 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, A22; then f1 . ((s ^\ k) . n) <= (f2 /* (s ^\ k)) . n by A23, A25, FUNCT_2:108, XBOOLE_1:1; hence (f1 /* (s ^\ k)) . n <= (f2 /* (s ^\ k)) . n by A22, A30, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; A32: f2 /* (s ^\ k) is convergent by A2, A3, A16, A18, A26, Def7; f1 /* (s ^\ k) is convergent by A1, A6, A16, A18, A28, Def7; hence lim_left (f1,x0) <= lim_left (f2,x0) by A29, A32, A27, A31, SEQ_2:18; ::_thesis: verum end; supposeA33: ( (dom f2) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ & ( for g being Real st g in (dom f2) /\ ].(x0 - r),x0.[ holds f1 . g <= f2 . g ) ) ; ::_thesis: lim_left (f1,x0) <= lim_left (f2,x0) defpred S1[ Element of NAT , real number ] means ( x0 - (1 / (\$1 + 1)) < \$2 & \$2 < x0 & \$2 in dom f2 ); A34: now__::_thesis:_for_n_being_Element_of_NAT_ex_g_being_Real_st_S1[n,g] let n be Element of NAT ; ::_thesis: ex g being Real st S1[n,g] 0 < 1 / (n + 1) by XREAL_1:139; then x0 - (1 / (n + 1)) < x0 - 0 by XREAL_1:15; then consider g being Real such that A35: x0 - (1 / (n + 1)) < g and A36: g < x0 and A37: g in dom f2 by A2, Def1; take g = g; ::_thesis: S1[n,g] thus S1[n,g] 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(A34); A39: lim s = x0 by A38, Th5; A40: rng s c= (dom f2) /\ (left_open_halfline x0) by A38, Th5; A41: ].(x0 - r),x0.[ c= left_open_halfline x0 by XXREAL_1:263; A42: s is convergent by A38, Th5; x0 - r < x0 by A4, Lm1; then consider k being Element of NAT such that A43: for n being Element of NAT st k <= n holds x0 - r < s . n by A42, A39, Th1; A44: lim (s ^\ k) = x0 by A42, A39, SEQ_4:20; A45: now__::_thesis:_for_x_being_set_st_x_in_rng_(s_^\_k)_holds_ x_in_(dom_f2)_/\_].(x0_-_r),x0.[ let x be set ; ::_thesis: ( x in rng (s ^\ k) implies x in (dom f2) /\ ].(x0 - r),x0.[ ) assume x in rng (s ^\ k) ; ::_thesis: x in (dom f2) /\ ].(x0 - r),x0.[ then consider n being Element of NAT such that A46: (s ^\ k) . n = x by FUNCT_2:113; s . (n + k) < x0 by A38; then A47: (s ^\ k) . n < x0 by NAT_1:def_3; s . (n + k) in dom f2 by A38; then A48: (s ^\ k) . n in dom f2 by NAT_1:def_3; x0 - r < s . (n + k) by A43, 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 A47; then (s ^\ k) . n in ].(x0 - r),x0.[ by RCOMP_1:def_2; hence x in (dom f2) /\ ].(x0 - r),x0.[ by A46, A48, XBOOLE_0:def_4; ::_thesis: verum end; then A49: rng (s ^\ k) c= (dom f2) /\ ].(x0 - r),x0.[ by TARSKI:def_3; then A50: rng (s ^\ k) c= (dom f1) /\ ].(x0 - r),x0.[ by A33, XBOOLE_1:1; (dom f1) /\ ].(x0 - r),x0.[ c= ].(x0 - r),x0.[ by XBOOLE_1:17; then rng (s ^\ k) c= ].(x0 - r),x0.[ by A50, XBOOLE_1:1; then A51: rng (s ^\ k) c= left_open_halfline x0 by A41, XBOOLE_1:1; A52: (dom f1) /\ ].(x0 - r),x0.[ c= dom f1 by XBOOLE_1:17; then rng (s ^\ k) c= dom f1 by A50, XBOOLE_1:1; then A53: rng (s ^\ k) c= (dom f1) /\ (left_open_halfline x0) by A51, XBOOLE_1:19; then A54: lim (f1 /* (s ^\ k)) = lim_left (f1,x0) by A1, A42, A44, Def7; rng (s ^\ k) c= rng s by VALUED_0:21; then A55: rng (s ^\ k) c= (dom f2) /\ (left_open_halfline x0) by A40, XBOOLE_1:1; then A56: lim (f2 /* (s ^\ k)) = lim_left (f2,x0) by A2, A42, A44, Def7; A57: (dom f2) /\ ].(x0 - r),x0.[ c= dom f2 by XBOOLE_1:17; A58: 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 A33, A45; then f1 . ((s ^\ k) . n) <= (f2 /* (s ^\ k)) . n by A49, A57, FUNCT_2:108, XBOOLE_1:1; hence (f1 /* (s ^\ k)) . n <= (f2 /* (s ^\ k)) . n by A50, A52, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; A59: f1 /* (s ^\ k) is convergent by A1, A6, A42, A44, A53, Def7; f2 /* (s ^\ k) is convergent by A2, A3, A42, A44, A55, Def7; hence lim_left (f1,x0) <= lim_left (f2,x0) by A56, A59, A54, A58, SEQ_2:18; ::_thesis: verum end; end; end; hence lim_left (f1,x0) <= lim_left (f2,x0) ; ::_thesis: verum end; theorem :: LIMFUNC2:68 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & ex r being Real st ( 0 < r & ( ( (dom f1) /\ ].x0,(x0 + r).[ c= (dom f2) /\ ].x0,(x0 + r).[ & ( for g being Real st g in (dom f1) /\ ].x0,(x0 + r).[ holds f1 . g <= f2 . g ) ) or ( (dom f2) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ & ( for g being Real st g in (dom f2) /\ ].x0,(x0 + r).[ holds f1 . g <= f2 . g ) ) ) ) holds lim_right (f1,x0) <= lim_right (f2,x0) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & ex r being Real st ( 0 < r & ( ( (dom f1) /\ ].x0,(x0 + r).[ c= (dom f2) /\ ].x0,(x0 + r).[ & ( for g being Real st g in (dom f1) /\ ].x0,(x0 + r).[ holds f1 . g <= f2 . g ) ) or ( (dom f2) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ & ( for g being Real st g in (dom f2) /\ ].x0,(x0 + r).[ holds f1 . g <= f2 . g ) ) ) ) holds lim_right (f1,x0) <= lim_right (f2,x0) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_right_convergent_in x0 & f2 is_right_convergent_in x0 & ex r being Real st ( 0 < r & ( ( (dom f1) /\ ].x0,(x0 + r).[ c= (dom f2) /\ ].x0,(x0 + r).[ & ( for g being Real st g in (dom f1) /\ ].x0,(x0 + r).[ holds f1 . g <= f2 . g ) ) or ( (dom f2) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ & ( for g being Real st g in (dom f2) /\ ].x0,(x0 + r).[ holds f1 . g <= f2 . g ) ) ) ) implies lim_right (f1,x0) <= lim_right (f2,x0) ) assume that A1: f1 is_right_convergent_in x0 and A2: f2 is_right_convergent_in x0 ; ::_thesis: ( for r being Real holds ( not 0 < r or ( not ( (dom f1) /\ ].x0,(x0 + r).[ c= (dom f2) /\ ].x0,(x0 + r).[ & ( for g being Real st g in (dom f1) /\ ].x0,(x0 + r).[ holds f1 . g <= f2 . g ) ) & not ( (dom f2) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ & ( for g being Real st g in (dom f2) /\ ].x0,(x0 + r).[ holds f1 . g <= f2 . g ) ) ) ) or lim_right (f1,x0) <= lim_right (f2,x0) ) A3: lim_right (f2,x0) = lim_right (f2,x0) ; given r being Real such that A4: 0 < r and A5: ( ( (dom f1) /\ ].x0,(x0 + r).[ c= (dom f2) /\ ].x0,(x0 + r).[ & ( for g being Real st g in (dom f1) /\ ].x0,(x0 + r).[ holds f1 . g <= f2 . g ) ) or ( (dom f2) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ & ( for g being Real st g in (dom f2) /\ ].x0,(x0 + r).[ holds f1 . g <= f2 . g ) ) ) ; ::_thesis: lim_right (f1,x0) <= lim_right (f2,x0) A6: lim_right (f1,x0) = lim_right (f1,x0) ; now__::_thesis:_lim_right_(f1,x0)_<=_lim_right_(f2,x0) percases ( ( (dom f1) /\ ].x0,(x0 + r).[ c= (dom f2) /\ ].x0,(x0 + r).[ & ( for g being Real st g in (dom f1) /\ ].x0,(x0 + r).[ holds f1 . g <= f2 . g ) ) or ( (dom f2) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ & ( for g being Real st g in (dom f2) /\ ].x0,(x0 + r).[ holds f1 . g <= f2 . g ) ) ) by A5; supposeA7: ( (dom f1) /\ ].x0,(x0 + r).[ c= (dom f2) /\ ].x0,(x0 + r).[ & ( for g being Real st g in (dom f1) /\ ].x0,(x0 + r).[ holds f1 . g <= f2 . g ) ) ; ::_thesis: lim_right (f1,x0) <= lim_right (f2,x0) defpred S1[ Element of NAT , real number ] means ( x0 < \$2 & \$2 < x0 + (1 / (\$1 + 1)) & \$2 in dom f1 ); A8: now__::_thesis:_for_n_being_Element_of_NAT_ex_g_being_Real_st_S1[n,g] let n be Element of NAT ; ::_thesis: ex g being Real st S1[n,g] x0 < x0 + (1 / (n + 1)) by Lm3; then consider g being Real such that A9: g < x0 + (1 / (n + 1)) and A10: x0 < g and A11: g in dom f1 by A1, Def4; take g = g; ::_thesis: S1[n,g] thus S1[n,g] by A9, A10, A11; ::_thesis: verum end; consider s being Real_Sequence such that A12: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A8); A13: lim s = x0 by A12, Th6; A14: rng s c= (dom f1) /\ (right_open_halfline x0) by A12, Th6; A15: ].x0,(x0 + r).[ c= right_open_halfline x0 by XXREAL_1:247; A16: s is convergent by A12, Th6; x0 < x0 + r by A4, Lm1; then consider k being Element of NAT such that A17: for n being Element of NAT st k <= n holds s . n < x0 + r by A16, A13, Th2; A18: lim (s ^\ k) = x0 by A16, A13, SEQ_4:20; now__::_thesis:_for_x_being_set_st_x_in_rng_(s_^\_k)_holds_ x_in_(dom_f1)_/\_].x0,(x0_+_r).[ let x be set ; ::_thesis: ( x in rng (s ^\ k) implies x in (dom f1) /\ ].x0,(x0 + r).[ ) assume x in rng (s ^\ k) ; ::_thesis: x in (dom f1) /\ ].x0,(x0 + r).[ then consider n being Element of NAT such that A19: (s ^\ k) . n = x by FUNCT_2:113; x0 < s . (n + k) by A12; then A20: x0 < (s ^\ k) . n by NAT_1:def_3; s . (n + k) in dom f1 by A12; then A21: (s ^\ k) . n in dom f1 by NAT_1:def_3; s . (n + k) < x0 + r by A17, NAT_1:12; then (s ^\ k) . n < x0 + r by NAT_1:def_3; then (s ^\ k) . n in { g2 where g2 is Real : ( x0 < g2 & g2 < x0 + r ) } by A20; then (s ^\ k) . n in ].x0,(x0 + r).[ by RCOMP_1:def_2; hence x in (dom f1) /\ ].x0,(x0 + r).[ by A19, A21, XBOOLE_0:def_4; ::_thesis: verum end; then A22: rng (s ^\ k) c= (dom f1) /\ ].x0,(x0 + r).[ by TARSKI:def_3; then A23: rng (s ^\ k) c= (dom f2) /\ ].x0,(x0 + r).[ by A7, XBOOLE_1:1; (dom f2) /\ ].x0,(x0 + r).[ c= ].x0,(x0 + r).[ by XBOOLE_1:17; then rng (s ^\ k) c= ].x0,(x0 + r).[ by A23, XBOOLE_1:1; then A24: rng (s ^\ k) c= right_open_halfline x0 by A15, XBOOLE_1:1; A25: (dom f2) /\ ].x0,(x0 + r).[ c= dom f2 by XBOOLE_1:17; then rng (s ^\ k) c= dom f2 by A23, XBOOLE_1:1; then A26: rng (s ^\ k) c= (dom f2) /\ (right_open_halfline x0) by A24, XBOOLE_1:19; then A27: lim (f2 /* (s ^\ k)) = lim_right (f2,x0) by A2, A16, A18, Def8; rng (s ^\ k) c= rng s by VALUED_0:21; then A28: rng (s ^\ k) c= (dom f1) /\ (right_open_halfline x0) by A14, XBOOLE_1:1; then A29: lim (f1 /* (s ^\ k)) = lim_right (f1,x0) by A1, A16, A18, Def8; A30: (dom f1) /\ ].x0,(x0 + r).[ c= dom f1 by XBOOLE_1:17; A31: 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, A22; then f1 . ((s ^\ k) . n) <= (f2 /* (s ^\ k)) . n by A23, A25, FUNCT_2:108, XBOOLE_1:1; hence (f1 /* (s ^\ k)) . n <= (f2 /* (s ^\ k)) . n by A22, A30, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; A32: f2 /* (s ^\ k) is convergent by A2, A3, A16, A18, A26, Def8; f1 /* (s ^\ k) is convergent by A1, A6, A16, A18, A28, Def8; hence lim_right (f1,x0) <= lim_right (f2,x0) by A29, A32, A27, A31, SEQ_2:18; ::_thesis: verum end; supposeA33: ( (dom f2) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ & ( for g being Real st g in (dom f2) /\ ].x0,(x0 + r).[ holds f1 . g <= f2 . g ) ) ; ::_thesis: lim_right (f1,x0) <= lim_right (f2,x0) defpred S1[ Element of NAT , real number ] means ( x0 < \$2 & \$2 < x0 + (1 / (\$1 + 1)) & \$2 in dom f2 ); A34: now__::_thesis:_for_n_being_Element_of_NAT_ex_g_being_Real_st_S1[n,g] let n be Element of NAT ; ::_thesis: ex g being Real st S1[n,g] 0 < 1 / (n + 1) by XREAL_1:139; then x0 + 0 < x0 + (1 / (n + 1)) by XREAL_1:8; then consider g being Real such that A35: g < x0 + (1 / (n + 1)) and A36: x0 < g and A37: g in dom f2 by A2, Def4; take g = g; ::_thesis: S1[n,g] thus S1[n,g] 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(A34); A39: lim s = x0 by A38, Th6; A40: rng s c= (dom f2) /\ (right_open_halfline x0) by A38, Th6; A41: ].x0,(x0 + r).[ c= right_open_halfline x0 by XXREAL_1:247; A42: s is convergent by A38, Th6; x0 < x0 + r by A4, Lm1; then consider k being Element of NAT such that A43: for n being Element of NAT st k <= n holds s . n < x0 + r by A42, A39, Th2; A44: lim (s ^\ k) = x0 by A42, A39, SEQ_4:20; A45: now__::_thesis:_for_x_being_set_st_x_in_rng_(s_^\_k)_holds_ x_in_(dom_f2)_/\_].x0,(x0_+_r).[ let x be set ; ::_thesis: ( x in rng (s ^\ k) implies x in (dom f2) /\ ].x0,(x0 + r).[ ) assume x in rng (s ^\ k) ; ::_thesis: x in (dom f2) /\ ].x0,(x0 + r).[ then consider n being Element of NAT such that A46: (s ^\ k) . n = x by FUNCT_2:113; x0 < s . (n + k) by A38; then A47: x0 < (s ^\ k) . n by NAT_1:def_3; s . (n + k) in dom f2 by A38; then A48: (s ^\ k) . n in dom f2 by NAT_1:def_3; s . (n + k) < x0 + r by A43, NAT_1:12; then (s ^\ k) . n < x0 + r by NAT_1:def_3; then (s ^\ k) . n in { g2 where g2 is Real : ( x0 < g2 & g2 < x0 + r ) } by A47; then (s ^\ k) . n in ].x0,(x0 + r).[ by RCOMP_1:def_2; hence x in (dom f2) /\ ].x0,(x0 + r).[ by A46, A48, XBOOLE_0:def_4; ::_thesis: verum end; then A49: rng (s ^\ k) c= (dom f2) /\ ].x0,(x0 + r).[ by TARSKI:def_3; then A50: rng (s ^\ k) c= (dom f1) /\ ].x0,(x0 + r).[ by A33, XBOOLE_1:1; (dom f1) /\ ].x0,(x0 + r).[ c= ].x0,(x0 + r).[ by XBOOLE_1:17; then rng (s ^\ k) c= ].x0,(x0 + r).[ by A50, XBOOLE_1:1; then A51: rng (s ^\ k) c= right_open_halfline x0 by A41, XBOOLE_1:1; A52: (dom f1) /\ ].x0,(x0 + r).[ c= dom f1 by XBOOLE_1:17; then rng (s ^\ k) c= dom f1 by A50, XBOOLE_1:1; then A53: rng (s ^\ k) c= (dom f1) /\ (right_open_halfline x0) by A51, XBOOLE_1:19; then A54: lim (f1 /* (s ^\ k)) = lim_right (f1,x0) by A1, A42, A44, Def8; rng (s ^\ k) c= rng s by VALUED_0:21; then A55: rng (s ^\ k) c= (dom f2) /\ (right_open_halfline x0) by A40, XBOOLE_1:1; then A56: lim (f2 /* (s ^\ k)) = lim_right (f2,x0) by A2, A42, A44, Def8; A57: (dom f2) /\ ].x0,(x0 + r).[ c= dom f2 by XBOOLE_1:17; A58: 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 A33, A45; then f1 . ((s ^\ k) . n) <= (f2 /* (s ^\ k)) . n by A49, A57, FUNCT_2:108, XBOOLE_1:1; hence (f1 /* (s ^\ k)) . n <= (f2 /* (s ^\ k)) . n by A50, A52, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; A59: f1 /* (s ^\ k) is convergent by A1, A6, A42, A44, A53, Def8; f2 /* (s ^\ k) is convergent by A2, A3, A42, A44, A55, Def8; hence lim_right (f1,x0) <= lim_right (f2,x0) by A56, A59, A54, A58, SEQ_2:18; ::_thesis: verum end; end; end; hence lim_right (f1,x0) <= lim_right (f2,x0) ; ::_thesis: verum end; theorem :: LIMFUNC2:69 for x0 being Real for f being PartFunc of REAL,REAL st ( f is_left_divergent_to+infty_in x0 or f is_left_divergent_to-infty_in x0 ) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f & f . g <> 0 ) ) holds ( f ^ is_left_convergent_in x0 & lim_left ((f ^),x0) = 0 ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st ( f is_left_divergent_to+infty_in x0 or f is_left_divergent_to-infty_in x0 ) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f & f . g <> 0 ) ) holds ( f ^ is_left_convergent_in x0 & lim_left ((f ^),x0) = 0 ) let f be PartFunc of REAL,REAL; ::_thesis: ( ( f is_left_divergent_to+infty_in x0 or f is_left_divergent_to-infty_in x0 ) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f & f . g <> 0 ) ) implies ( f ^ is_left_convergent_in x0 & lim_left ((f ^),x0) = 0 ) ) assume A1: ( f is_left_divergent_to+infty_in x0 or f is_left_divergent_to-infty_in x0 ) ; ::_thesis: ( ex r being Real st ( r < x0 & ( for g being Real holds ( not r < g or not g < x0 or not g in dom f or not f . g <> 0 ) ) ) or ( f ^ is_left_convergent_in x0 & lim_left ((f ^),x0) = 0 ) ) A2: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(f_^))_/\_(left_open_halfline_x0)_holds_ (_(f_^)_/*_seq_is_convergent_&_lim_((f_^)_/*_seq)_=_0_) dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def_2; then A3: dom (f ^) c= dom f by XBOOLE_1:36; let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f ^)) /\ (left_open_halfline 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 ^)) /\ (left_open_halfline x0) ; ::_thesis: ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) (dom (f ^)) /\ (left_open_halfline x0) c= left_open_halfline x0 by XBOOLE_1:17; then A7: rng seq c= left_open_halfline x0 by A6, XBOOLE_1:1; A8: (dom (f ^)) /\ (left_open_halfline x0) c= dom (f ^) by XBOOLE_1:17; then rng seq c= dom (f ^) by A6, XBOOLE_1:1; then rng seq c= dom f by A3, XBOOLE_1:1; then A9: rng seq c= (dom f) /\ (left_open_halfline x0) by A7, XBOOLE_1:19; now__::_thesis:_(_(f_^)_/*_seq_is_convergent_&_lim_((f_^)_/*_seq)_=_0_) percases ( f is_left_divergent_to+infty_in x0 or f is_left_divergent_to-infty_in x0 ) by A1; suppose f is_left_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, A9, 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, A8, A11, RFUNCT_2:12, XBOOLE_1:1; ::_thesis: verum end; suppose f is_left_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, A9, 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, A8, 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 r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f & f . g <> 0 ) ; ::_thesis: ( f ^ is_left_convergent_in x0 & lim_left ((f ^),x0) = 0 ) now__::_thesis:_for_r_being_Real_st_r_<_x0_holds_ ex_g_being_Real_st_ (_r_<_g_&_g_<_x0_&_g_in_dom_(f_^)_) 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 g being Real such that A15: r < g and A16: g < x0 and A17: g in dom f and A18: f . g <> 0 by A14; take g = g; ::_thesis: ( r < g & g < x0 & g in dom (f ^) ) thus ( r < g & g < x0 ) by A15, A16; ::_thesis: g in dom (f ^) not f . g in {0} by A18, TARSKI:def_1; then not g in f " {0} by FUNCT_1:def_7; then g in (dom f) \ (f " {0}) by A17, XBOOLE_0:def_5; hence g in dom (f ^) by RFUNCT_1:def_2; ::_thesis: verum end; hence f ^ is_left_convergent_in x0 by A2, Def1; ::_thesis: lim_left ((f ^),x0) = 0 hence lim_left ((f ^),x0) = 0 by A2, Def7; ::_thesis: verum end; theorem :: LIMFUNC2:70 for x0 being Real for f being PartFunc of REAL,REAL st ( f is_right_divergent_to+infty_in x0 or 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 & f . g <> 0 ) ) holds ( f ^ is_right_convergent_in x0 & lim_right ((f ^),x0) = 0 ) proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st ( f is_right_divergent_to+infty_in x0 or 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 & f . g <> 0 ) ) holds ( f ^ is_right_convergent_in x0 & lim_right ((f ^),x0) = 0 ) let f be PartFunc of REAL,REAL; ::_thesis: ( ( f is_right_divergent_to+infty_in x0 or 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 & f . g <> 0 ) ) implies ( f ^ is_right_convergent_in x0 & lim_right ((f ^),x0) = 0 ) ) assume A1: ( f is_right_divergent_to+infty_in x0 or f is_right_divergent_to-infty_in x0 ) ; ::_thesis: ( ex r being Real st ( x0 < r & ( for g being Real holds ( not g < r or not x0 < g or not g in dom f or not f . g <> 0 ) ) ) or ( f ^ is_right_convergent_in x0 & lim_right ((f ^),x0) = 0 ) ) A2: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_convergent_&_lim_seq_=_x0_&_rng_seq_c=_(dom_(f_^))_/\_(right_open_halfline_x0)_holds_ (_(f_^)_/*_seq_is_convergent_&_lim_((f_^)_/*_seq)_=_0_) dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def_2; then A3: dom (f ^) c= dom f by XBOOLE_1:36; let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f ^)) /\ (right_open_halfline 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 ^)) /\ (right_open_halfline x0) ; ::_thesis: ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) (dom (f ^)) /\ (right_open_halfline x0) c= right_open_halfline x0 by XBOOLE_1:17; then A7: rng seq c= right_open_halfline x0 by A6, XBOOLE_1:1; A8: (dom (f ^)) /\ (right_open_halfline x0) c= dom (f ^) by XBOOLE_1:17; then rng seq c= dom (f ^) by A6, XBOOLE_1:1; then rng seq c= dom f by A3, XBOOLE_1:1; then A9: rng seq c= (dom f) /\ (right_open_halfline x0) by A7, XBOOLE_1:19; now__::_thesis:_(_(f_^)_/*_seq_is_convergent_&_lim_((f_^)_/*_seq)_=_0_) percases ( f is_right_divergent_to+infty_in x0 or f is_right_divergent_to-infty_in x0 ) by A1; suppose f is_right_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, A9, Def5; 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, A8, A11, RFUNCT_2:12, XBOOLE_1:1; ::_thesis: verum end; suppose f is_right_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, A9, Def6; 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, A8, 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 r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f & f . g <> 0 ) ; ::_thesis: ( f ^ is_right_convergent_in x0 & lim_right ((f ^),x0) = 0 ) now__::_thesis:_for_r_being_Real_st_x0_<_r_holds_ ex_g_being_Real_st_ (_g_<_r_&_x0_<_g_&_g_in_dom_(f_^)_) 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 g being Real such that A15: g < r and A16: x0 < g and A17: g in dom f and A18: f . g <> 0 by A14; take g = g; ::_thesis: ( g < r & x0 < g & g in dom (f ^) ) thus ( g < r & x0 < g ) by A15, A16; ::_thesis: g in dom (f ^) not f . g in {0} by A18, TARSKI:def_1; then not g in f " {0} by FUNCT_1:def_7; then g in (dom f) \ (f " {0}) by A17, XBOOLE_0:def_5; hence g in dom (f ^) by RFUNCT_1:def_2; ::_thesis: verum end; hence f ^ is_right_convergent_in x0 by A2, Def4; ::_thesis: lim_right ((f ^),x0) = 0 hence lim_right ((f ^),x0) = 0 by A2, Def8; ::_thesis: verum end; theorem :: LIMFUNC2:71 for x0 being Real for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 & lim_left (f,x0) = 0 & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds 0 < f . g ) ) holds f ^ is_left_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 & lim_left (f,x0) = 0 & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds 0 < f . g ) ) holds f ^ is_left_divergent_to+infty_in x0 let f be PartFunc of REAL,REAL; ::_thesis: ( f is_left_convergent_in x0 & lim_left (f,x0) = 0 & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds 0 < f . g ) ) implies f ^ is_left_divergent_to+infty_in x0 ) assume that A1: f is_left_convergent_in x0 and A2: lim_left (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.[ & not 0 < f . g ) ) or f ^ is_left_divergent_to+infty_in x0 ) given r being Real such that A3: 0 < r and A4: for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds 0 < f . g ; ::_thesis: f ^ is_left_divergent_to+infty_in x0 thus for r1 being Real st r1 < x0 holds ex g1 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) ) :: according to LIMFUNC2:def_2 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom (f ^)) /\ (left_open_halfline x0) holds (f ^) /* seq is divergent_to+infty proof let r1 be Real; ::_thesis: ( r1 < x0 implies ex g1 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) ) ) assume r1 < x0 ; ::_thesis: ex g1 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) ) then consider g1 being Real such that A5: r1 < g1 and A6: g1 < x0 and g1 in dom f by A1, Def1; now__::_thesis:_ex_g2_being_Real_st_ (_r1_<_g2_&_g2_<_x0_&_g2_in_dom_(f_^)_) percases ( g1 <= x0 - r or x0 - r <= g1 ) ; supposeA7: g1 <= x0 - r ; ::_thesis: ex g2 being Real st ( r1 < g2 & g2 < x0 & g2 in dom (f ^) ) x0 - r < x0 by A3, Lm1; then consider g2 being Real such that A8: x0 - r < g2 and A9: g2 < x0 and A10: g2 in dom f by A1, Def1; take g2 = g2; ::_thesis: ( r1 < g2 & g2 < x0 & g2 in dom (f ^) ) g1 < g2 by A7, A8, XXREAL_0:2; hence ( r1 < g2 & g2 < x0 ) by A5, A9, XXREAL_0:2; ::_thesis: g2 in dom (f ^) g2 in { r2 where r2 is Real : ( x0 - r < r2 & r2 < x0 ) } by A8, A9; then g2 in ].(x0 - r),x0.[ by RCOMP_1:def_2; then g2 in (dom f) /\ ].(x0 - r),x0.[ by A10, XBOOLE_0:def_4; then 0 <> f . g2 by A4; then not f . g2 in {0} by TARSKI:def_1; then not g2 in f " {0} by FUNCT_1:def_7; then g2 in (dom f) \ (f " {0}) by A10, XBOOLE_0:def_5; hence g2 in dom (f ^) by RFUNCT_1:def_2; ::_thesis: verum end; supposeA11: x0 - r <= g1 ; ::_thesis: ex g2 being Real st ( r1 < g2 & g2 < x0 & g2 in dom (f ^) ) consider g2 being Real such that A12: g1 < g2 and A13: g2 < x0 and A14: g2 in dom f by A1, A6, Def1; take g2 = g2; ::_thesis: ( r1 < g2 & g2 < x0 & g2 in dom (f ^) ) thus ( r1 < g2 & g2 < x0 ) by A5, A12, A13, XXREAL_0:2; ::_thesis: g2 in dom (f ^) x0 - r < g2 by A11, A12, XXREAL_0:2; then g2 in { r2 where r2 is Real : ( x0 - r < r2 & r2 < x0 ) } by A13; then g2 in ].(x0 - r),x0.[ by RCOMP_1:def_2; then g2 in (dom f) /\ ].(x0 - r),x0.[ by A14, XBOOLE_0:def_4; then 0 <> f . g2 by A4; then not f . g2 in {0} by TARSKI:def_1; then not g2 in f " {0} by FUNCT_1:def_7; then g2 in (dom f) \ (f " {0}) by A14, XBOOLE_0:def_5; hence g2 in dom (f ^) by RFUNCT_1:def_2; ::_thesis: verum end; end; end; hence ex g1 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) ) ; ::_thesis: verum end; 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 A15: s is convergent and A16: lim s = x0 and A17: rng s c= (dom (f ^)) /\ (left_open_halfline x0) ; ::_thesis: (f ^) /* s is divergent_to+infty x0 - r < x0 by A3, Lm1; then consider k being Element of NAT such that A18: for n being Element of NAT st k <= n holds x0 - r < s . n by A15, A16, Th1; A19: lim (s ^\ k) = x0 by A15, A16, SEQ_4:20; dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def_2; then A20: dom (f ^) c= dom f by XBOOLE_1:36; A21: rng (s ^\ k) c= rng s by VALUED_0:21; (dom (f ^)) /\ (left_open_halfline x0) c= left_open_halfline x0 by XBOOLE_1:17; then rng s c= left_open_halfline x0 by A17, XBOOLE_1:1; then A22: rng (s ^\ k) c= left_open_halfline x0 by A21, XBOOLE_1:1; A23: (dom (f ^)) /\ (left_open_halfline x0) c= dom (f ^) by XBOOLE_1:17; then A24: rng s c= dom (f ^) by A17, XBOOLE_1:1; then A25: rng s c= dom f by A20, XBOOLE_1:1; then A26: rng (s ^\ k) c= dom f by A21, XBOOLE_1:1; then A27: rng (s ^\ k) c= (dom f) /\ (left_open_halfline x0) by A22, XBOOLE_1:19; then A28: lim (f /* (s ^\ k)) = 0 by A1, A2, A15, A19, Def7; 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 x0 - r < s . (n + k) by A18, NAT_1:12; then A29: x0 - r < (s ^\ k) . n by NAT_1:def_3; A30: (s ^\ k) . n in rng (s ^\ k) by VALUED_0:28; then (s ^\ k) . n in left_open_halfline x0 by A22; then (s ^\ k) . n in { g1 where g1 is Real : g1 < x0 } by XXREAL_1:229; then ex g1 being Real st ( g1 = (s ^\ k) . n & g1 < x0 ) ; then (s ^\ k) . n in { g2 where g2 is Real : ( x0 - r < g2 & g2 < x0 ) } by A29; then (s ^\ k) . n in ].(x0 - r),x0.[ by RCOMP_1:def_2; then (s ^\ k) . n in (dom f) /\ ].(x0 - r),x0.[ by A26, A30, XBOOLE_0:def_4; then 0 < f . ((s ^\ k) . n) by A4; hence 0 < (f /* (s ^\ k)) . n by A25, A21, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; then A31: for n being Element of NAT st 0 <= n holds 0 < (f /* (s ^\ k)) . n ; f /* (s ^\ k) is convergent by A1, A2, A15, A19, A27, Def7; then A32: (f /* (s ^\ k)) " is divergent_to+infty by A28, A31, LIMFUNC1:35; (f /* (s ^\ k)) " = ((f /* s) ^\ k) " by A24, A20, VALUED_0:27, XBOOLE_1:1 .= ((f /* s) ") ^\ k by SEQM_3:18 .= ((f ^) /* s) ^\ k by A17, A23, RFUNCT_2:12, XBOOLE_1:1 ; hence (f ^) /* s is divergent_to+infty by A32, LIMFUNC1:7; ::_thesis: verum end; theorem :: LIMFUNC2:72 for x0 being Real for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 & lim_left (f,x0) = 0 & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds f . g < 0 ) ) holds f ^ is_left_divergent_to-infty_in x0 proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 & lim_left (f,x0) = 0 & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds f . g < 0 ) ) holds f ^ is_left_divergent_to-infty_in x0 let f be PartFunc of REAL,REAL; ::_thesis: ( f is_left_convergent_in x0 & lim_left (f,x0) = 0 & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds f . g < 0 ) ) implies f ^ is_left_divergent_to-infty_in x0 ) assume that A1: f is_left_convergent_in x0 and A2: lim_left (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.[ & not f . g < 0 ) ) or f ^ is_left_divergent_to-infty_in x0 ) given r being Real such that A3: 0 < r and A4: for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds f . g < 0 ; ::_thesis: f ^ is_left_divergent_to-infty_in x0 thus for r1 being Real st r1 < x0 holds ex g1 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) ) :: according to LIMFUNC2:def_3 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom (f ^)) /\ (left_open_halfline x0) holds (f ^) /* seq is divergent_to-infty proof let r1 be Real; ::_thesis: ( r1 < x0 implies ex g1 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) ) ) assume r1 < x0 ; ::_thesis: ex g1 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) ) then consider g1 being Real such that A5: r1 < g1 and A6: g1 < x0 and g1 in dom f by A1, Def1; now__::_thesis:_ex_g2_being_Real_st_ (_r1_<_g2_&_g2_<_x0_&_g2_in_dom_(f_^)_) percases ( g1 <= x0 - r or x0 - r <= g1 ) ; supposeA7: g1 <= x0 - r ; ::_thesis: ex g2 being Real st ( r1 < g2 & g2 < x0 & g2 in dom (f ^) ) x0 - r < x0 by A3, Lm1; then consider g2 being Real such that A8: x0 - r < g2 and A9: g2 < x0 and A10: g2 in dom f by A1, Def1; take g2 = g2; ::_thesis: ( r1 < g2 & g2 < x0 & g2 in dom (f ^) ) g1 < g2 by A7, A8, XXREAL_0:2; hence ( r1 < g2 & g2 < x0 ) by A5, A9, XXREAL_0:2; ::_thesis: g2 in dom (f ^) g2 in { r2 where r2 is Real : ( x0 - r < r2 & r2 < x0 ) } by A8, A9; then g2 in ].(x0 - r),x0.[ by RCOMP_1:def_2; then g2 in (dom f) /\ ].(x0 - r),x0.[ by A10, XBOOLE_0:def_4; then not f . g2 in {0} by A4; then not g2 in f " {0} by FUNCT_1:def_7; then g2 in (dom f) \ (f " {0}) by A10, XBOOLE_0:def_5; hence g2 in dom (f ^) by RFUNCT_1:def_2; ::_thesis: verum end; supposeA11: x0 - r <= g1 ; ::_thesis: ex g2 being Real st ( r1 < g2 & g2 < x0 & g2 in dom (f ^) ) consider g2 being Real such that A12: g1 < g2 and A13: g2 < x0 and A14: g2 in dom f by A1, A6, Def1; take g2 = g2; ::_thesis: ( r1 < g2 & g2 < x0 & g2 in dom (f ^) ) thus ( r1 < g2 & g2 < x0 ) by A5, A12, A13, XXREAL_0:2; ::_thesis: g2 in dom (f ^) x0 - r < g2 by A11, A12, XXREAL_0:2; then g2 in { r2 where r2 is Real : ( x0 - r < r2 & r2 < x0 ) } by A13; then g2 in ].(x0 - r),x0.[ by RCOMP_1:def_2; then g2 in (dom f) /\ ].(x0 - r),x0.[ by A14, XBOOLE_0:def_4; then not f . g2 in {0} by A4; then not g2 in f " {0} by FUNCT_1:def_7; then g2 in (dom f) \ (f " {0}) by A14, XBOOLE_0:def_5; hence g2 in dom (f ^) by RFUNCT_1:def_2; ::_thesis: verum end; end; end; hence ex g1 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) ) ; ::_thesis: verum end; 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 A15: s is convergent and A16: lim s = x0 and A17: rng s c= (dom (f ^)) /\ (left_open_halfline x0) ; ::_thesis: (f ^) /* s is divergent_to-infty x0 - r < x0 by A3, Lm1; then consider k being Element of NAT such that A18: for n being Element of NAT st k <= n holds x0 - r < s . n by A15, A16, Th1; A19: lim (s ^\ k) = x0 by A15, A16, SEQ_4:20; dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def_2; then A20: dom (f ^) c= dom f by XBOOLE_1:36; A21: rng (s ^\ k) c= rng s by VALUED_0:21; (dom (f ^)) /\ (left_open_halfline x0) c= left_open_halfline x0 by XBOOLE_1:17; then rng s c= left_open_halfline x0 by A17, XBOOLE_1:1; then A22: rng (s ^\ k) c= left_open_halfline x0 by A21, XBOOLE_1:1; A23: (dom (f ^)) /\ (left_open_halfline x0) c= dom (f ^) by XBOOLE_1:17; then A24: rng s c= dom (f ^) by A17, XBOOLE_1:1; then A25: rng s c= dom f by A20, XBOOLE_1:1; then A26: rng (s ^\ k) c= dom f by A21, XBOOLE_1:1; then A27: rng (s ^\ k) c= (dom f) /\ (left_open_halfline x0) by A22, XBOOLE_1:19; then A28: lim (f /* (s ^\ k)) = 0 by A1, A2, A15, A19, Def7; 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 x0 - r < s . (n + k) by A18, NAT_1:12; then A29: x0 - r < (s ^\ k) . n by NAT_1:def_3; A30: (s ^\ k) . n in rng (s ^\ k) by VALUED_0:28; then (s ^\ k) . n in left_open_halfline x0 by A22; then (s ^\ k) . n in { g1 where g1 is Real : g1 < x0 } by XXREAL_1:229; then ex g1 being Real st ( g1 = (s ^\ k) . n & g1 < x0 ) ; then (s ^\ k) . n in { g2 where g2 is Real : ( x0 - r < g2 & g2 < x0 ) } by A29; then (s ^\ k) . n in ].(x0 - r),x0.[ by RCOMP_1:def_2; then (s ^\ k) . n in (dom f) /\ ].(x0 - r),x0.[ by A26, A30, XBOOLE_0:def_4; then f . ((s ^\ k) . n) < 0 by A4; hence (f /* (s ^\ k)) . n < 0 by A25, A21, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; then A31: for n being Element of NAT st 0 <= n holds (f /* (s ^\ k)) . n < 0 ; f /* (s ^\ k) is convergent by A1, A2, A15, A19, A27, Def7; then A32: (f /* (s ^\ k)) " is divergent_to-infty by A28, A31, LIMFUNC1:36; (f /* (s ^\ k)) " = ((f /* s) ^\ k) " by A24, A20, VALUED_0:27, XBOOLE_1:1 .= ((f /* s) ") ^\ k by SEQM_3:18 .= ((f ^) /* s) ^\ k by A17, A23, RFUNCT_2:12, XBOOLE_1:1 ; hence (f ^) /* s is divergent_to-infty by A32, LIMFUNC1:7; ::_thesis: verum end; theorem :: LIMFUNC2:73 for x0 being Real for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 & lim_right (f,x0) = 0 & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds 0 < f . g ) ) holds f ^ is_right_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 & lim_right (f,x0) = 0 & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds 0 < f . g ) ) holds f ^ is_right_divergent_to+infty_in x0 let f be PartFunc of REAL,REAL; ::_thesis: ( f is_right_convergent_in x0 & lim_right (f,x0) = 0 & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds 0 < f . g ) ) implies f ^ is_right_divergent_to+infty_in x0 ) assume that A1: f is_right_convergent_in x0 and A2: lim_right (f,x0) = 0 ; ::_thesis: ( for r being Real holds ( not 0 < r or ex g being Real st ( g in (dom f) /\ ].x0,(x0 + r).[ & not 0 < f . g ) ) or f ^ is_right_divergent_to+infty_in x0 ) given r being Real such that A3: 0 < r and A4: for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds 0 < f . g ; ::_thesis: f ^ is_right_divergent_to+infty_in x0 thus for r1 being Real st x0 < r1 holds ex g1 being Real st ( g1 < r1 & x0 < g1 & g1 in dom (f ^) ) :: according to LIMFUNC2:def_5 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom (f ^)) /\ (right_open_halfline x0) holds (f ^) /* seq is divergent_to+infty proof let r1 be Real; ::_thesis: ( x0 < r1 implies ex g1 being Real st ( g1 < r1 & x0 < g1 & g1 in dom (f ^) ) ) assume x0 < r1 ; ::_thesis: ex g1 being Real st ( g1 < r1 & x0 < g1 & g1 in dom (f ^) ) then consider g1 being Real such that A5: g1 < r1 and A6: x0 < g1 and g1 in dom f by A1, Def4; now__::_thesis:_ex_g2_being_Real_st_ (_g2_<_r1_&_x0_<_g2_&_g2_in_dom_(f_^)_) percases ( g1 <= x0 + r or x0 + r <= g1 ) ; supposeA7: g1 <= x0 + r ; ::_thesis: ex g2 being Real st ( g2 < r1 & x0 < g2 & g2 in dom (f ^) ) consider g2 being Real such that A8: g2 < g1 and A9: x0 < g2 and A10: g2 in dom f by A1, A6, Def4; take g2 = g2; ::_thesis: ( g2 < r1 & x0 < g2 & g2 in dom (f ^) ) thus ( g2 < r1 & x0 < g2 ) by A5, A8, A9, XXREAL_0:2; ::_thesis: g2 in dom (f ^) g2 < x0 + r by A7, A8, XXREAL_0:2; then g2 in { r2 where r2 is Real : ( x0 < r2 & r2 < x0 + r ) } by A9; then g2 in ].x0,(x0 + r).[ by RCOMP_1:def_2; then g2 in (dom f) /\ ].x0,(x0 + r).[ by A10, XBOOLE_0:def_4; then 0 <> f . g2 by A4; then not f . g2 in {0} by TARSKI:def_1; then not g2 in f " {0} by FUNCT_1:def_7; then g2 in (dom f) \ (f " {0}) by A10, XBOOLE_0:def_5; hence g2 in dom (f ^) by RFUNCT_1:def_2; ::_thesis: verum end; supposeA11: x0 + r <= g1 ; ::_thesis: ex g2 being Real st ( g2 < r1 & x0 < g2 & g2 in dom (f ^) ) x0 < x0 + r by A3, Lm1; then consider g2 being Real such that A12: g2 < x0 + r and A13: x0 < g2 and A14: g2 in dom f by A1, Def4; take g2 = g2; ::_thesis: ( g2 < r1 & x0 < g2 & g2 in dom (f ^) ) g2 < g1 by A11, A12, XXREAL_0:2; hence ( g2 < r1 & x0 < g2 ) by A5, A13, XXREAL_0:2; ::_thesis: g2 in dom (f ^) g2 in { r2 where r2 is Real : ( x0 < r2 & r2 < x0 + r ) } by A12, A13; then g2 in ].x0,(x0 + r).[ by RCOMP_1:def_2; then g2 in (dom f) /\ ].x0,(x0 + r).[ by A14, XBOOLE_0:def_4; then 0 <> f . g2 by A4; then not f . g2 in {0} by TARSKI:def_1; then not g2 in f " {0} by FUNCT_1:def_7; then g2 in (dom f) \ (f " {0}) by A14, XBOOLE_0:def_5; hence g2 in dom (f ^) by RFUNCT_1:def_2; ::_thesis: verum end; end; end; hence ex g1 being Real st ( g1 < r1 & x0 < g1 & g1 in dom (f ^) ) ; ::_thesis: verum end; 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 A15: s is convergent and A16: lim s = x0 and A17: rng s c= (dom (f ^)) /\ (right_open_halfline x0) ; ::_thesis: (f ^) /* s is divergent_to+infty x0 < x0 + r by A3, Lm1; then consider k being Element of NAT such that A18: for n being Element of NAT st k <= n holds s . n < x0 + r by A15, A16, Th2; A19: lim (s ^\ k) = x0 by A15, A16, SEQ_4:20; dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def_2; then A20: dom (f ^) c= dom f by XBOOLE_1:36; A21: rng (s ^\ k) c= rng s by VALUED_0:21; (dom (f ^)) /\ (right_open_halfline x0) c= right_open_halfline x0 by XBOOLE_1:17; then rng s c= right_open_halfline x0 by A17, XBOOLE_1:1; then A22: rng (s ^\ k) c= right_open_halfline x0 by A21, XBOOLE_1:1; A23: (dom (f ^)) /\ (right_open_halfline x0) c= dom (f ^) by XBOOLE_1:17; then A24: rng s c= dom (f ^) by A17, XBOOLE_1:1; then A25: rng s c= dom f by A20, XBOOLE_1:1; then A26: rng (s ^\ k) c= dom f by A21, XBOOLE_1:1; then A27: rng (s ^\ k) c= (dom f) /\ (right_open_halfline x0) by A22, XBOOLE_1:19; then A28: lim (f /* (s ^\ k)) = 0 by A1, A2, A15, A19, Def8; 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 s . (n + k) < x0 + r by A18, NAT_1:12; then A29: (s ^\ k) . n < x0 + r by NAT_1:def_3; A30: (s ^\ k) . n in rng (s ^\ k) by VALUED_0:28; then (s ^\ k) . n in right_open_halfline x0 by A22; then (s ^\ k) . n in { g1 where g1 is Real : x0 < g1 } by XXREAL_1:230; then ex g1 being Real st ( g1 = (s ^\ k) . n & x0 < g1 ) ; then (s ^\ k) . n in { g2 where g2 is Real : ( x0 < g2 & g2 < x0 + r ) } by A29; then (s ^\ k) . n in ].x0,(x0 + r).[ by RCOMP_1:def_2; then (s ^\ k) . n in (dom f) /\ ].x0,(x0 + r).[ by A26, A30, XBOOLE_0:def_4; then 0 < f . ((s ^\ k) . n) by A4; hence 0 < (f /* (s ^\ k)) . n by A25, A21, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; then A31: for n being Element of NAT st 0 <= n holds 0 < (f /* (s ^\ k)) . n ; f /* (s ^\ k) is convergent by A1, A2, A15, A19, A27, Def8; then A32: (f /* (s ^\ k)) " is divergent_to+infty by A28, A31, LIMFUNC1:35; (f /* (s ^\ k)) " = ((f /* s) ^\ k) " by A24, A20, VALUED_0:27, XBOOLE_1:1 .= ((f /* s) ") ^\ k by SEQM_3:18 .= ((f ^) /* s) ^\ k by A17, A23, RFUNCT_2:12, XBOOLE_1:1 ; hence (f ^) /* s is divergent_to+infty by A32, LIMFUNC1:7; ::_thesis: verum end; theorem :: LIMFUNC2:74 for x0 being Real for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 & lim_right (f,x0) = 0 & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds f . g < 0 ) ) holds f ^ is_right_divergent_to-infty_in x0 proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 & lim_right (f,x0) = 0 & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds f . g < 0 ) ) holds f ^ is_right_divergent_to-infty_in x0 let f be PartFunc of REAL,REAL; ::_thesis: ( f is_right_convergent_in x0 & lim_right (f,x0) = 0 & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds f . g < 0 ) ) implies f ^ is_right_divergent_to-infty_in x0 ) assume that A1: f is_right_convergent_in x0 and A2: lim_right (f,x0) = 0 ; ::_thesis: ( for r being Real holds ( not 0 < r or ex g being Real st ( g in (dom f) /\ ].x0,(x0 + r).[ & not f . g < 0 ) ) or f ^ is_right_divergent_to-infty_in x0 ) given r being Real such that A3: 0 < r and A4: for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds f . g < 0 ; ::_thesis: f ^ is_right_divergent_to-infty_in x0 thus for r1 being Real st x0 < r1 holds ex g1 being Real st ( g1 < r1 & x0 < g1 & g1 in dom (f ^) ) :: according to LIMFUNC2:def_6 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom (f ^)) /\ (right_open_halfline x0) holds (f ^) /* seq is divergent_to-infty proof let r1 be Real; ::_thesis: ( x0 < r1 implies ex g1 being Real st ( g1 < r1 & x0 < g1 & g1 in dom (f ^) ) ) assume x0 < r1 ; ::_thesis: ex g1 being Real st ( g1 < r1 & x0 < g1 & g1 in dom (f ^) ) then consider g1 being Real such that A5: g1 < r1 and A6: x0 < g1 and g1 in dom f by A1, Def4; now__::_thesis:_ex_g2_being_Real_st_ (_g2_<_r1_&_x0_<_g2_&_g2_in_dom_(f_^)_) percases ( g1 <= x0 + r or x0 + r <= g1 ) ; supposeA7: g1 <= x0 + r ; ::_thesis: ex g2 being Real st ( g2 < r1 & x0 < g2 & g2 in dom (f ^) ) consider g2 being Real such that A8: g2 < g1 and A9: x0 < g2 and A10: g2 in dom f by A1, A6, Def4; take g2 = g2; ::_thesis: ( g2 < r1 & x0 < g2 & g2 in dom (f ^) ) thus ( g2 < r1 & x0 < g2 ) by A5, A8, A9, XXREAL_0:2; ::_thesis: g2 in dom (f ^) g2 < x0 + r by A7, A8, XXREAL_0:2; then g2 in { r2 where r2 is Real : ( x0 < r2 & r2 < x0 + r ) } by A9; then g2 in ].x0,(x0 + r).[ by RCOMP_1:def_2; then g2 in (dom f) /\ ].x0,(x0 + r).[ by A10, XBOOLE_0:def_4; then not f . g2 in {0} by A4; then not g2 in f " {0} by FUNCT_1:def_7; then g2 in (dom f) \ (f " {0}) by A10, XBOOLE_0:def_5; hence g2 in dom (f ^) by RFUNCT_1:def_2; ::_thesis: verum end; supposeA11: x0 + r <= g1 ; ::_thesis: ex g2 being Real st ( g2 < r1 & x0 < g2 & g2 in dom (f ^) ) x0 < x0 + r by A3, Lm1; then consider g2 being Real such that A12: g2 < x0 + r and A13: x0 < g2 and A14: g2 in dom f by A1, Def4; take g2 = g2; ::_thesis: ( g2 < r1 & x0 < g2 & g2 in dom (f ^) ) g2 < g1 by A11, A12, XXREAL_0:2; hence ( g2 < r1 & x0 < g2 ) by A5, A13, XXREAL_0:2; ::_thesis: g2 in dom (f ^) g2 in { r2 where r2 is Real : ( x0 < r2 & r2 < x0 + r ) } by A12, A13; then g2 in ].x0,(x0 + r).[ by RCOMP_1:def_2; then g2 in (dom f) /\ ].x0,(x0 + r).[ by A14, XBOOLE_0:def_4; then not f . g2 in {0} by A4; then not g2 in f " {0} by FUNCT_1:def_7; then g2 in (dom f) \ (f " {0}) by A14, XBOOLE_0:def_5; hence g2 in dom (f ^) by RFUNCT_1:def_2; ::_thesis: verum end; end; end; hence ex g1 being Real st ( g1 < r1 & x0 < g1 & g1 in dom (f ^) ) ; ::_thesis: verum end; 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 A15: s is convergent and A16: lim s = x0 and A17: rng s c= (dom (f ^)) /\ (right_open_halfline x0) ; ::_thesis: (f ^) /* s is divergent_to-infty x0 < x0 + r by A3, Lm1; then consider k being Element of NAT such that A18: for n being Element of NAT st k <= n holds s . n < x0 + r by A15, A16, Th2; A19: lim (s ^\ k) = x0 by A15, A16, SEQ_4:20; dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def_2; then A20: dom (f ^) c= dom f by XBOOLE_1:36; A21: rng (s ^\ k) c= rng s by VALUED_0:21; (dom (f ^)) /\ (right_open_halfline x0) c= right_open_halfline x0 by XBOOLE_1:17; then rng s c= right_open_halfline x0 by A17, XBOOLE_1:1; then A22: rng (s ^\ k) c= right_open_halfline x0 by A21, XBOOLE_1:1; A23: (dom (f ^)) /\ (right_open_halfline x0) c= dom (f ^) by XBOOLE_1:17; then A24: rng s c= dom (f ^) by A17, XBOOLE_1:1; then A25: rng s c= dom f by A20, XBOOLE_1:1; then A26: rng (s ^\ k) c= dom f by A21, XBOOLE_1:1; then A27: rng (s ^\ k) c= (dom f) /\ (right_open_halfline x0) by A22, XBOOLE_1:19; then A28: lim (f /* (s ^\ k)) = 0 by A1, A2, A15, A19, Def8; 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 s . (n + k) < x0 + r by A18, NAT_1:12; then A29: (s ^\ k) . n < x0 + r by NAT_1:def_3; A30: (s ^\ k) . n in rng (s ^\ k) by VALUED_0:28; then (s ^\ k) . n in right_open_halfline x0 by A22; then (s ^\ k) . n in { g1 where g1 is Real : x0 < g1 } by XXREAL_1:230; then ex g1 being Real st ( g1 = (s ^\ k) . n & x0 < g1 ) ; then (s ^\ k) . n in { g2 where g2 is Real : ( x0 < g2 & g2 < x0 + r ) } by A29; then (s ^\ k) . n in ].x0,(x0 + r).[ by RCOMP_1:def_2; then (s ^\ k) . n in (dom f) /\ ].x0,(x0 + r).[ by A26, A30, XBOOLE_0:def_4; then f . ((s ^\ k) . n) < 0 by A4; hence (f /* (s ^\ k)) . n < 0 by A25, A21, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; then A31: for n being Element of NAT st 0 <= n holds (f /* (s ^\ k)) . n < 0 ; f /* (s ^\ k) is convergent by A1, A2, A15, A19, A27, Def8; then A32: (f /* (s ^\ k)) " is divergent_to-infty by A28, A31, LIMFUNC1:36; (f /* (s ^\ k)) " = ((f /* s) ^\ k) " by A24, A20, VALUED_0:27, XBOOLE_1:1 .= ((f /* s) ") ^\ k by SEQM_3:18 .= ((f ^) /* s) ^\ k by A17, A23, RFUNCT_2:12, XBOOLE_1:1 ; hence (f ^) /* s is divergent_to-infty by A32, LIMFUNC1:7; ::_thesis: verum end; theorem :: LIMFUNC2:75 for x0 being Real for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 & lim_left (f,x0) = 0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f & f . g <> 0 ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds 0 <= f . g ) ) holds f ^ is_left_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 & lim_left (f,x0) = 0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f & f . g <> 0 ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds 0 <= f . g ) ) holds f ^ is_left_divergent_to+infty_in x0 let f be PartFunc of REAL,REAL; ::_thesis: ( f is_left_convergent_in x0 & lim_left (f,x0) = 0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f & f . g <> 0 ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds 0 <= f . g ) ) implies f ^ is_left_divergent_to+infty_in x0 ) assume that A1: f is_left_convergent_in x0 and A2: lim_left (f,x0) = 0 and A3: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f & f . g <> 0 ) ; ::_thesis: ( for r being Real holds ( not 0 < r or ex g being Real st ( g in (dom f) /\ ].(x0 - r),x0.[ & not 0 <= f . g ) ) or f ^ is_left_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.[ holds 0 <= f . g ; ::_thesis: f ^ is_left_divergent_to+infty_in x0 thus for r1 being Real st r1 < x0 holds ex g1 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) ) :: according to LIMFUNC2:def_2 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom (f ^)) /\ (left_open_halfline x0) holds (f ^) /* seq is divergent_to+infty proof let r1 be Real; ::_thesis: ( r1 < x0 implies ex g1 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) ) ) assume r1 < x0 ; ::_thesis: ex g1 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) ) then consider g1 being Real such that A6: r1 < g1 and A7: g1 < x0 and A8: g1 in dom f and A9: f . g1 <> 0 by A3; take g1 ; ::_thesis: ( r1 < g1 & g1 < x0 & g1 in dom (f ^) ) thus ( r1 < g1 & g1 < x0 ) by A6, A7; ::_thesis: g1 in dom (f ^) not f . g1 in {0} by A9, TARSKI:def_1; then not g1 in f " {0} by FUNCT_1:def_7; then g1 in (dom f) \ (f " {0}) by A8, XBOOLE_0:def_5; hence g1 in dom (f ^) by RFUNCT_1:def_2; ::_thesis: verum end; 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 A10: s is convergent and A11: lim s = x0 and A12: rng s c= (dom (f ^)) /\ (left_open_halfline x0) ; ::_thesis: (f ^) /* s is divergent_to+infty x0 - r < x0 by A4, Lm1; then consider k being Element of NAT such that A13: for n being Element of NAT st k <= n holds x0 - r < s . n by A10, A11, Th1; A14: lim (s ^\ k) = x0 by A10, A11, SEQ_4:20; A15: (dom (f ^)) /\ (left_open_halfline x0) c= dom (f ^) by XBOOLE_1:17; then A16: rng s c= dom (f ^) by A12, XBOOLE_1:1; A17: rng (s ^\ k) c= rng s by VALUED_0:21; dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def_2; then A18: dom (f ^) c= dom f by XBOOLE_1:36; then A19: rng s c= dom f by A16, XBOOLE_1:1; then A20: rng (s ^\ k) c= dom f by A17, XBOOLE_1:1; (dom (f ^)) /\ (left_open_halfline x0) c= left_open_halfline x0 by XBOOLE_1:17; then rng s c= left_open_halfline x0 by A12, XBOOLE_1:1; then A21: rng (s ^\ k) c= left_open_halfline x0 by A17, XBOOLE_1:1; then A22: rng (s ^\ k) c= (dom f) /\ (left_open_halfline x0) by A20, XBOOLE_1:19; then A23: lim (f /* (s ^\ k)) = 0 by A1, A2, A10, A14, Def7; A24: f /* (s ^\ k) is non-zero by A16, A17, 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 x0 - r < s . (n + k) by A13, NAT_1:12; then A25: x0 - r < (s ^\ k) . n by NAT_1:def_3; A26: (s ^\ k) . n in rng (s ^\ k) by VALUED_0:28; then (s ^\ k) . n in left_open_halfline x0 by A21; then (s ^\ k) . n in { g1 where g1 is Real : g1 < x0 } by XXREAL_1:229; then ex g1 being Real st ( g1 = (s ^\ k) . n & g1 < x0 ) ; then (s ^\ k) . n in { g2 where g2 is Real : ( x0 - r < g2 & g2 < x0 ) } by A25; then (s ^\ k) . n in ].(x0 - r),x0.[ by RCOMP_1:def_2; then (s ^\ k) . n in (dom f) /\ ].(x0 - r),x0.[ by A20, A26, XBOOLE_0:def_4; then A27: 0 <= f . ((s ^\ k) . n) by A5; (f /* (s ^\ k)) . n <> 0 by A24, SEQ_1:5; hence 0 < (f /* (s ^\ k)) . n by A19, A17, A27, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; then A28: for n being Element of NAT st 0 <= n holds 0 < (f /* (s ^\ k)) . n ; f /* (s ^\ k) is convergent by A1, A2, A10, A14, A22, Def7; then A29: (f /* (s ^\ k)) " is divergent_to+infty by A23, A28, LIMFUNC1:35; (f /* (s ^\ k)) " = ((f /* s) ^\ k) " by A16, A18, VALUED_0:27, XBOOLE_1:1 .= ((f /* s) ") ^\ k by SEQM_3:18 .= ((f ^) /* s) ^\ k by A12, A15, RFUNCT_2:12, XBOOLE_1:1 ; hence (f ^) /* s is divergent_to+infty by A29, LIMFUNC1:7; ::_thesis: verum end; theorem :: LIMFUNC2:76 for x0 being Real for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 & lim_left (f,x0) = 0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f & f . g <> 0 ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds f . g <= 0 ) ) holds f ^ is_left_divergent_to-infty_in x0 proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 & lim_left (f,x0) = 0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f & f . g <> 0 ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds f . g <= 0 ) ) holds f ^ is_left_divergent_to-infty_in x0 let f be PartFunc of REAL,REAL; ::_thesis: ( f is_left_convergent_in x0 & lim_left (f,x0) = 0 & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f & f . g <> 0 ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].(x0 - r),x0.[ holds f . g <= 0 ) ) implies f ^ is_left_divergent_to-infty_in x0 ) assume that A1: f is_left_convergent_in x0 and A2: lim_left (f,x0) = 0 and A3: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom f & f . g <> 0 ) ; ::_thesis: ( for r being Real holds ( not 0 < r or ex g being Real st ( g in (dom f) /\ ].(x0 - r),x0.[ & not f . g <= 0 ) ) or f ^ is_left_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.[ holds f . g <= 0 ; ::_thesis: f ^ is_left_divergent_to-infty_in x0 thus for r1 being Real st r1 < x0 holds ex g1 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) ) :: according to LIMFUNC2:def_3 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom (f ^)) /\ (left_open_halfline x0) holds (f ^) /* seq is divergent_to-infty proof let r1 be Real; ::_thesis: ( r1 < x0 implies ex g1 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) ) ) assume r1 < x0 ; ::_thesis: ex g1 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f ^) ) then consider g1 being Real such that A6: r1 < g1 and A7: g1 < x0 and A8: g1 in dom f and A9: f . g1 <> 0 by A3; take g1 ; ::_thesis: ( r1 < g1 & g1 < x0 & g1 in dom (f ^) ) thus ( r1 < g1 & g1 < x0 ) by A6, A7; ::_thesis: g1 in dom (f ^) not f . g1 in {0} by A9, TARSKI:def_1; then not g1 in f " {0} by FUNCT_1:def_7; then g1 in (dom f) \ (f " {0}) by A8, XBOOLE_0:def_5; hence g1 in dom (f ^) by RFUNCT_1:def_2; ::_thesis: verum end; 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 A10: s is convergent and A11: lim s = x0 and A12: rng s c= (dom (f ^)) /\ (left_open_halfline x0) ; ::_thesis: (f ^) /* s is divergent_to-infty x0 - r < x0 by A4, Lm1; then consider k being Element of NAT such that A13: for n being Element of NAT st k <= n holds x0 - r < s . n by A10, A11, Th1; A14: lim (s ^\ k) = x0 by A10, A11, SEQ_4:20; A15: (dom (f ^)) /\ (left_open_halfline x0) c= dom (f ^) by XBOOLE_1:17; then A16: rng s c= dom (f ^) by A12, XBOOLE_1:1; A17: rng (s ^\ k) c= rng s by VALUED_0:21; dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def_2; then A18: dom (f ^) c= dom f by XBOOLE_1:36; then A19: rng s c= dom f by A16, XBOOLE_1:1; then A20: rng (s ^\ k) c= dom f by A17, XBOOLE_1:1; (dom (f ^)) /\ (left_open_halfline x0) c= left_open_halfline x0 by XBOOLE_1:17; then rng s c= left_open_halfline x0 by A12, XBOOLE_1:1; then A21: rng (s ^\ k) c= left_open_halfline x0 by A17, XBOOLE_1:1; then A22: rng (s ^\ k) c= (dom f) /\ (left_open_halfline x0) by A20, XBOOLE_1:19; then A23: lim (f /* (s ^\ k)) = 0 by A1, A2, A10, A14, Def7; A24: f /* (s ^\ k) is non-zero by A16, A17, 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 x0 - r < s . (n + k) by A13, NAT_1:12; then A25: x0 - r < (s ^\ k) . n by NAT_1:def_3; A26: (s ^\ k) . n in rng (s ^\ k) by VALUED_0:28; then (s ^\ k) . n in left_open_halfline x0 by A21; then (s ^\ k) . n in { g1 where g1 is Real : g1 < x0 } by XXREAL_1:229; then ex g1 being Real st ( g1 = (s ^\ k) . n & g1 < x0 ) ; then (s ^\ k) . n in { g2 where g2 is Real : ( x0 - r < g2 & g2 < x0 ) } by A25; then (s ^\ k) . n in ].(x0 - r),x0.[ by RCOMP_1:def_2; then (s ^\ k) . n in (dom f) /\ ].(x0 - r),x0.[ by A20, A26, XBOOLE_0:def_4; then A27: f . ((s ^\ k) . n) <= 0 by A5; (f /* (s ^\ k)) . n <> 0 by A24, SEQ_1:5; hence (f /* (s ^\ k)) . n < 0 by A19, A17, A27, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; then A28: for n being Element of NAT st 0 <= n holds (f /* (s ^\ k)) . n < 0 ; f /* (s ^\ k) is convergent by A1, A2, A10, A14, A22, Def7; then A29: (f /* (s ^\ k)) " is divergent_to-infty by A23, A28, LIMFUNC1:36; (f /* (s ^\ k)) " = ((f /* s) ^\ k) " by A16, A18, VALUED_0:27, XBOOLE_1:1 .= ((f /* s) ") ^\ k by SEQM_3:18 .= ((f ^) /* s) ^\ k by A12, A15, RFUNCT_2:12, XBOOLE_1:1 ; hence (f ^) /* s is divergent_to-infty by A29, LIMFUNC1:7; ::_thesis: verum end; theorem :: LIMFUNC2:77 for x0 being Real for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 & lim_right (f,x0) = 0 & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f & f . g <> 0 ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds 0 <= f . g ) ) holds f ^ is_right_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 & lim_right (f,x0) = 0 & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f & f . g <> 0 ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds 0 <= f . g ) ) holds f ^ is_right_divergent_to+infty_in x0 let f be PartFunc of REAL,REAL; ::_thesis: ( f is_right_convergent_in x0 & lim_right (f,x0) = 0 & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f & f . g <> 0 ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds 0 <= f . g ) ) implies f ^ is_right_divergent_to+infty_in x0 ) assume that A1: f is_right_convergent_in x0 and A2: lim_right (f,x0) = 0 and A3: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f & f . g <> 0 ) ; ::_thesis: ( for r being Real holds ( not 0 < r or ex g being Real st ( g in (dom f) /\ ].x0,(x0 + r).[ & not 0 <= f . g ) ) or f ^ is_right_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,(x0 + r).[ holds 0 <= f . g ; ::_thesis: f ^ is_right_divergent_to+infty_in x0 thus for r1 being Real st x0 < r1 holds ex g1 being Real st ( g1 < r1 & x0 < g1 & g1 in dom (f ^) ) :: according to LIMFUNC2:def_5 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom (f ^)) /\ (right_open_halfline x0) holds (f ^) /* seq is divergent_to+infty proof let r1 be Real; ::_thesis: ( x0 < r1 implies ex g1 being Real st ( g1 < r1 & x0 < g1 & g1 in dom (f ^) ) ) assume x0 < r1 ; ::_thesis: ex g1 being Real st ( g1 < r1 & x0 < g1 & g1 in dom (f ^) ) then consider g1 being Real such that A6: g1 < r1 and A7: x0 < g1 and A8: g1 in dom f and A9: f . g1 <> 0 by A3; take g1 ; ::_thesis: ( g1 < r1 & x0 < g1 & g1 in dom (f ^) ) thus ( g1 < r1 & x0 < g1 ) by A6, A7; ::_thesis: g1 in dom (f ^) not f . g1 in {0} by A9, TARSKI:def_1; then not g1 in f " {0} by FUNCT_1:def_7; then g1 in (dom f) \ (f " {0}) by A8, XBOOLE_0:def_5; hence g1 in dom (f ^) by RFUNCT_1:def_2; ::_thesis: verum end; 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 A10: s is convergent and A11: lim s = x0 and A12: rng s c= (dom (f ^)) /\ (right_open_halfline x0) ; ::_thesis: (f ^) /* s is divergent_to+infty x0 < x0 + r by A4, Lm1; then consider k being Element of NAT such that A13: for n being Element of NAT st k <= n holds s . n < x0 + r by A10, A11, Th2; A14: lim (s ^\ k) = x0 by A10, A11, SEQ_4:20; A15: (dom (f ^)) /\ (right_open_halfline x0) c= dom (f ^) by XBOOLE_1:17; then A16: rng s c= dom (f ^) by A12, XBOOLE_1:1; A17: rng (s ^\ k) c= rng s by VALUED_0:21; dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def_2; then A18: dom (f ^) c= dom f by XBOOLE_1:36; then A19: rng s c= dom f by A16, XBOOLE_1:1; then A20: rng (s ^\ k) c= dom f by A17, XBOOLE_1:1; (dom (f ^)) /\ (right_open_halfline x0) c= right_open_halfline x0 by XBOOLE_1:17; then rng s c= right_open_halfline x0 by A12, XBOOLE_1:1; then A21: rng (s ^\ k) c= right_open_halfline x0 by A17, XBOOLE_1:1; then A22: rng (s ^\ k) c= (dom f) /\ (right_open_halfline x0) by A20, XBOOLE_1:19; then A23: lim (f /* (s ^\ k)) = 0 by A1, A2, A10, A14, Def8; A24: f /* (s ^\ k) is non-zero by A16, A17, 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 s . (n + k) < x0 + r by A13, NAT_1:12; then A25: (s ^\ k) . n < x0 + r by NAT_1:def_3; A26: (s ^\ k) . n in rng (s ^\ k) by VALUED_0:28; then (s ^\ k) . n in right_open_halfline x0 by A21; then (s ^\ k) . n in { g1 where g1 is Real : x0 < g1 } by XXREAL_1:230; then ex g1 being Real st ( g1 = (s ^\ k) . n & x0 < g1 ) ; then (s ^\ k) . n in { g2 where g2 is Real : ( x0 < g2 & g2 < x0 + r ) } by A25; then (s ^\ k) . n in ].x0,(x0 + r).[ by RCOMP_1:def_2; then (s ^\ k) . n in (dom f) /\ ].x0,(x0 + r).[ by A20, A26, XBOOLE_0:def_4; then A27: 0 <= f . ((s ^\ k) . n) by A5; 0 <> (f /* (s ^\ k)) . n by A24, SEQ_1:5; hence 0 < (f /* (s ^\ k)) . n by A19, A17, A27, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; then A28: for n being Element of NAT st 0 <= n holds 0 < (f /* (s ^\ k)) . n ; f /* (s ^\ k) is convergent by A1, A2, A10, A14, A22, Def8; then A29: (f /* (s ^\ k)) " is divergent_to+infty by A23, A28, LIMFUNC1:35; (f /* (s ^\ k)) " = ((f /* s) ^\ k) " by A16, A18, VALUED_0:27, XBOOLE_1:1 .= ((f /* s) ") ^\ k by SEQM_3:18 .= ((f ^) /* s) ^\ k by A12, A15, RFUNCT_2:12, XBOOLE_1:1 ; hence (f ^) /* s is divergent_to+infty by A29, LIMFUNC1:7; ::_thesis: verum end; theorem :: LIMFUNC2:78 for x0 being Real for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 & lim_right (f,x0) = 0 & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f & f . g <> 0 ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds f . g <= 0 ) ) holds f ^ is_right_divergent_to-infty_in x0 proof let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_right_convergent_in x0 & lim_right (f,x0) = 0 & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f & f . g <> 0 ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds f . g <= 0 ) ) holds f ^ is_right_divergent_to-infty_in x0 let f be PartFunc of REAL,REAL; ::_thesis: ( f is_right_convergent_in x0 & lim_right (f,x0) = 0 & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f & f . g <> 0 ) ) & ex r being Real st ( 0 < r & ( for g being Real st g in (dom f) /\ ].x0,(x0 + r).[ holds f . g <= 0 ) ) implies f ^ is_right_divergent_to-infty_in x0 ) assume that A1: f is_right_convergent_in x0 and A2: lim_right (f,x0) = 0 and A3: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom f & f . g <> 0 ) ; ::_thesis: ( for r being Real holds ( not 0 < r or ex g being Real st ( g in (dom f) /\ ].x0,(x0 + r).[ & not f . g <= 0 ) ) or f ^ is_right_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,(x0 + r).[ holds f . g <= 0 ; ::_thesis: f ^ is_right_divergent_to-infty_in x0 thus for r1 being Real st x0 < r1 holds ex g1 being Real st ( g1 < r1 & x0 < g1 & g1 in dom (f ^) ) :: according to LIMFUNC2:def_6 ::_thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom (f ^)) /\ (right_open_halfline x0) holds (f ^) /* seq is divergent_to-infty proof let r1 be Real; ::_thesis: ( x0 < r1 implies ex g1 being Real st ( g1 < r1 & x0 < g1 & g1 in dom (f ^) ) ) assume x0 < r1 ; ::_thesis: ex g1 being Real st ( g1 < r1 & x0 < g1 & g1 in dom (f ^) ) then consider g1 being Real such that A6: g1 < r1 and A7: x0 < g1 and A8: g1 in dom f and A9: f . g1 <> 0 by A3; take g1 ; ::_thesis: ( g1 < r1 & x0 < g1 & g1 in dom (f ^) ) thus ( g1 < r1 & x0 < g1 ) by A6, A7; ::_thesis: g1 in dom (f ^) not f . g1 in {0} by A9, TARSKI:def_1; then not g1 in f " {0} by FUNCT_1:def_7; then g1 in (dom f) \ (f " {0}) by A8, XBOOLE_0:def_5; hence g1 in dom (f ^) by RFUNCT_1:def_2; ::_thesis: verum end; 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 A10: s is convergent and A11: lim s = x0 and A12: rng s c= (dom (f ^)) /\ (right_open_halfline x0) ; ::_thesis: (f ^) /* s is divergent_to-infty x0 < x0 + r by A4, Lm1; then consider k being Element of NAT such that A13: for n being Element of NAT st k <= n holds s . n < x0 + r by A10, A11, Th2; A14: lim (s ^\ k) = x0 by A10, A11, SEQ_4:20; A15: (dom (f ^)) /\ (right_open_halfline x0) c= dom (f ^) by XBOOLE_1:17; then A16: rng s c= dom (f ^) by A12, XBOOLE_1:1; A17: rng (s ^\ k) c= rng s by VALUED_0:21; dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def_2; then A18: dom (f ^) c= dom f by XBOOLE_1:36; then A19: rng s c= dom f by A16, XBOOLE_1:1; then A20: rng (s ^\ k) c= dom f by A17, XBOOLE_1:1; (dom (f ^)) /\ (right_open_halfline x0) c= right_open_halfline x0 by XBOOLE_1:17; then rng s c= right_open_halfline x0 by A12, XBOOLE_1:1; then A21: rng (s ^\ k) c= right_open_halfline x0 by A17, XBOOLE_1:1; then A22: rng (s ^\ k) c= (dom f) /\ (right_open_halfline x0) by A20, XBOOLE_1:19; then A23: lim (f /* (s ^\ k)) = 0 by A1, A2, A10, A14, Def8; A24: f /* (s ^\ k) is non-zero by A16, A17, 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 s . (n + k) < x0 + r by A13, NAT_1:12; then A25: (s ^\ k) . n < x0 + r by NAT_1:def_3; A26: (s ^\ k) . n in rng (s ^\ k) by VALUED_0:28; then (s ^\ k) . n in right_open_halfline x0 by A21; then (s ^\ k) . n in { g1 where g1 is Real : x0 < g1 } by XXREAL_1:230; then ex g1 being Real st ( g1 = (s ^\ k) . n & x0 < g1 ) ; then (s ^\ k) . n in { g2 where g2 is Real : ( x0 < g2 & g2 < x0 + r ) } by A25; then (s ^\ k) . n in ].x0,(x0 + r).[ by RCOMP_1:def_2; then (s ^\ k) . n in (dom f) /\ ].x0,(x0 + r).[ by A20, A26, XBOOLE_0:def_4; then A27: f . ((s ^\ k) . n) <= 0 by A5; (f /* (s ^\ k)) . n <> 0 by A24, SEQ_1:5; hence (f /* (s ^\ k)) . n < 0 by A19, A17, A27, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; then A28: for n being Element of NAT st 0 <= n holds (f /* (s ^\ k)) . n < 0 ; f /* (s ^\ k) is convergent by A1, A2, A10, A14, A22, Def8; then A29: (f /* (s ^\ k)) " is divergent_to-infty by A23, A28, LIMFUNC1:36; (f /* (s ^\ k)) " = ((f /* s) ^\ k) " by A16, A18, VALUED_0:27, XBOOLE_1:1 .= ((f /* s) ") ^\ k by SEQM_3:18 .= ((f ^) /* s) ^\ k by A12, A15, RFUNCT_2:12, XBOOLE_1:1 ; hence (f ^) /* s is divergent_to-infty by A29, LIMFUNC1:7; ::_thesis: verum end;