:: LIMFUNC4 semantic presentation begin Lm1: for g, r being Real st 0 < g holds ( r - g < r & r < r + g ) proof let g, r be Real; ::_thesis: ( 0 < g implies ( r - g < r & r < r + g ) ) assume A1: 0 < g ; ::_thesis: ( r - g < r & r < r + g ) then r - g < r - 0 by XREAL_1:15; hence r - g < r ; ::_thesis: r < r + g r + 0 < r + g by A1, XREAL_1:8; hence r < r + g ; ::_thesis: verum end; Lm2: for f2, f1 being PartFunc of REAL,REAL for s being Real_Sequence st rng s c= dom (f2 * f1) holds ( rng s c= dom f1 & rng (f1 /* s) c= dom f2 ) proof let f2, f1 be PartFunc of REAL,REAL; ::_thesis: for s being Real_Sequence st rng s c= dom (f2 * f1) holds ( rng s c= dom f1 & rng (f1 /* s) c= dom f2 ) let s be Real_Sequence; ::_thesis: ( rng s c= dom (f2 * f1) implies ( rng s c= dom f1 & rng (f1 /* s) c= dom f2 ) ) assume A1: rng s c= dom (f2 * f1) ; ::_thesis: ( rng s c= dom f1 & rng (f1 /* s) c= dom f2 ) A2: dom (f2 * f1) c= dom f1 by RELAT_1:25; hence rng s c= dom f1 by A1, XBOOLE_1:1; ::_thesis: rng (f1 /* s) c= dom f2 let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (f1 /* s) or x in dom f2 ) assume x in rng (f1 /* s) ; ::_thesis: x in dom f2 then consider n being Element of NAT such that A3: (f1 /* s) . n = x by FUNCT_2:113; s . n in rng s by VALUED_0:28; then f1 . (s . n) in dom f2 by A1, FUNCT_1:11; hence x in dom f2 by A1, A2, A3, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; theorem Th1: :: LIMFUNC4:1 for f2, f1 being PartFunc of REAL,REAL for s being Real_Sequence for X being set st rng s c= (dom (f2 * f1)) /\ X holds ( rng s c= dom (f2 * f1) & rng s c= X & rng s c= dom f1 & rng s c= (dom f1) /\ X & rng (f1 /* s) c= dom f2 ) proof let f2, f1 be PartFunc of REAL,REAL; ::_thesis: for s being Real_Sequence for X being set st rng s c= (dom (f2 * f1)) /\ X holds ( rng s c= dom (f2 * f1) & rng s c= X & rng s c= dom f1 & rng s c= (dom f1) /\ X & rng (f1 /* s) c= dom f2 ) let s be Real_Sequence; ::_thesis: for X being set st rng s c= (dom (f2 * f1)) /\ X holds ( rng s c= dom (f2 * f1) & rng s c= X & rng s c= dom f1 & rng s c= (dom f1) /\ X & rng (f1 /* s) c= dom f2 ) let X be set ; ::_thesis: ( rng s c= (dom (f2 * f1)) /\ X implies ( rng s c= dom (f2 * f1) & rng s c= X & rng s c= dom f1 & rng s c= (dom f1) /\ X & rng (f1 /* s) c= dom f2 ) ) assume rng s c= (dom (f2 * f1)) /\ X ; ::_thesis: ( rng s c= dom (f2 * f1) & rng s c= X & rng s c= dom f1 & rng s c= (dom f1) /\ X & rng (f1 /* s) c= dom f2 ) hence A1: ( rng s c= dom (f2 * f1) & rng s c= X ) by XBOOLE_1:18; ::_thesis: ( rng s c= dom f1 & rng s c= (dom f1) /\ X & rng (f1 /* s) c= dom f2 ) A2: dom (f2 * f1) c= dom f1 by RELAT_1:25; hence rng s c= dom f1 by A1, XBOOLE_1:1; ::_thesis: ( rng s c= (dom f1) /\ X & rng (f1 /* s) c= dom f2 ) hence rng s c= (dom f1) /\ X by A1, XBOOLE_1:19; ::_thesis: rng (f1 /* s) c= dom f2 let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (f1 /* s) or x in dom f2 ) assume x in rng (f1 /* s) ; ::_thesis: x in dom f2 then consider n being Element of NAT such that A3: (f1 /* s) . n = x by FUNCT_2:113; s . n in rng s by VALUED_0:28; then f1 . (s . n) in dom f2 by A1, FUNCT_1:11; hence x in dom f2 by A1, A2, A3, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; theorem Th2: :: LIMFUNC4:2 for f2, f1 being PartFunc of REAL,REAL for s being Real_Sequence for X being set st rng s c= (dom (f2 * f1)) \ X holds ( rng s c= dom (f2 * f1) & rng s c= dom f1 & rng s c= (dom f1) \ X & rng (f1 /* s) c= dom f2 ) proof let f2, f1 be PartFunc of REAL,REAL; ::_thesis: for s being Real_Sequence for X being set st rng s c= (dom (f2 * f1)) \ X holds ( rng s c= dom (f2 * f1) & rng s c= dom f1 & rng s c= (dom f1) \ X & rng (f1 /* s) c= dom f2 ) let s be Real_Sequence; ::_thesis: for X being set st rng s c= (dom (f2 * f1)) \ X holds ( rng s c= dom (f2 * f1) & rng s c= dom f1 & rng s c= (dom f1) \ X & rng (f1 /* s) c= dom f2 ) let X be set ; ::_thesis: ( rng s c= (dom (f2 * f1)) \ X implies ( rng s c= dom (f2 * f1) & rng s c= dom f1 & rng s c= (dom f1) \ X & rng (f1 /* s) c= dom f2 ) ) assume A1: rng s c= (dom (f2 * f1)) \ X ; ::_thesis: ( rng s c= dom (f2 * f1) & rng s c= dom f1 & rng s c= (dom f1) \ X & rng (f1 /* s) c= dom f2 ) (dom (f2 * f1)) \ X c= dom (f2 * f1) by XBOOLE_1:36; hence A2: rng s c= dom (f2 * f1) by A1, XBOOLE_1:1; ::_thesis: ( rng s c= dom f1 & rng s c= (dom f1) \ X & rng (f1 /* s) c= dom f2 ) A3: dom (f2 * f1) c= dom f1 by RELAT_1:25; hence A4: rng s c= dom f1 by A2, XBOOLE_1:1; ::_thesis: ( rng s c= (dom f1) \ X & rng (f1 /* s) c= dom f2 ) thus rng s c= (dom f1) \ X ::_thesis: rng (f1 /* s) c= dom f2 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng s or x in (dom f1) \ X ) assume A5: x in rng s ; ::_thesis: x in (dom f1) \ X then not x in X by A1, XBOOLE_0:def_5; hence x in (dom f1) \ X by A4, A5, XBOOLE_0:def_5; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (f1 /* s) or x in dom f2 ) assume x in rng (f1 /* s) ; ::_thesis: x in dom f2 then consider n being Element of NAT such that A6: (f1 /* s) . n = x by FUNCT_2:113; s . n in rng s by VALUED_0:28; then f1 . (s . n) in dom f2 by A2, FUNCT_1:11; hence x in dom f2 by A2, A3, A6, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; theorem :: LIMFUNC4:3 for f1, f2 being PartFunc of REAL,REAL st f1 is divergent_in+infty_to+infty & f2 is divergent_in+infty_to+infty & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) holds f2 * f1 is divergent_in+infty_to+infty proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in+infty_to+infty & f2 is divergent_in+infty_to+infty & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) implies f2 * f1 is divergent_in+infty_to+infty ) assume that A1: f1 is divergent_in+infty_to+infty and A2: f2 is divergent_in+infty_to+infty and A3: for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ; ::_thesis: f2 * f1 is divergent_in+infty_to+infty now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to+infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (f2_*_f1)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is divergent_to+infty & rng s c= dom (f2 * f1) implies (f2 * f1) /* s is divergent_to+infty ) assume that A4: s is divergent_to+infty and A5: rng s c= dom (f2 * f1) ; ::_thesis: (f2 * f1) /* s is divergent_to+infty rng s c= dom f1 by A5, Lm2; then A6: f1 /* s is divergent_to+infty by A1, A4, LIMFUNC1:def_7; rng (f1 /* s) c= dom f2 by A5, Lm2; then f2 /* (f1 /* s) is divergent_to+infty by A2, A6, LIMFUNC1:def_7; hence (f2 * f1) /* s is divergent_to+infty by A5, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is divergent_in+infty_to+infty by A3, LIMFUNC1:def_7; ::_thesis: verum end; theorem :: LIMFUNC4:4 for f1, f2 being PartFunc of REAL,REAL st f1 is divergent_in+infty_to+infty & f2 is divergent_in+infty_to-infty & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) holds f2 * f1 is divergent_in+infty_to-infty proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in+infty_to+infty & f2 is divergent_in+infty_to-infty & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) implies f2 * f1 is divergent_in+infty_to-infty ) assume that A1: f1 is divergent_in+infty_to+infty and A2: f2 is divergent_in+infty_to-infty and A3: for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ; ::_thesis: f2 * f1 is divergent_in+infty_to-infty now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to+infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (f2_*_f1)_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is divergent_to+infty & rng s c= dom (f2 * f1) implies (f2 * f1) /* s is divergent_to-infty ) assume that A4: s is divergent_to+infty and A5: rng s c= dom (f2 * f1) ; ::_thesis: (f2 * f1) /* s is divergent_to-infty rng s c= dom f1 by A5, Lm2; then A6: f1 /* s is divergent_to+infty by A1, A4, LIMFUNC1:def_7; rng (f1 /* s) c= dom f2 by A5, Lm2; then f2 /* (f1 /* s) is divergent_to-infty by A2, A6, LIMFUNC1:def_8; hence (f2 * f1) /* s is divergent_to-infty by A5, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is divergent_in+infty_to-infty by A3, LIMFUNC1:def_8; ::_thesis: verum end; theorem :: LIMFUNC4:5 for f1, f2 being PartFunc of REAL,REAL st f1 is divergent_in+infty_to-infty & f2 is divergent_in-infty_to+infty & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) holds f2 * f1 is divergent_in+infty_to+infty proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in+infty_to-infty & f2 is divergent_in-infty_to+infty & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) implies f2 * f1 is divergent_in+infty_to+infty ) assume that A1: f1 is divergent_in+infty_to-infty and A2: f2 is divergent_in-infty_to+infty and A3: for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ; ::_thesis: f2 * f1 is divergent_in+infty_to+infty now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to+infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (f2_*_f1)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is divergent_to+infty & rng s c= dom (f2 * f1) implies (f2 * f1) /* s is divergent_to+infty ) assume that A4: s is divergent_to+infty and A5: rng s c= dom (f2 * f1) ; ::_thesis: (f2 * f1) /* s is divergent_to+infty rng s c= dom f1 by A5, Lm2; then A6: f1 /* s is divergent_to-infty by A1, A4, LIMFUNC1:def_8; rng (f1 /* s) c= dom f2 by A5, Lm2; then f2 /* (f1 /* s) is divergent_to+infty by A2, A6, LIMFUNC1:def_10; hence (f2 * f1) /* s is divergent_to+infty by A5, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is divergent_in+infty_to+infty by A3, LIMFUNC1:def_7; ::_thesis: verum end; theorem :: LIMFUNC4:6 for f1, f2 being PartFunc of REAL,REAL st f1 is divergent_in+infty_to-infty & f2 is divergent_in-infty_to-infty & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) holds f2 * f1 is divergent_in+infty_to-infty proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in+infty_to-infty & f2 is divergent_in-infty_to-infty & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) implies f2 * f1 is divergent_in+infty_to-infty ) assume that A1: f1 is divergent_in+infty_to-infty and A2: f2 is divergent_in-infty_to-infty and A3: for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ; ::_thesis: f2 * f1 is divergent_in+infty_to-infty now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to+infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (f2_*_f1)_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is divergent_to+infty & rng s c= dom (f2 * f1) implies (f2 * f1) /* s is divergent_to-infty ) assume that A4: s is divergent_to+infty and A5: rng s c= dom (f2 * f1) ; ::_thesis: (f2 * f1) /* s is divergent_to-infty rng s c= dom f1 by A5, Lm2; then A6: f1 /* s is divergent_to-infty by A1, A4, LIMFUNC1:def_8; rng (f1 /* s) c= dom f2 by A5, Lm2; then f2 /* (f1 /* s) is divergent_to-infty by A2, A6, LIMFUNC1:def_11; hence (f2 * f1) /* s is divergent_to-infty by A5, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is divergent_in+infty_to-infty by A3, LIMFUNC1:def_8; ::_thesis: verum end; theorem :: LIMFUNC4:7 for f1, f2 being PartFunc of REAL,REAL st f1 is divergent_in-infty_to+infty & f2 is divergent_in+infty_to+infty & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) holds f2 * f1 is divergent_in-infty_to+infty proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in-infty_to+infty & f2 is divergent_in+infty_to+infty & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) implies f2 * f1 is divergent_in-infty_to+infty ) assume that A1: f1 is divergent_in-infty_to+infty and A2: f2 is divergent_in+infty_to+infty and A3: for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ; ::_thesis: f2 * f1 is divergent_in-infty_to+infty now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to-infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (f2_*_f1)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is divergent_to-infty & rng s c= dom (f2 * f1) implies (f2 * f1) /* s is divergent_to+infty ) assume that A4: s is divergent_to-infty and A5: rng s c= dom (f2 * f1) ; ::_thesis: (f2 * f1) /* s is divergent_to+infty rng s c= dom f1 by A5, Lm2; then A6: f1 /* s is divergent_to+infty by A1, A4, LIMFUNC1:def_10; rng (f1 /* s) c= dom f2 by A5, Lm2; then f2 /* (f1 /* s) is divergent_to+infty by A2, A6, LIMFUNC1:def_7; hence (f2 * f1) /* s is divergent_to+infty by A5, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is divergent_in-infty_to+infty by A3, LIMFUNC1:def_10; ::_thesis: verum end; theorem :: LIMFUNC4:8 for f1, f2 being PartFunc of REAL,REAL st f1 is divergent_in-infty_to+infty & f2 is divergent_in+infty_to-infty & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) holds f2 * f1 is divergent_in-infty_to-infty proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in-infty_to+infty & f2 is divergent_in+infty_to-infty & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) implies f2 * f1 is divergent_in-infty_to-infty ) assume that A1: f1 is divergent_in-infty_to+infty and A2: f2 is divergent_in+infty_to-infty and A3: for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ; ::_thesis: f2 * f1 is divergent_in-infty_to-infty now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to-infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (f2_*_f1)_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is divergent_to-infty & rng s c= dom (f2 * f1) implies (f2 * f1) /* s is divergent_to-infty ) assume that A4: s is divergent_to-infty and A5: rng s c= dom (f2 * f1) ; ::_thesis: (f2 * f1) /* s is divergent_to-infty rng s c= dom f1 by A5, Lm2; then A6: f1 /* s is divergent_to+infty by A1, A4, LIMFUNC1:def_10; rng (f1 /* s) c= dom f2 by A5, Lm2; then f2 /* (f1 /* s) is divergent_to-infty by A2, A6, LIMFUNC1:def_8; hence (f2 * f1) /* s is divergent_to-infty by A5, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is divergent_in-infty_to-infty by A3, LIMFUNC1:def_11; ::_thesis: verum end; theorem :: LIMFUNC4:9 for f1, f2 being PartFunc of REAL,REAL st f1 is divergent_in-infty_to-infty & f2 is divergent_in-infty_to+infty & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) holds f2 * f1 is divergent_in-infty_to+infty proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in-infty_to-infty & f2 is divergent_in-infty_to+infty & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) implies f2 * f1 is divergent_in-infty_to+infty ) assume that A1: f1 is divergent_in-infty_to-infty and A2: f2 is divergent_in-infty_to+infty and A3: for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ; ::_thesis: f2 * f1 is divergent_in-infty_to+infty now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to-infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (f2_*_f1)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is divergent_to-infty & rng s c= dom (f2 * f1) implies (f2 * f1) /* s is divergent_to+infty ) assume that A4: s is divergent_to-infty and A5: rng s c= dom (f2 * f1) ; ::_thesis: (f2 * f1) /* s is divergent_to+infty rng s c= dom f1 by A5, Lm2; then A6: f1 /* s is divergent_to-infty by A1, A4, LIMFUNC1:def_11; rng (f1 /* s) c= dom f2 by A5, Lm2; then f2 /* (f1 /* s) is divergent_to+infty by A2, A6, LIMFUNC1:def_10; hence (f2 * f1) /* s is divergent_to+infty by A5, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is divergent_in-infty_to+infty by A3, LIMFUNC1:def_10; ::_thesis: verum end; theorem :: LIMFUNC4:10 for f1, f2 being PartFunc of REAL,REAL st f1 is divergent_in-infty_to-infty & f2 is divergent_in-infty_to-infty & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) holds f2 * f1 is divergent_in-infty_to-infty proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in-infty_to-infty & f2 is divergent_in-infty_to-infty & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) implies f2 * f1 is divergent_in-infty_to-infty ) assume that A1: f1 is divergent_in-infty_to-infty and A2: f2 is divergent_in-infty_to-infty and A3: for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ; ::_thesis: f2 * f1 is divergent_in-infty_to-infty now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to-infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (f2_*_f1)_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is divergent_to-infty & rng s c= dom (f2 * f1) implies (f2 * f1) /* s is divergent_to-infty ) assume that A4: s is divergent_to-infty and A5: rng s c= dom (f2 * f1) ; ::_thesis: (f2 * f1) /* s is divergent_to-infty rng s c= dom f1 by A5, Lm2; then A6: f1 /* s is divergent_to-infty by A1, A4, LIMFUNC1:def_11; rng (f1 /* s) c= dom f2 by A5, Lm2; then f2 /* (f1 /* s) is divergent_to-infty by A2, A6, LIMFUNC1:def_11; hence (f2 * f1) /* s is divergent_to-infty by A5, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is divergent_in-infty_to-infty by A3, LIMFUNC1:def_11; ::_thesis: verum end; theorem :: LIMFUNC4:11 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_left_divergent_to+infty_in x0 & f2 is divergent_in+infty_to+infty & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) holds f2 * f1 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 divergent_in+infty_to+infty & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) holds f2 * f1 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 divergent_in+infty_to+infty & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) implies f2 * f1 is_left_divergent_to+infty_in x0 ) assume that A1: f1 is_left_divergent_to+infty_in x0 and A2: f2 is divergent_in+infty_to+infty and A3: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ; ::_thesis: f2 * f1 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_(f2_*_f1))_/\_(left_open_halfline_x0)_holds_ (f2_*_f1)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) implies (f2 * f1) /* s is divergent_to+infty ) assume that A4: ( s is convergent & lim s = x0 ) and A5: rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) ; ::_thesis: (f2 * f1) /* s is divergent_to+infty A6: rng s c= dom (f2 * f1) by A5, Th1; rng s c= (dom f1) /\ (left_open_halfline x0) by A5, Th1; then A7: f1 /* s is divergent_to+infty by A1, A4, LIMFUNC2:def_2; rng (f1 /* s) c= dom f2 by A5, Th1; then f2 /* (f1 /* s) is divergent_to+infty by A2, A7, LIMFUNC1:def_7; hence (f2 * f1) /* s is divergent_to+infty by A6, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is_left_divergent_to+infty_in x0 by A3, LIMFUNC2:def_2; ::_thesis: verum end; theorem :: LIMFUNC4:12 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_left_divergent_to+infty_in x0 & f2 is divergent_in+infty_to-infty & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) holds f2 * f1 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 divergent_in+infty_to-infty & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) holds f2 * f1 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 divergent_in+infty_to-infty & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) implies f2 * f1 is_left_divergent_to-infty_in x0 ) assume that A1: f1 is_left_divergent_to+infty_in x0 and A2: f2 is divergent_in+infty_to-infty and A3: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ; ::_thesis: f2 * f1 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_(f2_*_f1))_/\_(left_open_halfline_x0)_holds_ (f2_*_f1)_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) implies (f2 * f1) /* s is divergent_to-infty ) assume that A4: ( s is convergent & lim s = x0 ) and A5: rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) ; ::_thesis: (f2 * f1) /* s is divergent_to-infty A6: rng s c= dom (f2 * f1) by A5, Th1; rng s c= (dom f1) /\ (left_open_halfline x0) by A5, Th1; then A7: f1 /* s is divergent_to+infty by A1, A4, LIMFUNC2:def_2; rng (f1 /* s) c= dom f2 by A5, Th1; then f2 /* (f1 /* s) is divergent_to-infty by A2, A7, LIMFUNC1:def_8; hence (f2 * f1) /* s is divergent_to-infty by A6, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is_left_divergent_to-infty_in x0 by A3, LIMFUNC2:def_3; ::_thesis: verum end; theorem :: LIMFUNC4:13 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_left_divergent_to-infty_in x0 & f2 is divergent_in-infty_to+infty & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) holds f2 * f1 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 divergent_in-infty_to+infty & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) holds f2 * f1 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 divergent_in-infty_to+infty & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) implies f2 * f1 is_left_divergent_to+infty_in x0 ) assume that A1: f1 is_left_divergent_to-infty_in x0 and A2: f2 is divergent_in-infty_to+infty and A3: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ; ::_thesis: f2 * f1 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_(f2_*_f1))_/\_(left_open_halfline_x0)_holds_ (f2_*_f1)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) implies (f2 * f1) /* s is divergent_to+infty ) assume that A4: ( s is convergent & lim s = x0 ) and A5: rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) ; ::_thesis: (f2 * f1) /* s is divergent_to+infty A6: rng s c= dom (f2 * f1) by A5, Th1; rng s c= (dom f1) /\ (left_open_halfline x0) by A5, Th1; then A7: f1 /* s is divergent_to-infty by A1, A4, LIMFUNC2:def_3; rng (f1 /* s) c= dom f2 by A5, Th1; then f2 /* (f1 /* s) is divergent_to+infty by A2, A7, LIMFUNC1:def_10; hence (f2 * f1) /* s is divergent_to+infty by A6, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is_left_divergent_to+infty_in x0 by A3, LIMFUNC2:def_2; ::_thesis: verum end; theorem :: LIMFUNC4:14 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_left_divergent_to-infty_in x0 & f2 is divergent_in-infty_to-infty & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) holds f2 * f1 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 divergent_in-infty_to-infty & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) holds f2 * f1 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 divergent_in-infty_to-infty & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) implies f2 * f1 is_left_divergent_to-infty_in x0 ) assume that A1: f1 is_left_divergent_to-infty_in x0 and A2: f2 is divergent_in-infty_to-infty and A3: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ; ::_thesis: f2 * f1 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_(f2_*_f1))_/\_(left_open_halfline_x0)_holds_ (f2_*_f1)_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) implies (f2 * f1) /* s is divergent_to-infty ) assume that A4: ( s is convergent & lim s = x0 ) and A5: rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) ; ::_thesis: (f2 * f1) /* s is divergent_to-infty A6: rng s c= dom (f2 * f1) by A5, Th1; rng s c= (dom f1) /\ (left_open_halfline x0) by A5, Th1; then A7: f1 /* s is divergent_to-infty by A1, A4, LIMFUNC2:def_3; rng (f1 /* s) c= dom f2 by A5, Th1; then f2 /* (f1 /* s) is divergent_to-infty by A2, A7, LIMFUNC1:def_11; hence (f2 * f1) /* s is divergent_to-infty by A6, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is_left_divergent_to-infty_in x0 by A3, LIMFUNC2:def_3; ::_thesis: verum end; theorem :: LIMFUNC4:15 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_right_divergent_to+infty_in x0 & f2 is divergent_in+infty_to+infty & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) holds f2 * f1 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 divergent_in+infty_to+infty & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) holds f2 * f1 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 divergent_in+infty_to+infty & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) implies f2 * f1 is_right_divergent_to+infty_in x0 ) assume that A1: f1 is_right_divergent_to+infty_in x0 and A2: f2 is divergent_in+infty_to+infty and A3: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ; ::_thesis: f2 * f1 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_(f2_*_f1))_/\_(right_open_halfline_x0)_holds_ (f2_*_f1)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) implies (f2 * f1) /* s is divergent_to+infty ) assume that A4: ( s is convergent & lim s = x0 ) and A5: rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) ; ::_thesis: (f2 * f1) /* s is divergent_to+infty A6: rng s c= dom (f2 * f1) by A5, Th1; rng s c= (dom f1) /\ (right_open_halfline x0) by A5, Th1; then A7: f1 /* s is divergent_to+infty by A1, A4, LIMFUNC2:def_5; rng (f1 /* s) c= dom f2 by A5, Th1; then f2 /* (f1 /* s) is divergent_to+infty by A2, A7, LIMFUNC1:def_7; hence (f2 * f1) /* s is divergent_to+infty by A6, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is_right_divergent_to+infty_in x0 by A3, LIMFUNC2:def_5; ::_thesis: verum end; theorem :: LIMFUNC4:16 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_right_divergent_to+infty_in x0 & f2 is divergent_in+infty_to-infty & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) holds f2 * f1 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 divergent_in+infty_to-infty & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) holds f2 * f1 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 divergent_in+infty_to-infty & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) implies f2 * f1 is_right_divergent_to-infty_in x0 ) assume that A1: f1 is_right_divergent_to+infty_in x0 and A2: f2 is divergent_in+infty_to-infty and A3: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ; ::_thesis: f2 * f1 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_(f2_*_f1))_/\_(right_open_halfline_x0)_holds_ (f2_*_f1)_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) implies (f2 * f1) /* s is divergent_to-infty ) assume that A4: ( s is convergent & lim s = x0 ) and A5: rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) ; ::_thesis: (f2 * f1) /* s is divergent_to-infty A6: rng s c= dom (f2 * f1) by A5, Th1; rng s c= (dom f1) /\ (right_open_halfline x0) by A5, Th1; then A7: f1 /* s is divergent_to+infty by A1, A4, LIMFUNC2:def_5; rng (f1 /* s) c= dom f2 by A5, Th1; then f2 /* (f1 /* s) is divergent_to-infty by A2, A7, LIMFUNC1:def_8; hence (f2 * f1) /* s is divergent_to-infty by A6, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is_right_divergent_to-infty_in x0 by A3, LIMFUNC2:def_6; ::_thesis: verum end; theorem :: LIMFUNC4:17 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_right_divergent_to-infty_in x0 & f2 is divergent_in-infty_to+infty & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) holds f2 * f1 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 divergent_in-infty_to+infty & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) holds f2 * f1 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 divergent_in-infty_to+infty & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) implies f2 * f1 is_right_divergent_to+infty_in x0 ) assume that A1: f1 is_right_divergent_to-infty_in x0 and A2: f2 is divergent_in-infty_to+infty and A3: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ; ::_thesis: f2 * f1 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_(f2_*_f1))_/\_(right_open_halfline_x0)_holds_ (f2_*_f1)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) implies (f2 * f1) /* s is divergent_to+infty ) assume that A4: ( s is convergent & lim s = x0 ) and A5: rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) ; ::_thesis: (f2 * f1) /* s is divergent_to+infty A6: rng s c= dom (f2 * f1) by A5, Th1; rng s c= (dom f1) /\ (right_open_halfline x0) by A5, Th1; then A7: f1 /* s is divergent_to-infty by A1, A4, LIMFUNC2:def_6; rng (f1 /* s) c= dom f2 by A5, Th1; then f2 /* (f1 /* s) is divergent_to+infty by A2, A7, LIMFUNC1:def_10; hence (f2 * f1) /* s is divergent_to+infty by A6, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is_right_divergent_to+infty_in x0 by A3, LIMFUNC2:def_5; ::_thesis: verum end; theorem :: LIMFUNC4:18 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_right_divergent_to-infty_in x0 & f2 is divergent_in-infty_to-infty & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) holds f2 * f1 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 divergent_in-infty_to-infty & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) holds f2 * f1 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 divergent_in-infty_to-infty & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) implies f2 * f1 is_right_divergent_to-infty_in x0 ) assume that A1: f1 is_right_divergent_to-infty_in x0 and A2: f2 is divergent_in-infty_to-infty and A3: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ; ::_thesis: f2 * f1 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_(f2_*_f1))_/\_(right_open_halfline_x0)_holds_ (f2_*_f1)_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) implies (f2 * f1) /* s is divergent_to-infty ) assume that A4: ( s is convergent & lim s = x0 ) and A5: rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) ; ::_thesis: (f2 * f1) /* s is divergent_to-infty A6: rng s c= dom (f2 * f1) by A5, Th1; rng s c= (dom f1) /\ (right_open_halfline x0) by A5, Th1; then A7: f1 /* s is divergent_to-infty by A1, A4, LIMFUNC2:def_6; rng (f1 /* s) c= dom f2 by A5, Th1; then f2 /* (f1 /* s) is divergent_to-infty by A2, A7, LIMFUNC1:def_11; hence (f2 * f1) /* s is divergent_to-infty by A6, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is_right_divergent_to-infty_in x0 by A3, LIMFUNC2:def_6; ::_thesis: verum end; theorem :: LIMFUNC4:19 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_divergent_to+infty_in lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds f1 . r < lim_left (f1,x0) ) ) holds f2 * f1 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_convergent_in x0 & f2 is_left_divergent_to+infty_in lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds f1 . r < lim_left (f1,x0) ) ) holds f2 * f1 is_left_divergent_to+infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_convergent_in x0 & f2 is_left_divergent_to+infty_in lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds f1 . r < lim_left (f1,x0) ) ) implies f2 * f1 is_left_divergent_to+infty_in x0 ) assume that A1: f1 is_left_convergent_in x0 and A2: f2 is_left_divergent_to+infty_in lim_left (f1,x0) and A3: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ ].(x0 - g),x0.[ & not f1 . r < lim_left (f1,x0) ) ) or f2 * f1 is_left_divergent_to+infty_in x0 ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds f1 . r < lim_left (f1,x0) ; ::_thesis: f2 * f1 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_(f2_*_f1))_/\_(left_open_halfline_x0)_holds_ (f2_*_f1)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) implies (f2 * f1) /* s is divergent_to+infty ) assume that A6: ( s is convergent & lim s = x0 ) and A7: rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) ; ::_thesis: (f2 * f1) /* s is divergent_to+infty consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds x0 - g < s . n by A4, A6, Lm1, LIMFUNC2:1; set q = (f1 /* s) ^\ k; A9: rng s c= dom (f2 * f1) by A7, Th1; rng (f1 /* s) c= dom f2 by A7, Th1; then A10: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A9, VALUED_0:31 ; A11: rng s c= (dom f1) /\ (left_open_halfline x0) by A7, Th1; then A12: f1 /* s is convergent by A1, A2, A6, LIMFUNC2:def_7; A13: rng s c= dom f1 by A7, Th1; A14: rng s c= left_open_halfline x0 by A7, Th1; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_/\_(left_open_halfline_(lim_left_(f1,x0))) let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) /\ (left_open_halfline (lim_left (f1,x0))) ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) /\ (left_open_halfline (lim_left (f1,x0))) then consider n being Element of NAT such that A15: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A16: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A13, FUNCT_2:108 .= x by A15, NAT_1:def_3 ; A17: x0 - g < s . (n + k) by A8, NAT_1:12; A18: s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in left_open_halfline x0 by A14; then s . (n + k) in { g1 where g1 is Real : g1 < x0 } by XXREAL_1:229; then ex g1 being Real st ( g1 = s . (n + k) & g1 < x0 ) ; then s . (n + k) in { g2 where g2 is Real : ( x0 - g < g2 & g2 < x0 ) } by A17; then s . (n + k) in ].(x0 - g),x0.[ by RCOMP_1:def_2; then s . (n + k) in (dom f1) /\ ].(x0 - g),x0.[ by A13, A18, XBOOLE_0:def_4; then f1 . (s . (n + k)) < lim_left (f1,x0) by A5; then f1 . (s . (n + k)) in { r1 where r1 is Real : r1 < lim_left (f1,x0) } ; then A19: f1 . (s . (n + k)) in left_open_halfline (lim_left (f1,x0)) by XXREAL_1:229; f1 . (s . (n + k)) in dom f2 by A9, A18, FUNCT_1:11; hence x in (dom f2) /\ (left_open_halfline (lim_left (f1,x0))) by A19, A16, XBOOLE_0:def_4; ::_thesis: verum end; then A20: rng ((f1 /* s) ^\ k) c= (dom f2) /\ (left_open_halfline (lim_left (f1,x0))) by TARSKI:def_3; lim (f1 /* s) = lim_left (f1,x0) by A1, A6, A11, LIMFUNC2:def_7; then lim ((f1 /* s) ^\ k) = lim_left (f1,x0) by A12, SEQ_4:20; then f2 /* ((f1 /* s) ^\ k) is divergent_to+infty by A2, A12, A20, LIMFUNC2:def_2; hence (f2 * f1) /* s is divergent_to+infty by A10, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is_left_divergent_to+infty_in x0 by A3, LIMFUNC2:def_2; ::_thesis: verum end; theorem :: LIMFUNC4:20 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_divergent_to-infty_in lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds f1 . r < lim_left (f1,x0) ) ) holds f2 * f1 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_convergent_in x0 & f2 is_left_divergent_to-infty_in lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds f1 . r < lim_left (f1,x0) ) ) holds f2 * f1 is_left_divergent_to-infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_convergent_in x0 & f2 is_left_divergent_to-infty_in lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds f1 . r < lim_left (f1,x0) ) ) implies f2 * f1 is_left_divergent_to-infty_in x0 ) assume that A1: f1 is_left_convergent_in x0 and A2: f2 is_left_divergent_to-infty_in lim_left (f1,x0) and A3: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ ].(x0 - g),x0.[ & not f1 . r < lim_left (f1,x0) ) ) or f2 * f1 is_left_divergent_to-infty_in x0 ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds f1 . r < lim_left (f1,x0) ; ::_thesis: f2 * f1 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_(f2_*_f1))_/\_(left_open_halfline_x0)_holds_ (f2_*_f1)_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) implies (f2 * f1) /* s is divergent_to-infty ) assume that A6: ( s is convergent & lim s = x0 ) and A7: rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) ; ::_thesis: (f2 * f1) /* s is divergent_to-infty consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds x0 - g < s . n by A4, A6, Lm1, LIMFUNC2:1; set q = (f1 /* s) ^\ k; A9: rng s c= dom (f2 * f1) by A7, Th1; rng (f1 /* s) c= dom f2 by A7, Th1; then A10: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A9, VALUED_0:31 ; A11: rng s c= (dom f1) /\ (left_open_halfline x0) by A7, Th1; then A12: f1 /* s is convergent by A1, A2, A6, LIMFUNC2:def_7; A13: rng s c= dom f1 by A7, Th1; A14: rng s c= left_open_halfline x0 by A7, Th1; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_/\_(left_open_halfline_(lim_left_(f1,x0))) let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) /\ (left_open_halfline (lim_left (f1,x0))) ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) /\ (left_open_halfline (lim_left (f1,x0))) then consider n being Element of NAT such that A15: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A16: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A13, FUNCT_2:108 .= x by A15, NAT_1:def_3 ; A17: x0 - g < s . (n + k) by A8, NAT_1:12; A18: s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in left_open_halfline x0 by A14; then s . (n + k) in { g1 where g1 is Real : g1 < x0 } by XXREAL_1:229; then ex g1 being Real st ( g1 = s . (n + k) & g1 < x0 ) ; then s . (n + k) in { g2 where g2 is Real : ( x0 - g < g2 & g2 < x0 ) } by A17; then s . (n + k) in ].(x0 - g),x0.[ by RCOMP_1:def_2; then s . (n + k) in (dom f1) /\ ].(x0 - g),x0.[ by A13, A18, XBOOLE_0:def_4; then f1 . (s . (n + k)) < lim_left (f1,x0) by A5; then f1 . (s . (n + k)) in { r1 where r1 is Real : r1 < lim_left (f1,x0) } ; then A19: f1 . (s . (n + k)) in left_open_halfline (lim_left (f1,x0)) by XXREAL_1:229; f1 . (s . (n + k)) in dom f2 by A9, A18, FUNCT_1:11; hence x in (dom f2) /\ (left_open_halfline (lim_left (f1,x0))) by A19, A16, XBOOLE_0:def_4; ::_thesis: verum end; then A20: rng ((f1 /* s) ^\ k) c= (dom f2) /\ (left_open_halfline (lim_left (f1,x0))) by TARSKI:def_3; lim (f1 /* s) = lim_left (f1,x0) by A1, A6, A11, LIMFUNC2:def_7; then lim ((f1 /* s) ^\ k) = lim_left (f1,x0) by A12, SEQ_4:20; then f2 /* ((f1 /* s) ^\ k) is divergent_to-infty by A2, A12, A20, LIMFUNC2:def_3; hence (f2 * f1) /* s is divergent_to-infty by A10, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is_left_divergent_to-infty_in x0 by A3, LIMFUNC2:def_3; ::_thesis: verum end; theorem :: LIMFUNC4:21 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_right_divergent_to+infty_in lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds lim_left (f1,x0) < f1 . r ) ) holds f2 * f1 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_convergent_in x0 & f2 is_right_divergent_to+infty_in lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds lim_left (f1,x0) < f1 . r ) ) holds f2 * f1 is_left_divergent_to+infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_convergent_in x0 & f2 is_right_divergent_to+infty_in lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds lim_left (f1,x0) < f1 . r ) ) implies f2 * f1 is_left_divergent_to+infty_in x0 ) assume that A1: f1 is_left_convergent_in x0 and A2: f2 is_right_divergent_to+infty_in lim_left (f1,x0) and A3: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ ].(x0 - g),x0.[ & not lim_left (f1,x0) < f1 . r ) ) or f2 * f1 is_left_divergent_to+infty_in x0 ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds lim_left (f1,x0) < f1 . r ; ::_thesis: f2 * f1 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_(f2_*_f1))_/\_(left_open_halfline_x0)_holds_ (f2_*_f1)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) implies (f2 * f1) /* s is divergent_to+infty ) assume that A6: ( s is convergent & lim s = x0 ) and A7: rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) ; ::_thesis: (f2 * f1) /* s is divergent_to+infty consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds x0 - g < s . n by A4, A6, Lm1, LIMFUNC2:1; set q = (f1 /* s) ^\ k; A9: rng s c= dom (f2 * f1) by A7, Th1; rng (f1 /* s) c= dom f2 by A7, Th1; then A10: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A9, VALUED_0:31 ; A11: rng s c= (dom f1) /\ (left_open_halfline x0) by A7, Th1; then A12: f1 /* s is convergent by A1, A2, A6, LIMFUNC2:def_7; A13: rng s c= dom f1 by A7, Th1; A14: rng s c= left_open_halfline x0 by A7, Th1; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_/\_(right_open_halfline_(lim_left_(f1,x0))) let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) /\ (right_open_halfline (lim_left (f1,x0))) ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) /\ (right_open_halfline (lim_left (f1,x0))) then consider n being Element of NAT such that A15: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A16: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A13, FUNCT_2:108 .= x by A15, NAT_1:def_3 ; A17: x0 - g < s . (n + k) by A8, NAT_1:12; A18: s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in left_open_halfline x0 by A14; then s . (n + k) in { g1 where g1 is Real : g1 < x0 } by XXREAL_1:229; then ex g1 being Real st ( g1 = s . (n + k) & g1 < x0 ) ; then s . (n + k) in { g2 where g2 is Real : ( x0 - g < g2 & g2 < x0 ) } by A17; then s . (n + k) in ].(x0 - g),x0.[ by RCOMP_1:def_2; then s . (n + k) in (dom f1) /\ ].(x0 - g),x0.[ by A13, A18, XBOOLE_0:def_4; then lim_left (f1,x0) < f1 . (s . (n + k)) by A5; then f1 . (s . (n + k)) in { r1 where r1 is Real : lim_left (f1,x0) < r1 } ; then A19: f1 . (s . (n + k)) in right_open_halfline (lim_left (f1,x0)) by XXREAL_1:230; f1 . (s . (n + k)) in dom f2 by A9, A18, FUNCT_1:11; hence x in (dom f2) /\ (right_open_halfline (lim_left (f1,x0))) by A19, A16, XBOOLE_0:def_4; ::_thesis: verum end; then A20: rng ((f1 /* s) ^\ k) c= (dom f2) /\ (right_open_halfline (lim_left (f1,x0))) by TARSKI:def_3; lim (f1 /* s) = lim_left (f1,x0) by A1, A6, A11, LIMFUNC2:def_7; then lim ((f1 /* s) ^\ k) = lim_left (f1,x0) by A12, SEQ_4:20; then f2 /* ((f1 /* s) ^\ k) is divergent_to+infty by A2, A12, A20, LIMFUNC2:def_5; hence (f2 * f1) /* s is divergent_to+infty by A10, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is_left_divergent_to+infty_in x0 by A3, LIMFUNC2:def_2; ::_thesis: verum end; theorem :: LIMFUNC4:22 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_right_divergent_to-infty_in lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds lim_left (f1,x0) < f1 . r ) ) holds f2 * f1 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_convergent_in x0 & f2 is_right_divergent_to-infty_in lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds lim_left (f1,x0) < f1 . r ) ) holds f2 * f1 is_left_divergent_to-infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_convergent_in x0 & f2 is_right_divergent_to-infty_in lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds lim_left (f1,x0) < f1 . r ) ) implies f2 * f1 is_left_divergent_to-infty_in x0 ) assume that A1: f1 is_left_convergent_in x0 and A2: f2 is_right_divergent_to-infty_in lim_left (f1,x0) and A3: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ ].(x0 - g),x0.[ & not lim_left (f1,x0) < f1 . r ) ) or f2 * f1 is_left_divergent_to-infty_in x0 ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds lim_left (f1,x0) < f1 . r ; ::_thesis: f2 * f1 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_(f2_*_f1))_/\_(left_open_halfline_x0)_holds_ (f2_*_f1)_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) implies (f2 * f1) /* s is divergent_to-infty ) assume that A6: ( s is convergent & lim s = x0 ) and A7: rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) ; ::_thesis: (f2 * f1) /* s is divergent_to-infty consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds x0 - g < s . n by A4, A6, Lm1, LIMFUNC2:1; set q = (f1 /* s) ^\ k; A9: rng s c= dom (f2 * f1) by A7, Th1; rng (f1 /* s) c= dom f2 by A7, Th1; then A10: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A9, VALUED_0:31 ; A11: rng s c= (dom f1) /\ (left_open_halfline x0) by A7, Th1; then A12: f1 /* s is convergent by A1, A2, A6, LIMFUNC2:def_7; A13: rng s c= dom f1 by A7, Th1; A14: rng s c= left_open_halfline x0 by A7, Th1; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_/\_(right_open_halfline_(lim_left_(f1,x0))) let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) /\ (right_open_halfline (lim_left (f1,x0))) ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) /\ (right_open_halfline (lim_left (f1,x0))) then consider n being Element of NAT such that A15: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A16: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A13, FUNCT_2:108 .= x by A15, NAT_1:def_3 ; A17: x0 - g < s . (n + k) by A8, NAT_1:12; A18: s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in left_open_halfline x0 by A14; then s . (n + k) in { g1 where g1 is Real : g1 < x0 } by XXREAL_1:229; then ex g1 being Real st ( g1 = s . (n + k) & g1 < x0 ) ; then s . (n + k) in { g2 where g2 is Real : ( x0 - g < g2 & g2 < x0 ) } by A17; then s . (n + k) in ].(x0 - g),x0.[ by RCOMP_1:def_2; then s . (n + k) in (dom f1) /\ ].(x0 - g),x0.[ by A13, A18, XBOOLE_0:def_4; then lim_left (f1,x0) < f1 . (s . (n + k)) by A5; then f1 . (s . (n + k)) in { r1 where r1 is Real : lim_left (f1,x0) < r1 } ; then A19: f1 . (s . (n + k)) in right_open_halfline (lim_left (f1,x0)) by XXREAL_1:230; f1 . (s . (n + k)) in dom f2 by A9, A18, FUNCT_1:11; hence x in (dom f2) /\ (right_open_halfline (lim_left (f1,x0))) by A19, A16, XBOOLE_0:def_4; ::_thesis: verum end; then A20: rng ((f1 /* s) ^\ k) c= (dom f2) /\ (right_open_halfline (lim_left (f1,x0))) by TARSKI:def_3; lim (f1 /* s) = lim_left (f1,x0) by A1, A6, A11, LIMFUNC2:def_7; then lim ((f1 /* s) ^\ k) = lim_left (f1,x0) by A12, SEQ_4:20; then f2 /* ((f1 /* s) ^\ k) is divergent_to-infty by A2, A12, A20, LIMFUNC2:def_6; hence (f2 * f1) /* s is divergent_to-infty by A10, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is_left_divergent_to-infty_in x0 by A3, LIMFUNC2:def_3; ::_thesis: verum end; theorem :: LIMFUNC4:23 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_right_divergent_to+infty_in lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds lim_right (f1,x0) < f1 . r ) ) holds f2 * f1 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_convergent_in x0 & f2 is_right_divergent_to+infty_in lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds lim_right (f1,x0) < f1 . r ) ) holds f2 * f1 is_right_divergent_to+infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_right_convergent_in x0 & f2 is_right_divergent_to+infty_in lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds lim_right (f1,x0) < f1 . r ) ) implies f2 * f1 is_right_divergent_to+infty_in x0 ) assume that A1: f1 is_right_convergent_in x0 and A2: f2 is_right_divergent_to+infty_in lim_right (f1,x0) and A3: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ ].x0,(x0 + g).[ & not lim_right (f1,x0) < f1 . r ) ) or f2 * f1 is_right_divergent_to+infty_in x0 ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds lim_right (f1,x0) < f1 . r ; ::_thesis: f2 * f1 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_(f2_*_f1))_/\_(right_open_halfline_x0)_holds_ (f2_*_f1)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) implies (f2 * f1) /* s is divergent_to+infty ) assume that A6: ( s is convergent & lim s = x0 ) and A7: rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) ; ::_thesis: (f2 * f1) /* s is divergent_to+infty consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds s . n < x0 + g by A4, A6, Lm1, LIMFUNC2:2; set q = (f1 /* s) ^\ k; A9: rng s c= dom (f2 * f1) by A7, Th1; rng (f1 /* s) c= dom f2 by A7, Th1; then A10: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A9, VALUED_0:31 ; A11: rng s c= (dom f1) /\ (right_open_halfline x0) by A7, Th1; then A12: f1 /* s is convergent by A1, A2, A6, LIMFUNC2:def_8; A13: rng s c= dom f1 by A7, Th1; A14: rng s c= right_open_halfline x0 by A7, Th1; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_/\_(right_open_halfline_(lim_right_(f1,x0))) let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) /\ (right_open_halfline (lim_right (f1,x0))) ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) /\ (right_open_halfline (lim_right (f1,x0))) then consider n being Element of NAT such that A15: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A16: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A13, FUNCT_2:108 .= x by A15, NAT_1:def_3 ; A17: s . (n + k) < x0 + g by A8, NAT_1:12; A18: s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in right_open_halfline x0 by A14; then s . (n + k) in { g1 where g1 is Real : x0 < g1 } by XXREAL_1:230; then ex g1 being Real st ( g1 = s . (n + k) & x0 < g1 ) ; then s . (n + k) in { g2 where g2 is Real : ( x0 < g2 & g2 < x0 + g ) } by A17; then s . (n + k) in ].x0,(x0 + g).[ by RCOMP_1:def_2; then s . (n + k) in (dom f1) /\ ].x0,(x0 + g).[ by A13, A18, XBOOLE_0:def_4; then lim_right (f1,x0) < f1 . (s . (n + k)) by A5; then f1 . (s . (n + k)) in { r1 where r1 is Real : lim_right (f1,x0) < r1 } ; then A19: f1 . (s . (n + k)) in right_open_halfline (lim_right (f1,x0)) by XXREAL_1:230; f1 . (s . (n + k)) in dom f2 by A9, A18, FUNCT_1:11; hence x in (dom f2) /\ (right_open_halfline (lim_right (f1,x0))) by A19, A16, XBOOLE_0:def_4; ::_thesis: verum end; then A20: rng ((f1 /* s) ^\ k) c= (dom f2) /\ (right_open_halfline (lim_right (f1,x0))) by TARSKI:def_3; lim (f1 /* s) = lim_right (f1,x0) by A1, A6, A11, LIMFUNC2:def_8; then lim ((f1 /* s) ^\ k) = lim_right (f1,x0) by A12, SEQ_4:20; then f2 /* ((f1 /* s) ^\ k) is divergent_to+infty by A2, A12, A20, LIMFUNC2:def_5; hence (f2 * f1) /* s is divergent_to+infty by A10, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is_right_divergent_to+infty_in x0 by A3, LIMFUNC2:def_5; ::_thesis: verum end; theorem :: LIMFUNC4:24 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_right_divergent_to-infty_in lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds lim_right (f1,x0) < f1 . r ) ) holds f2 * f1 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_convergent_in x0 & f2 is_right_divergent_to-infty_in lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds lim_right (f1,x0) < f1 . r ) ) holds f2 * f1 is_right_divergent_to-infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_right_convergent_in x0 & f2 is_right_divergent_to-infty_in lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds lim_right (f1,x0) < f1 . r ) ) implies f2 * f1 is_right_divergent_to-infty_in x0 ) assume that A1: f1 is_right_convergent_in x0 and A2: f2 is_right_divergent_to-infty_in lim_right (f1,x0) and A3: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ ].x0,(x0 + g).[ & not lim_right (f1,x0) < f1 . r ) ) or f2 * f1 is_right_divergent_to-infty_in x0 ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds lim_right (f1,x0) < f1 . r ; ::_thesis: f2 * f1 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_(f2_*_f1))_/\_(right_open_halfline_x0)_holds_ (f2_*_f1)_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) implies (f2 * f1) /* s is divergent_to-infty ) assume that A6: ( s is convergent & lim s = x0 ) and A7: rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) ; ::_thesis: (f2 * f1) /* s is divergent_to-infty consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds s . n < x0 + g by A4, A6, Lm1, LIMFUNC2:2; set q = (f1 /* s) ^\ k; A9: rng s c= dom (f2 * f1) by A7, Th1; rng (f1 /* s) c= dom f2 by A7, Th1; then A10: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A9, VALUED_0:31 ; A11: rng s c= (dom f1) /\ (right_open_halfline x0) by A7, Th1; then A12: f1 /* s is convergent by A1, A2, A6, LIMFUNC2:def_8; A13: rng s c= dom f1 by A7, Th1; A14: rng s c= right_open_halfline x0 by A7, Th1; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_/\_(right_open_halfline_(lim_right_(f1,x0))) let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) /\ (right_open_halfline (lim_right (f1,x0))) ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) /\ (right_open_halfline (lim_right (f1,x0))) then consider n being Element of NAT such that A15: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A16: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A13, FUNCT_2:108 .= x by A15, NAT_1:def_3 ; A17: s . (n + k) < x0 + g by A8, NAT_1:12; A18: s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in right_open_halfline x0 by A14; then s . (n + k) in { g1 where g1 is Real : x0 < g1 } by XXREAL_1:230; then ex g1 being Real st ( g1 = s . (n + k) & x0 < g1 ) ; then s . (n + k) in { g2 where g2 is Real : ( x0 < g2 & g2 < x0 + g ) } by A17; then s . (n + k) in ].x0,(x0 + g).[ by RCOMP_1:def_2; then s . (n + k) in (dom f1) /\ ].x0,(x0 + g).[ by A13, A18, XBOOLE_0:def_4; then lim_right (f1,x0) < f1 . (s . (n + k)) by A5; then f1 . (s . (n + k)) in { r1 where r1 is Real : lim_right (f1,x0) < r1 } ; then A19: f1 . (s . (n + k)) in right_open_halfline (lim_right (f1,x0)) by XXREAL_1:230; f1 . (s . (n + k)) in dom f2 by A9, A18, FUNCT_1:11; hence x in (dom f2) /\ (right_open_halfline (lim_right (f1,x0))) by A19, A16, XBOOLE_0:def_4; ::_thesis: verum end; then A20: rng ((f1 /* s) ^\ k) c= (dom f2) /\ (right_open_halfline (lim_right (f1,x0))) by TARSKI:def_3; lim (f1 /* s) = lim_right (f1,x0) by A1, A6, A11, LIMFUNC2:def_8; then lim ((f1 /* s) ^\ k) = lim_right (f1,x0) by A12, SEQ_4:20; then f2 /* ((f1 /* s) ^\ k) is divergent_to-infty by A2, A12, A20, LIMFUNC2:def_6; hence (f2 * f1) /* s is divergent_to-infty by A10, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is_right_divergent_to-infty_in x0 by A3, LIMFUNC2:def_6; ::_thesis: verum end; theorem :: LIMFUNC4:25 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_left_divergent_to+infty_in lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds f1 . r < lim_right (f1,x0) ) ) holds f2 * f1 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_convergent_in x0 & f2 is_left_divergent_to+infty_in lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds f1 . r < lim_right (f1,x0) ) ) holds f2 * f1 is_right_divergent_to+infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_right_convergent_in x0 & f2 is_left_divergent_to+infty_in lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds f1 . r < lim_right (f1,x0) ) ) implies f2 * f1 is_right_divergent_to+infty_in x0 ) assume that A1: f1 is_right_convergent_in x0 and A2: f2 is_left_divergent_to+infty_in lim_right (f1,x0) and A3: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ ].x0,(x0 + g).[ & not f1 . r < lim_right (f1,x0) ) ) or f2 * f1 is_right_divergent_to+infty_in x0 ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds f1 . r < lim_right (f1,x0) ; ::_thesis: f2 * f1 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_(f2_*_f1))_/\_(right_open_halfline_x0)_holds_ (f2_*_f1)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) implies (f2 * f1) /* s is divergent_to+infty ) assume that A6: ( s is convergent & lim s = x0 ) and A7: rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) ; ::_thesis: (f2 * f1) /* s is divergent_to+infty consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds s . n < x0 + g by A4, A6, Lm1, LIMFUNC2:2; set q = (f1 /* s) ^\ k; A9: rng s c= dom (f2 * f1) by A7, Th1; rng (f1 /* s) c= dom f2 by A7, Th1; then A10: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A9, VALUED_0:31 ; A11: rng s c= (dom f1) /\ (right_open_halfline x0) by A7, Th1; then A12: f1 /* s is convergent by A1, A2, A6, LIMFUNC2:def_8; A13: rng s c= dom f1 by A7, Th1; A14: rng s c= right_open_halfline x0 by A7, Th1; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_/\_(left_open_halfline_(lim_right_(f1,x0))) let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) /\ (left_open_halfline (lim_right (f1,x0))) ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) /\ (left_open_halfline (lim_right (f1,x0))) then consider n being Element of NAT such that A15: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A16: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A13, FUNCT_2:108 .= x by A15, NAT_1:def_3 ; A17: s . (n + k) < x0 + g by A8, NAT_1:12; A18: s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in right_open_halfline x0 by A14; then s . (n + k) in { g1 where g1 is Real : x0 < g1 } by XXREAL_1:230; then ex g1 being Real st ( g1 = s . (n + k) & x0 < g1 ) ; then s . (n + k) in { g2 where g2 is Real : ( x0 < g2 & g2 < x0 + g ) } by A17; then s . (n + k) in ].x0,(x0 + g).[ by RCOMP_1:def_2; then s . (n + k) in (dom f1) /\ ].x0,(x0 + g).[ by A13, A18, XBOOLE_0:def_4; then f1 . (s . (n + k)) < lim_right (f1,x0) by A5; then f1 . (s . (n + k)) in { r1 where r1 is Real : r1 < lim_right (f1,x0) } ; then A19: f1 . (s . (n + k)) in left_open_halfline (lim_right (f1,x0)) by XXREAL_1:229; f1 . (s . (n + k)) in dom f2 by A9, A18, FUNCT_1:11; hence x in (dom f2) /\ (left_open_halfline (lim_right (f1,x0))) by A19, A16, XBOOLE_0:def_4; ::_thesis: verum end; then A20: rng ((f1 /* s) ^\ k) c= (dom f2) /\ (left_open_halfline (lim_right (f1,x0))) by TARSKI:def_3; lim (f1 /* s) = lim_right (f1,x0) by A1, A6, A11, LIMFUNC2:def_8; then lim ((f1 /* s) ^\ k) = lim_right (f1,x0) by A12, SEQ_4:20; then f2 /* ((f1 /* s) ^\ k) is divergent_to+infty by A2, A12, A20, LIMFUNC2:def_2; hence (f2 * f1) /* s is divergent_to+infty by A10, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is_right_divergent_to+infty_in x0 by A3, LIMFUNC2:def_5; ::_thesis: verum end; theorem :: LIMFUNC4:26 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_left_divergent_to-infty_in lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds f1 . r < lim_right (f1,x0) ) ) holds f2 * f1 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_convergent_in x0 & f2 is_left_divergent_to-infty_in lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds f1 . r < lim_right (f1,x0) ) ) holds f2 * f1 is_right_divergent_to-infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_right_convergent_in x0 & f2 is_left_divergent_to-infty_in lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds f1 . r < lim_right (f1,x0) ) ) implies f2 * f1 is_right_divergent_to-infty_in x0 ) assume that A1: f1 is_right_convergent_in x0 and A2: f2 is_left_divergent_to-infty_in lim_right (f1,x0) and A3: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ ].x0,(x0 + g).[ & not f1 . r < lim_right (f1,x0) ) ) or f2 * f1 is_right_divergent_to-infty_in x0 ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds f1 . r < lim_right (f1,x0) ; ::_thesis: f2 * f1 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_(f2_*_f1))_/\_(right_open_halfline_x0)_holds_ (f2_*_f1)_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) implies (f2 * f1) /* s is divergent_to-infty ) assume that A6: ( s is convergent & lim s = x0 ) and A7: rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) ; ::_thesis: (f2 * f1) /* s is divergent_to-infty consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds s . n < x0 + g by A4, A6, Lm1, LIMFUNC2:2; set q = (f1 /* s) ^\ k; A9: rng s c= dom (f2 * f1) by A7, Th1; rng (f1 /* s) c= dom f2 by A7, Th1; then A10: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A9, VALUED_0:31 ; A11: rng s c= (dom f1) /\ (right_open_halfline x0) by A7, Th1; then A12: f1 /* s is convergent by A1, A2, A6, LIMFUNC2:def_8; A13: rng s c= dom f1 by A7, Th1; A14: rng s c= right_open_halfline x0 by A7, Th1; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_/\_(left_open_halfline_(lim_right_(f1,x0))) let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) /\ (left_open_halfline (lim_right (f1,x0))) ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) /\ (left_open_halfline (lim_right (f1,x0))) then consider n being Element of NAT such that A15: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A16: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A13, FUNCT_2:108 .= x by A15, NAT_1:def_3 ; A17: s . (n + k) < x0 + g by A8, NAT_1:12; A18: s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in right_open_halfline x0 by A14; then s . (n + k) in { g1 where g1 is Real : x0 < g1 } by XXREAL_1:230; then ex g1 being Real st ( g1 = s . (n + k) & x0 < g1 ) ; then s . (n + k) in { g2 where g2 is Real : ( x0 < g2 & g2 < x0 + g ) } by A17; then s . (n + k) in ].x0,(x0 + g).[ by RCOMP_1:def_2; then s . (n + k) in (dom f1) /\ ].x0,(x0 + g).[ by A13, A18, XBOOLE_0:def_4; then f1 . (s . (n + k)) < lim_right (f1,x0) by A5; then f1 . (s . (n + k)) in { r1 where r1 is Real : r1 < lim_right (f1,x0) } ; then A19: f1 . (s . (n + k)) in left_open_halfline (lim_right (f1,x0)) by XXREAL_1:229; f1 . (s . (n + k)) in dom f2 by A9, A18, FUNCT_1:11; hence x in (dom f2) /\ (left_open_halfline (lim_right (f1,x0))) by A19, A16, XBOOLE_0:def_4; ::_thesis: verum end; then A20: rng ((f1 /* s) ^\ k) c= (dom f2) /\ (left_open_halfline (lim_right (f1,x0))) by TARSKI:def_3; lim (f1 /* s) = lim_right (f1,x0) by A1, A6, A11, LIMFUNC2:def_8; then lim ((f1 /* s) ^\ k) = lim_right (f1,x0) by A12, SEQ_4:20; then f2 /* ((f1 /* s) ^\ k) is divergent_to-infty by A2, A12, A20, LIMFUNC2:def_3; hence (f2 * f1) /* s is divergent_to-infty by A10, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is_right_divergent_to-infty_in x0 by A3, LIMFUNC2:def_6; ::_thesis: verum end; theorem :: LIMFUNC4:27 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in+infty & f2 is_left_divergent_to+infty_in lim_in+infty f1 & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (right_open_halfline r) holds f1 . g < lim_in+infty f1 holds f2 * f1 is divergent_in+infty_to+infty proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in+infty & f2 is_left_divergent_to+infty_in lim_in+infty f1 & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (right_open_halfline r) holds f1 . g < lim_in+infty f1 implies f2 * f1 is divergent_in+infty_to+infty ) assume that A1: f1 is convergent_in+infty and A2: f2 is_left_divergent_to+infty_in lim_in+infty f1 and A3: for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ; ::_thesis: ( for r being Real ex g being Real st ( g in (dom f1) /\ (right_open_halfline r) & not f1 . g < lim_in+infty f1 ) or f2 * f1 is divergent_in+infty_to+infty ) given r being Real such that A4: for g being Real st g in (dom f1) /\ (right_open_halfline r) holds f1 . g < lim_in+infty f1 ; ::_thesis: f2 * f1 is divergent_in+infty_to+infty now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to+infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (f2_*_f1)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is divergent_to+infty & rng s c= dom (f2 * f1) implies (f2 * f1) /* s is divergent_to+infty ) assume that A5: s is divergent_to+infty and A6: rng s c= dom (f2 * f1) ; ::_thesis: (f2 * f1) /* s is divergent_to+infty consider k being Element of NAT such that A7: for n being Element of NAT st k <= n holds r < s . n by A5, LIMFUNC1:def_4; A8: rng (f1 /* s) c= dom f2 by A6, Lm2; set q = s ^\ k; A9: s ^\ k is divergent_to+infty by A5, LIMFUNC1:26; A10: rng s c= dom f1 by A6, Lm2; A11: rng (s ^\ k) c= rng s by VALUED_0:21; A12: rng (f1 /* (s ^\ k)) c= (dom f2) /\ (left_open_halfline (lim_in+infty f1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (f1 /* (s ^\ k)) or x in (dom f2) /\ (left_open_halfline (lim_in+infty f1)) ) assume x in rng (f1 /* (s ^\ k)) ; ::_thesis: x in (dom f2) /\ (left_open_halfline (lim_in+infty f1)) then consider n being Element of NAT such that A13: (f1 /* (s ^\ k)) . n = x by FUNCT_2:113; A14: x = f1 . ((s ^\ k) . n) by A10, A11, A13, FUNCT_2:108, XBOOLE_1:1 .= f1 . (s . (n + k)) by NAT_1:def_3 ; r < s . (n + k) by A7, NAT_1:12; then s . (n + k) in { r2 where r2 is Real : r < r2 } ; then A15: s . (n + k) in right_open_halfline r by XXREAL_1:230; s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in (dom f1) /\ (right_open_halfline r) by A10, A15, XBOOLE_0:def_4; then f1 . (s . (n + k)) < lim_in+infty f1 by A4; then x in { g1 where g1 is Real : g1 < lim_in+infty f1 } by A14; then A16: x in left_open_halfline (lim_in+infty f1) by XXREAL_1:229; (f1 /* s) . (n + k) in rng (f1 /* s) by VALUED_0:28; then (f1 /* s) . (n + k) in dom f2 by A8; then x in dom f2 by A10, A14, FUNCT_2:108; hence x in (dom f2) /\ (left_open_halfline (lim_in+infty f1)) by A16, XBOOLE_0:def_4; ::_thesis: verum end; rng (s ^\ k) c= dom f1 by A10, A11, XBOOLE_1:1; then ( f1 /* (s ^\ k) is convergent & lim (f1 /* (s ^\ k)) = lim_in+infty f1 ) by A1, A2, A9, LIMFUNC1:def_12; then A17: f2 /* (f1 /* (s ^\ k)) is divergent_to+infty by A2, A12, LIMFUNC2:def_2; f2 /* (f1 /* (s ^\ k)) = f2 /* ((f1 /* s) ^\ k) by A10, VALUED_0:27 .= (f2 /* (f1 /* s)) ^\ k by A8, VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A6, VALUED_0:31 ; hence (f2 * f1) /* s is divergent_to+infty by A17, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is divergent_in+infty_to+infty by A3, LIMFUNC1:def_7; ::_thesis: verum end; theorem :: LIMFUNC4:28 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in+infty & f2 is_left_divergent_to-infty_in lim_in+infty f1 & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (right_open_halfline r) holds f1 . g < lim_in+infty f1 holds f2 * f1 is divergent_in+infty_to-infty proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in+infty & f2 is_left_divergent_to-infty_in lim_in+infty f1 & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (right_open_halfline r) holds f1 . g < lim_in+infty f1 implies f2 * f1 is divergent_in+infty_to-infty ) assume that A1: f1 is convergent_in+infty and A2: f2 is_left_divergent_to-infty_in lim_in+infty f1 and A3: for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ; ::_thesis: ( for r being Real ex g being Real st ( g in (dom f1) /\ (right_open_halfline r) & not f1 . g < lim_in+infty f1 ) or f2 * f1 is divergent_in+infty_to-infty ) given r being Real such that A4: for g being Real st g in (dom f1) /\ (right_open_halfline r) holds f1 . g < lim_in+infty f1 ; ::_thesis: f2 * f1 is divergent_in+infty_to-infty now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to+infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (f2_*_f1)_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is divergent_to+infty & rng s c= dom (f2 * f1) implies (f2 * f1) /* s is divergent_to-infty ) assume that A5: s is divergent_to+infty and A6: rng s c= dom (f2 * f1) ; ::_thesis: (f2 * f1) /* s is divergent_to-infty consider k being Element of NAT such that A7: for n being Element of NAT st k <= n holds r < s . n by A5, LIMFUNC1:def_4; A8: rng (f1 /* s) c= dom f2 by A6, Lm2; set q = s ^\ k; A9: s ^\ k is divergent_to+infty by A5, LIMFUNC1:26; A10: rng s c= dom f1 by A6, Lm2; A11: rng (s ^\ k) c= rng s by VALUED_0:21; A12: rng (f1 /* (s ^\ k)) c= (dom f2) /\ (left_open_halfline (lim_in+infty f1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (f1 /* (s ^\ k)) or x in (dom f2) /\ (left_open_halfline (lim_in+infty f1)) ) assume x in rng (f1 /* (s ^\ k)) ; ::_thesis: x in (dom f2) /\ (left_open_halfline (lim_in+infty f1)) then consider n being Element of NAT such that A13: (f1 /* (s ^\ k)) . n = x by FUNCT_2:113; A14: x = f1 . ((s ^\ k) . n) by A10, A11, A13, FUNCT_2:108, XBOOLE_1:1 .= f1 . (s . (n + k)) by NAT_1:def_3 ; r < s . (n + k) by A7, NAT_1:12; then s . (n + k) in { r2 where r2 is Real : r < r2 } ; then A15: s . (n + k) in right_open_halfline r by XXREAL_1:230; s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in (dom f1) /\ (right_open_halfline r) by A10, A15, XBOOLE_0:def_4; then f1 . (s . (n + k)) < lim_in+infty f1 by A4; then x in { g1 where g1 is Real : g1 < lim_in+infty f1 } by A14; then A16: x in left_open_halfline (lim_in+infty f1) by XXREAL_1:229; (f1 /* s) . (n + k) in rng (f1 /* s) by VALUED_0:28; then (f1 /* s) . (n + k) in dom f2 by A8; then x in dom f2 by A10, A14, FUNCT_2:108; hence x in (dom f2) /\ (left_open_halfline (lim_in+infty f1)) by A16, XBOOLE_0:def_4; ::_thesis: verum end; rng (s ^\ k) c= dom f1 by A10, A11, XBOOLE_1:1; then ( f1 /* (s ^\ k) is convergent & lim (f1 /* (s ^\ k)) = lim_in+infty f1 ) by A1, A2, A9, LIMFUNC1:def_12; then A17: f2 /* (f1 /* (s ^\ k)) is divergent_to-infty by A2, A12, LIMFUNC2:def_3; f2 /* (f1 /* (s ^\ k)) = f2 /* ((f1 /* s) ^\ k) by A10, VALUED_0:27 .= (f2 /* (f1 /* s)) ^\ k by A8, VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A6, VALUED_0:31 ; hence (f2 * f1) /* s is divergent_to-infty by A17, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is divergent_in+infty_to-infty by A3, LIMFUNC1:def_8; ::_thesis: verum end; theorem :: LIMFUNC4:29 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in+infty & f2 is_right_divergent_to+infty_in lim_in+infty f1 & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (right_open_halfline r) holds lim_in+infty f1 < f1 . g holds f2 * f1 is divergent_in+infty_to+infty proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in+infty & f2 is_right_divergent_to+infty_in lim_in+infty f1 & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (right_open_halfline r) holds lim_in+infty f1 < f1 . g implies f2 * f1 is divergent_in+infty_to+infty ) assume that A1: f1 is convergent_in+infty and A2: f2 is_right_divergent_to+infty_in lim_in+infty f1 and A3: for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ; ::_thesis: ( for r being Real ex g being Real st ( g in (dom f1) /\ (right_open_halfline r) & not lim_in+infty f1 < f1 . g ) or f2 * f1 is divergent_in+infty_to+infty ) given r being Real such that A4: for g being Real st g in (dom f1) /\ (right_open_halfline r) holds lim_in+infty f1 < f1 . g ; ::_thesis: f2 * f1 is divergent_in+infty_to+infty now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to+infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (f2_*_f1)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is divergent_to+infty & rng s c= dom (f2 * f1) implies (f2 * f1) /* s is divergent_to+infty ) assume that A5: s is divergent_to+infty and A6: rng s c= dom (f2 * f1) ; ::_thesis: (f2 * f1) /* s is divergent_to+infty consider k being Element of NAT such that A7: for n being Element of NAT st k <= n holds r < s . n by A5, LIMFUNC1:def_4; A8: rng (f1 /* s) c= dom f2 by A6, Lm2; set q = s ^\ k; A9: s ^\ k is divergent_to+infty by A5, LIMFUNC1:26; A10: rng s c= dom f1 by A6, Lm2; A11: rng (s ^\ k) c= rng s by VALUED_0:21; A12: rng (f1 /* (s ^\ k)) c= (dom f2) /\ (right_open_halfline (lim_in+infty f1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (f1 /* (s ^\ k)) or x in (dom f2) /\ (right_open_halfline (lim_in+infty f1)) ) assume x in rng (f1 /* (s ^\ k)) ; ::_thesis: x in (dom f2) /\ (right_open_halfline (lim_in+infty f1)) then consider n being Element of NAT such that A13: (f1 /* (s ^\ k)) . n = x by FUNCT_2:113; A14: x = f1 . ((s ^\ k) . n) by A10, A11, A13, FUNCT_2:108, XBOOLE_1:1 .= f1 . (s . (n + k)) by NAT_1:def_3 ; r < s . (n + k) by A7, NAT_1:12; then s . (n + k) in { r2 where r2 is Real : r < r2 } ; then A15: s . (n + k) in right_open_halfline r by XXREAL_1:230; s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in (dom f1) /\ (right_open_halfline r) by A10, A15, XBOOLE_0:def_4; then lim_in+infty f1 < f1 . (s . (n + k)) by A4; then x in { g1 where g1 is Real : lim_in+infty f1 < g1 } by A14; then A16: x in right_open_halfline (lim_in+infty f1) by XXREAL_1:230; (f1 /* s) . (n + k) in rng (f1 /* s) by VALUED_0:28; then (f1 /* s) . (n + k) in dom f2 by A8; then x in dom f2 by A10, A14, FUNCT_2:108; hence x in (dom f2) /\ (right_open_halfline (lim_in+infty f1)) by A16, XBOOLE_0:def_4; ::_thesis: verum end; rng (s ^\ k) c= dom f1 by A10, A11, XBOOLE_1:1; then ( f1 /* (s ^\ k) is convergent & lim (f1 /* (s ^\ k)) = lim_in+infty f1 ) by A1, A2, A9, LIMFUNC1:def_12; then A17: f2 /* (f1 /* (s ^\ k)) is divergent_to+infty by A2, A12, LIMFUNC2:def_5; f2 /* (f1 /* (s ^\ k)) = f2 /* ((f1 /* s) ^\ k) by A10, VALUED_0:27 .= (f2 /* (f1 /* s)) ^\ k by A8, VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A6, VALUED_0:31 ; hence (f2 * f1) /* s is divergent_to+infty by A17, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is divergent_in+infty_to+infty by A3, LIMFUNC1:def_7; ::_thesis: verum end; theorem :: LIMFUNC4:30 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in+infty & f2 is_right_divergent_to-infty_in lim_in+infty f1 & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (right_open_halfline r) holds lim_in+infty f1 < f1 . g holds f2 * f1 is divergent_in+infty_to-infty proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in+infty & f2 is_right_divergent_to-infty_in lim_in+infty f1 & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (right_open_halfline r) holds lim_in+infty f1 < f1 . g implies f2 * f1 is divergent_in+infty_to-infty ) assume that A1: f1 is convergent_in+infty and A2: f2 is_right_divergent_to-infty_in lim_in+infty f1 and A3: for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ; ::_thesis: ( for r being Real ex g being Real st ( g in (dom f1) /\ (right_open_halfline r) & not lim_in+infty f1 < f1 . g ) or f2 * f1 is divergent_in+infty_to-infty ) given r being Real such that A4: for g being Real st g in (dom f1) /\ (right_open_halfline r) holds lim_in+infty f1 < f1 . g ; ::_thesis: f2 * f1 is divergent_in+infty_to-infty now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to+infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (f2_*_f1)_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is divergent_to+infty & rng s c= dom (f2 * f1) implies (f2 * f1) /* s is divergent_to-infty ) assume that A5: s is divergent_to+infty and A6: rng s c= dom (f2 * f1) ; ::_thesis: (f2 * f1) /* s is divergent_to-infty consider k being Element of NAT such that A7: for n being Element of NAT st k <= n holds r < s . n by A5, LIMFUNC1:def_4; A8: rng (f1 /* s) c= dom f2 by A6, Lm2; set q = s ^\ k; A9: s ^\ k is divergent_to+infty by A5, LIMFUNC1:26; A10: rng s c= dom f1 by A6, Lm2; A11: rng (s ^\ k) c= rng s by VALUED_0:21; A12: rng (f1 /* (s ^\ k)) c= (dom f2) /\ (right_open_halfline (lim_in+infty f1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (f1 /* (s ^\ k)) or x in (dom f2) /\ (right_open_halfline (lim_in+infty f1)) ) assume x in rng (f1 /* (s ^\ k)) ; ::_thesis: x in (dom f2) /\ (right_open_halfline (lim_in+infty f1)) then consider n being Element of NAT such that A13: (f1 /* (s ^\ k)) . n = x by FUNCT_2:113; A14: x = f1 . ((s ^\ k) . n) by A10, A11, A13, FUNCT_2:108, XBOOLE_1:1 .= f1 . (s . (n + k)) by NAT_1:def_3 ; r < s . (n + k) by A7, NAT_1:12; then s . (n + k) in { r2 where r2 is Real : r < r2 } ; then A15: s . (n + k) in right_open_halfline r by XXREAL_1:230; s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in (dom f1) /\ (right_open_halfline r) by A10, A15, XBOOLE_0:def_4; then lim_in+infty f1 < f1 . (s . (n + k)) by A4; then x in { g1 where g1 is Real : lim_in+infty f1 < g1 } by A14; then A16: x in right_open_halfline (lim_in+infty f1) by XXREAL_1:230; (f1 /* s) . (n + k) in rng (f1 /* s) by VALUED_0:28; then (f1 /* s) . (n + k) in dom f2 by A8; then x in dom f2 by A10, A14, FUNCT_2:108; hence x in (dom f2) /\ (right_open_halfline (lim_in+infty f1)) by A16, XBOOLE_0:def_4; ::_thesis: verum end; rng (s ^\ k) c= dom f1 by A10, A11, XBOOLE_1:1; then ( f1 /* (s ^\ k) is convergent & lim (f1 /* (s ^\ k)) = lim_in+infty f1 ) by A1, A2, A9, LIMFUNC1:def_12; then A17: f2 /* (f1 /* (s ^\ k)) is divergent_to-infty by A2, A12, LIMFUNC2:def_6; f2 /* (f1 /* (s ^\ k)) = f2 /* ((f1 /* s) ^\ k) by A10, VALUED_0:27 .= (f2 /* (f1 /* s)) ^\ k by A8, VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A6, VALUED_0:31 ; hence (f2 * f1) /* s is divergent_to-infty by A17, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is divergent_in+infty_to-infty by A3, LIMFUNC1:def_8; ::_thesis: verum end; theorem :: LIMFUNC4:31 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in-infty & f2 is_left_divergent_to+infty_in lim_in-infty f1 & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (left_open_halfline r) holds f1 . g < lim_in-infty f1 holds f2 * f1 is divergent_in-infty_to+infty proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in-infty & f2 is_left_divergent_to+infty_in lim_in-infty f1 & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (left_open_halfline r) holds f1 . g < lim_in-infty f1 implies f2 * f1 is divergent_in-infty_to+infty ) assume that A1: f1 is convergent_in-infty and A2: f2 is_left_divergent_to+infty_in lim_in-infty f1 and A3: for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ; ::_thesis: ( for r being Real ex g being Real st ( g in (dom f1) /\ (left_open_halfline r) & not f1 . g < lim_in-infty f1 ) or f2 * f1 is divergent_in-infty_to+infty ) given r being Real such that A4: for g being Real st g in (dom f1) /\ (left_open_halfline r) holds f1 . g < lim_in-infty f1 ; ::_thesis: f2 * f1 is divergent_in-infty_to+infty now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to-infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (f2_*_f1)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is divergent_to-infty & rng s c= dom (f2 * f1) implies (f2 * f1) /* s is divergent_to+infty ) assume that A5: s is divergent_to-infty and A6: rng s c= dom (f2 * f1) ; ::_thesis: (f2 * f1) /* s is divergent_to+infty consider k being Element of NAT such that A7: for n being Element of NAT st k <= n holds s . n < r by A5, LIMFUNC1:def_5; A8: rng (f1 /* s) c= dom f2 by A6, Lm2; set q = s ^\ k; A9: s ^\ k is divergent_to-infty by A5, LIMFUNC1:27; A10: rng s c= dom f1 by A6, Lm2; A11: rng (s ^\ k) c= rng s by VALUED_0:21; A12: rng (f1 /* (s ^\ k)) c= (dom f2) /\ (left_open_halfline (lim_in-infty f1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (f1 /* (s ^\ k)) or x in (dom f2) /\ (left_open_halfline (lim_in-infty f1)) ) assume x in rng (f1 /* (s ^\ k)) ; ::_thesis: x in (dom f2) /\ (left_open_halfline (lim_in-infty f1)) then consider n being Element of NAT such that A13: (f1 /* (s ^\ k)) . n = x by FUNCT_2:113; A14: x = f1 . ((s ^\ k) . n) by A10, A11, A13, FUNCT_2:108, XBOOLE_1:1 .= f1 . (s . (n + k)) by NAT_1:def_3 ; s . (n + k) < r by A7, NAT_1:12; then s . (n + k) in { r2 where r2 is Real : r2 < r } ; then A15: s . (n + k) in left_open_halfline r by XXREAL_1:229; s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in (dom f1) /\ (left_open_halfline r) by A10, A15, XBOOLE_0:def_4; then f1 . (s . (n + k)) < lim_in-infty f1 by A4; then x in { g1 where g1 is Real : g1 < lim_in-infty f1 } by A14; then A16: x in left_open_halfline (lim_in-infty f1) by XXREAL_1:229; (f1 /* s) . (n + k) in rng (f1 /* s) by VALUED_0:28; then (f1 /* s) . (n + k) in dom f2 by A8; then x in dom f2 by A10, A14, FUNCT_2:108; hence x in (dom f2) /\ (left_open_halfline (lim_in-infty f1)) by A16, XBOOLE_0:def_4; ::_thesis: verum end; rng (s ^\ k) c= dom f1 by A10, A11, XBOOLE_1:1; then ( f1 /* (s ^\ k) is convergent & lim (f1 /* (s ^\ k)) = lim_in-infty f1 ) by A1, A2, A9, LIMFUNC1:def_13; then A17: f2 /* (f1 /* (s ^\ k)) is divergent_to+infty by A2, A12, LIMFUNC2:def_2; f2 /* (f1 /* (s ^\ k)) = f2 /* ((f1 /* s) ^\ k) by A10, VALUED_0:27 .= (f2 /* (f1 /* s)) ^\ k by A8, VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A6, VALUED_0:31 ; hence (f2 * f1) /* s is divergent_to+infty by A17, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is divergent_in-infty_to+infty by A3, LIMFUNC1:def_10; ::_thesis: verum end; theorem :: LIMFUNC4:32 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in-infty & f2 is_left_divergent_to-infty_in lim_in-infty f1 & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (left_open_halfline r) holds f1 . g < lim_in-infty f1 holds f2 * f1 is divergent_in-infty_to-infty proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in-infty & f2 is_left_divergent_to-infty_in lim_in-infty f1 & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (left_open_halfline r) holds f1 . g < lim_in-infty f1 implies f2 * f1 is divergent_in-infty_to-infty ) assume that A1: f1 is convergent_in-infty and A2: f2 is_left_divergent_to-infty_in lim_in-infty f1 and A3: for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ; ::_thesis: ( for r being Real ex g being Real st ( g in (dom f1) /\ (left_open_halfline r) & not f1 . g < lim_in-infty f1 ) or f2 * f1 is divergent_in-infty_to-infty ) given r being Real such that A4: for g being Real st g in (dom f1) /\ (left_open_halfline r) holds f1 . g < lim_in-infty f1 ; ::_thesis: f2 * f1 is divergent_in-infty_to-infty now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to-infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (f2_*_f1)_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is divergent_to-infty & rng s c= dom (f2 * f1) implies (f2 * f1) /* s is divergent_to-infty ) assume that A5: s is divergent_to-infty and A6: rng s c= dom (f2 * f1) ; ::_thesis: (f2 * f1) /* s is divergent_to-infty consider k being Element of NAT such that A7: for n being Element of NAT st k <= n holds s . n < r by A5, LIMFUNC1:def_5; A8: rng (f1 /* s) c= dom f2 by A6, Lm2; set q = s ^\ k; A9: s ^\ k is divergent_to-infty by A5, LIMFUNC1:27; A10: rng s c= dom f1 by A6, Lm2; A11: rng (s ^\ k) c= rng s by VALUED_0:21; A12: rng (f1 /* (s ^\ k)) c= (dom f2) /\ (left_open_halfline (lim_in-infty f1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (f1 /* (s ^\ k)) or x in (dom f2) /\ (left_open_halfline (lim_in-infty f1)) ) assume x in rng (f1 /* (s ^\ k)) ; ::_thesis: x in (dom f2) /\ (left_open_halfline (lim_in-infty f1)) then consider n being Element of NAT such that A13: (f1 /* (s ^\ k)) . n = x by FUNCT_2:113; A14: x = f1 . ((s ^\ k) . n) by A10, A11, A13, FUNCT_2:108, XBOOLE_1:1 .= f1 . (s . (n + k)) by NAT_1:def_3 ; s . (n + k) < r by A7, NAT_1:12; then s . (n + k) in { r2 where r2 is Real : r2 < r } ; then A15: s . (n + k) in left_open_halfline r by XXREAL_1:229; s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in (dom f1) /\ (left_open_halfline r) by A10, A15, XBOOLE_0:def_4; then f1 . (s . (n + k)) < lim_in-infty f1 by A4; then x in { g1 where g1 is Real : g1 < lim_in-infty f1 } by A14; then A16: x in left_open_halfline (lim_in-infty f1) by XXREAL_1:229; (f1 /* s) . (n + k) in rng (f1 /* s) by VALUED_0:28; then (f1 /* s) . (n + k) in dom f2 by A8; then x in dom f2 by A10, A14, FUNCT_2:108; hence x in (dom f2) /\ (left_open_halfline (lim_in-infty f1)) by A16, XBOOLE_0:def_4; ::_thesis: verum end; rng (s ^\ k) c= dom f1 by A10, A11, XBOOLE_1:1; then ( f1 /* (s ^\ k) is convergent & lim (f1 /* (s ^\ k)) = lim_in-infty f1 ) by A1, A2, A9, LIMFUNC1:def_13; then A17: f2 /* (f1 /* (s ^\ k)) is divergent_to-infty by A2, A12, LIMFUNC2:def_3; f2 /* (f1 /* (s ^\ k)) = f2 /* ((f1 /* s) ^\ k) by A10, VALUED_0:27 .= (f2 /* (f1 /* s)) ^\ k by A8, VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A6, VALUED_0:31 ; hence (f2 * f1) /* s is divergent_to-infty by A17, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is divergent_in-infty_to-infty by A3, LIMFUNC1:def_11; ::_thesis: verum end; theorem :: LIMFUNC4:33 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in-infty & f2 is_right_divergent_to+infty_in lim_in-infty f1 & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (left_open_halfline r) holds lim_in-infty f1 < f1 . g holds f2 * f1 is divergent_in-infty_to+infty proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in-infty & f2 is_right_divergent_to+infty_in lim_in-infty f1 & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (left_open_halfline r) holds lim_in-infty f1 < f1 . g implies f2 * f1 is divergent_in-infty_to+infty ) assume that A1: f1 is convergent_in-infty and A2: f2 is_right_divergent_to+infty_in lim_in-infty f1 and A3: for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ; ::_thesis: ( for r being Real ex g being Real st ( g in (dom f1) /\ (left_open_halfline r) & not lim_in-infty f1 < f1 . g ) or f2 * f1 is divergent_in-infty_to+infty ) given r being Real such that A4: for g being Real st g in (dom f1) /\ (left_open_halfline r) holds lim_in-infty f1 < f1 . g ; ::_thesis: f2 * f1 is divergent_in-infty_to+infty now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to-infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (f2_*_f1)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is divergent_to-infty & rng s c= dom (f2 * f1) implies (f2 * f1) /* s is divergent_to+infty ) assume that A5: s is divergent_to-infty and A6: rng s c= dom (f2 * f1) ; ::_thesis: (f2 * f1) /* s is divergent_to+infty consider k being Element of NAT such that A7: for n being Element of NAT st k <= n holds s . n < r by A5, LIMFUNC1:def_5; A8: rng (f1 /* s) c= dom f2 by A6, Lm2; set q = s ^\ k; A9: s ^\ k is divergent_to-infty by A5, LIMFUNC1:27; A10: rng s c= dom f1 by A6, Lm2; A11: rng (s ^\ k) c= rng s by VALUED_0:21; A12: rng (f1 /* (s ^\ k)) c= (dom f2) /\ (right_open_halfline (lim_in-infty f1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (f1 /* (s ^\ k)) or x in (dom f2) /\ (right_open_halfline (lim_in-infty f1)) ) assume x in rng (f1 /* (s ^\ k)) ; ::_thesis: x in (dom f2) /\ (right_open_halfline (lim_in-infty f1)) then consider n being Element of NAT such that A13: (f1 /* (s ^\ k)) . n = x by FUNCT_2:113; A14: x = f1 . ((s ^\ k) . n) by A10, A11, A13, FUNCT_2:108, XBOOLE_1:1 .= f1 . (s . (n + k)) by NAT_1:def_3 ; s . (n + k) < r by A7, NAT_1:12; then s . (n + k) in { r2 where r2 is Real : r2 < r } ; then A15: s . (n + k) in left_open_halfline r by XXREAL_1:229; s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in (dom f1) /\ (left_open_halfline r) by A10, A15, XBOOLE_0:def_4; then lim_in-infty f1 < f1 . (s . (n + k)) by A4; then x in { g1 where g1 is Real : lim_in-infty f1 < g1 } by A14; then A16: x in right_open_halfline (lim_in-infty f1) by XXREAL_1:230; (f1 /* s) . (n + k) in rng (f1 /* s) by VALUED_0:28; then (f1 /* s) . (n + k) in dom f2 by A8; then x in dom f2 by A10, A14, FUNCT_2:108; hence x in (dom f2) /\ (right_open_halfline (lim_in-infty f1)) by A16, XBOOLE_0:def_4; ::_thesis: verum end; rng (s ^\ k) c= dom f1 by A10, A11, XBOOLE_1:1; then ( f1 /* (s ^\ k) is convergent & lim (f1 /* (s ^\ k)) = lim_in-infty f1 ) by A1, A2, A9, LIMFUNC1:def_13; then A17: f2 /* (f1 /* (s ^\ k)) is divergent_to+infty by A2, A12, LIMFUNC2:def_5; f2 /* (f1 /* (s ^\ k)) = f2 /* ((f1 /* s) ^\ k) by A10, VALUED_0:27 .= (f2 /* (f1 /* s)) ^\ k by A8, VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A6, VALUED_0:31 ; hence (f2 * f1) /* s is divergent_to+infty by A17, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is divergent_in-infty_to+infty by A3, LIMFUNC1:def_10; ::_thesis: verum end; theorem :: LIMFUNC4:34 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in-infty & f2 is_right_divergent_to-infty_in lim_in-infty f1 & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (left_open_halfline r) holds lim_in-infty f1 < f1 . g holds f2 * f1 is divergent_in-infty_to-infty proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in-infty & f2 is_right_divergent_to-infty_in lim_in-infty f1 & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (left_open_halfline r) holds lim_in-infty f1 < f1 . g implies f2 * f1 is divergent_in-infty_to-infty ) assume that A1: f1 is convergent_in-infty and A2: f2 is_right_divergent_to-infty_in lim_in-infty f1 and A3: for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ; ::_thesis: ( for r being Real ex g being Real st ( g in (dom f1) /\ (left_open_halfline r) & not lim_in-infty f1 < f1 . g ) or f2 * f1 is divergent_in-infty_to-infty ) given r being Real such that A4: for g being Real st g in (dom f1) /\ (left_open_halfline r) holds lim_in-infty f1 < f1 . g ; ::_thesis: f2 * f1 is divergent_in-infty_to-infty now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to-infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (f2_*_f1)_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is divergent_to-infty & rng s c= dom (f2 * f1) implies (f2 * f1) /* s is divergent_to-infty ) assume that A5: s is divergent_to-infty and A6: rng s c= dom (f2 * f1) ; ::_thesis: (f2 * f1) /* s is divergent_to-infty consider k being Element of NAT such that A7: for n being Element of NAT st k <= n holds s . n < r by A5, LIMFUNC1:def_5; A8: rng (f1 /* s) c= dom f2 by A6, Lm2; set q = s ^\ k; A9: s ^\ k is divergent_to-infty by A5, LIMFUNC1:27; A10: rng s c= dom f1 by A6, Lm2; A11: rng (s ^\ k) c= rng s by VALUED_0:21; A12: rng (f1 /* (s ^\ k)) c= (dom f2) /\ (right_open_halfline (lim_in-infty f1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (f1 /* (s ^\ k)) or x in (dom f2) /\ (right_open_halfline (lim_in-infty f1)) ) assume x in rng (f1 /* (s ^\ k)) ; ::_thesis: x in (dom f2) /\ (right_open_halfline (lim_in-infty f1)) then consider n being Element of NAT such that A13: (f1 /* (s ^\ k)) . n = x by FUNCT_2:113; A14: x = f1 . ((s ^\ k) . n) by A10, A11, A13, FUNCT_2:108, XBOOLE_1:1 .= f1 . (s . (n + k)) by NAT_1:def_3 ; s . (n + k) < r by A7, NAT_1:12; then s . (n + k) in { r2 where r2 is Real : r2 < r } ; then A15: s . (n + k) in left_open_halfline r by XXREAL_1:229; s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in (dom f1) /\ (left_open_halfline r) by A10, A15, XBOOLE_0:def_4; then lim_in-infty f1 < f1 . (s . (n + k)) by A4; then x in { g1 where g1 is Real : lim_in-infty f1 < g1 } by A14; then A16: x in right_open_halfline (lim_in-infty f1) by XXREAL_1:230; (f1 /* s) . (n + k) in rng (f1 /* s) by VALUED_0:28; then (f1 /* s) . (n + k) in dom f2 by A8; then x in dom f2 by A10, A14, FUNCT_2:108; hence x in (dom f2) /\ (right_open_halfline (lim_in-infty f1)) by A16, XBOOLE_0:def_4; ::_thesis: verum end; rng (s ^\ k) c= dom f1 by A10, A11, XBOOLE_1:1; then ( f1 /* (s ^\ k) is convergent & lim (f1 /* (s ^\ k)) = lim_in-infty f1 ) by A1, A2, A9, LIMFUNC1:def_13; then A17: f2 /* (f1 /* (s ^\ k)) is divergent_to-infty by A2, A12, LIMFUNC2:def_6; f2 /* (f1 /* (s ^\ k)) = f2 /* ((f1 /* s) ^\ k) by A10, VALUED_0:27 .= (f2 /* (f1 /* s)) ^\ k by A8, VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A6, VALUED_0:31 ; hence (f2 * f1) /* s is divergent_to-infty by A17, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is divergent_in-infty_to-infty by A3, LIMFUNC1:def_11; ::_thesis: verum end; theorem :: LIMFUNC4:35 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_divergent_to+infty_in x0 & f2 is divergent_in+infty_to+infty & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) holds f2 * f1 is_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_divergent_to+infty_in x0 & f2 is divergent_in+infty_to+infty & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) holds f2 * f1 is_divergent_to+infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_divergent_to+infty_in x0 & f2 is divergent_in+infty_to+infty & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) implies f2 * f1 is_divergent_to+infty_in x0 ) assume that A1: f1 is_divergent_to+infty_in x0 and A2: f2 is divergent_in+infty_to+infty and A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ; ::_thesis: f2 * f1 is_divergent_to+infty_in x0 now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f2_*_f1))_\_{x0}_holds_ (f2_*_f1)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) \ {x0} implies (f2 * f1) /* s is divergent_to+infty ) assume that A4: ( s is convergent & lim s = x0 ) and A5: rng s c= (dom (f2 * f1)) \ {x0} ; ::_thesis: (f2 * f1) /* s is divergent_to+infty A6: rng s c= dom (f2 * f1) by A5, Th2; rng s c= (dom f1) \ {x0} by A5, Th2; then A7: f1 /* s is divergent_to+infty by A1, A4, LIMFUNC3:def_2; rng (f1 /* s) c= dom f2 by A5, Th2; then f2 /* (f1 /* s) is divergent_to+infty by A2, A7, LIMFUNC1:def_7; hence (f2 * f1) /* s is divergent_to+infty by A6, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is_divergent_to+infty_in x0 by A3, LIMFUNC3:def_2; ::_thesis: verum end; theorem :: LIMFUNC4:36 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_divergent_to+infty_in x0 & f2 is divergent_in+infty_to-infty & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) holds f2 * f1 is_divergent_to-infty_in x0 proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_divergent_to+infty_in x0 & f2 is divergent_in+infty_to-infty & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) holds f2 * f1 is_divergent_to-infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_divergent_to+infty_in x0 & f2 is divergent_in+infty_to-infty & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) implies f2 * f1 is_divergent_to-infty_in x0 ) assume that A1: f1 is_divergent_to+infty_in x0 and A2: f2 is divergent_in+infty_to-infty and A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ; ::_thesis: f2 * f1 is_divergent_to-infty_in x0 now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f2_*_f1))_\_{x0}_holds_ (f2_*_f1)_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) \ {x0} implies (f2 * f1) /* s is divergent_to-infty ) assume that A4: ( s is convergent & lim s = x0 ) and A5: rng s c= (dom (f2 * f1)) \ {x0} ; ::_thesis: (f2 * f1) /* s is divergent_to-infty A6: rng s c= dom (f2 * f1) by A5, Th2; rng s c= (dom f1) \ {x0} by A5, Th2; then A7: f1 /* s is divergent_to+infty by A1, A4, LIMFUNC3:def_2; rng (f1 /* s) c= dom f2 by A5, Th2; then f2 /* (f1 /* s) is divergent_to-infty by A2, A7, LIMFUNC1:def_8; hence (f2 * f1) /* s is divergent_to-infty by A6, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is_divergent_to-infty_in x0 by A3, LIMFUNC3:def_3; ::_thesis: verum end; theorem :: LIMFUNC4:37 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_divergent_to-infty_in x0 & f2 is divergent_in-infty_to+infty & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) holds f2 * f1 is_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_divergent_to-infty_in x0 & f2 is divergent_in-infty_to+infty & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) holds f2 * f1 is_divergent_to+infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_divergent_to-infty_in x0 & f2 is divergent_in-infty_to+infty & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) implies f2 * f1 is_divergent_to+infty_in x0 ) assume that A1: f1 is_divergent_to-infty_in x0 and A2: f2 is divergent_in-infty_to+infty and A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ; ::_thesis: f2 * f1 is_divergent_to+infty_in x0 now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f2_*_f1))_\_{x0}_holds_ (f2_*_f1)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) \ {x0} implies (f2 * f1) /* s is divergent_to+infty ) assume that A4: ( s is convergent & lim s = x0 ) and A5: rng s c= (dom (f2 * f1)) \ {x0} ; ::_thesis: (f2 * f1) /* s is divergent_to+infty A6: rng s c= dom (f2 * f1) by A5, Th2; rng s c= (dom f1) \ {x0} by A5, Th2; then A7: f1 /* s is divergent_to-infty by A1, A4, LIMFUNC3:def_3; rng (f1 /* s) c= dom f2 by A5, Th2; then f2 /* (f1 /* s) is divergent_to+infty by A2, A7, LIMFUNC1:def_10; hence (f2 * f1) /* s is divergent_to+infty by A6, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is_divergent_to+infty_in x0 by A3, LIMFUNC3:def_2; ::_thesis: verum end; theorem :: LIMFUNC4:38 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_divergent_to-infty_in x0 & f2 is divergent_in-infty_to-infty & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) holds f2 * f1 is_divergent_to-infty_in x0 proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_divergent_to-infty_in x0 & f2 is divergent_in-infty_to-infty & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) holds f2 * f1 is_divergent_to-infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_divergent_to-infty_in x0 & f2 is divergent_in-infty_to-infty & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) implies f2 * f1 is_divergent_to-infty_in x0 ) assume that A1: f1 is_divergent_to-infty_in x0 and A2: f2 is divergent_in-infty_to-infty and A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ; ::_thesis: f2 * f1 is_divergent_to-infty_in x0 now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f2_*_f1))_\_{x0}_holds_ (f2_*_f1)_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) \ {x0} implies (f2 * f1) /* s is divergent_to-infty ) assume that A4: ( s is convergent & lim s = x0 ) and A5: rng s c= (dom (f2 * f1)) \ {x0} ; ::_thesis: (f2 * f1) /* s is divergent_to-infty A6: rng s c= dom (f2 * f1) by A5, Th2; rng s c= (dom f1) \ {x0} by A5, Th2; then A7: f1 /* s is divergent_to-infty by A1, A4, LIMFUNC3:def_3; rng (f1 /* s) c= dom f2 by A5, Th2; then f2 /* (f1 /* s) is divergent_to-infty by A2, A7, LIMFUNC1:def_11; hence (f2 * f1) /* s is divergent_to-infty by A6, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is_divergent_to-infty_in x0 by A3, LIMFUNC3:def_3; ::_thesis: verum end; theorem :: LIMFUNC4:39 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_divergent_to+infty_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r <> lim (f1,x0) ) ) holds f2 * f1 is_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_divergent_to+infty_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r <> lim (f1,x0) ) ) holds f2 * f1 is_divergent_to+infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_convergent_in x0 & f2 is_divergent_to+infty_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r <> lim (f1,x0) ) ) implies f2 * f1 is_divergent_to+infty_in x0 ) assume that A1: f1 is_convergent_in x0 and A2: f2 is_divergent_to+infty_in lim (f1,x0) and A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) & not f1 . r <> lim (f1,x0) ) ) or f2 * f1 is_divergent_to+infty_in x0 ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r <> lim (f1,x0) ; ::_thesis: f2 * f1 is_divergent_to+infty_in x0 now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f2_*_f1))_\_{x0}_holds_ (f2_*_f1)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) \ {x0} implies (f2 * f1) /* s is divergent_to+infty ) assume that A6: ( s is convergent & lim s = x0 ) and A7: rng s c= (dom (f2 * f1)) \ {x0} ; ::_thesis: (f2 * f1) /* s is divergent_to+infty consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds ( x0 - g < s . n & s . n < x0 + g ) by A4, A6, LIMFUNC3:7; set q = (f1 /* s) ^\ k; A9: rng s c= dom (f2 * f1) by A7, Th2; rng (f1 /* s) c= dom f2 by A7, Th2; then A10: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A9, VALUED_0:31 ; A11: rng s c= (dom f1) \ {x0} by A7, Th2; then A12: f1 /* s is convergent by A1, A2, A6, LIMFUNC3:def_4; A13: rng s c= dom f1 by A7, Th2; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_\_{(lim_(f1,x0))} let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) \ {(lim (f1,x0))} ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) \ {(lim (f1,x0))} then consider n being Element of NAT such that A14: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A15: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A13, FUNCT_2:108 .= x by A14, NAT_1:def_3 ; k <= n + k by NAT_1:12; then ( x0 - g < s . (n + k) & s . (n + k) < x0 + g ) by A8; then s . (n + k) in { g1 where g1 is Real : ( x0 - g < g1 & g1 < x0 + g ) } ; then A16: s . (n + k) in ].(x0 - g),(x0 + g).[ by RCOMP_1:def_2; A17: s . (n + k) in rng s by VALUED_0:28; then not s . (n + k) in {x0} by A7, XBOOLE_0:def_5; then s . (n + k) in ].(x0 - g),(x0 + g).[ \ {x0} by A16, XBOOLE_0:def_5; then s . (n + k) in ].(x0 - g),x0.[ \/ ].x0,(x0 + g).[ by A4, LIMFUNC3:4; then s . (n + k) in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) by A13, A17, XBOOLE_0:def_4; then f1 . (s . (n + k)) <> lim (f1,x0) by A5; then A18: not f1 . (s . (n + k)) in {(lim (f1,x0))} by TARSKI:def_1; f1 . (s . (n + k)) in dom f2 by A9, A17, FUNCT_1:11; hence x in (dom f2) \ {(lim (f1,x0))} by A18, A15, XBOOLE_0:def_5; ::_thesis: verum end; then A19: rng ((f1 /* s) ^\ k) c= (dom f2) \ {(lim (f1,x0))} by TARSKI:def_3; lim (f1 /* s) = lim (f1,x0) by A1, A6, A11, LIMFUNC3:def_4; then lim ((f1 /* s) ^\ k) = lim (f1,x0) by A12, SEQ_4:20; then f2 /* ((f1 /* s) ^\ k) is divergent_to+infty by A2, A12, A19, LIMFUNC3:def_2; hence (f2 * f1) /* s is divergent_to+infty by A10, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is_divergent_to+infty_in x0 by A3, LIMFUNC3:def_2; ::_thesis: verum end; theorem :: LIMFUNC4:40 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_divergent_to-infty_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r <> lim (f1,x0) ) ) holds f2 * f1 is_divergent_to-infty_in x0 proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_divergent_to-infty_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r <> lim (f1,x0) ) ) holds f2 * f1 is_divergent_to-infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_convergent_in x0 & f2 is_divergent_to-infty_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r <> lim (f1,x0) ) ) implies f2 * f1 is_divergent_to-infty_in x0 ) assume that A1: f1 is_convergent_in x0 and A2: f2 is_divergent_to-infty_in lim (f1,x0) and A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) & not f1 . r <> lim (f1,x0) ) ) or f2 * f1 is_divergent_to-infty_in x0 ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r <> lim (f1,x0) ; ::_thesis: f2 * f1 is_divergent_to-infty_in x0 now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f2_*_f1))_\_{x0}_holds_ (f2_*_f1)_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) \ {x0} implies (f2 * f1) /* s is divergent_to-infty ) assume that A6: ( s is convergent & lim s = x0 ) and A7: rng s c= (dom (f2 * f1)) \ {x0} ; ::_thesis: (f2 * f1) /* s is divergent_to-infty consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds ( x0 - g < s . n & s . n < x0 + g ) by A4, A6, LIMFUNC3:7; set q = (f1 /* s) ^\ k; A9: rng s c= dom (f2 * f1) by A7, Th2; rng (f1 /* s) c= dom f2 by A7, Th2; then A10: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A9, VALUED_0:31 ; A11: rng s c= (dom f1) \ {x0} by A7, Th2; then A12: f1 /* s is convergent by A1, A2, A6, LIMFUNC3:def_4; A13: rng s c= dom f1 by A7, Th2; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_\_{(lim_(f1,x0))} let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) \ {(lim (f1,x0))} ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) \ {(lim (f1,x0))} then consider n being Element of NAT such that A14: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A15: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A13, FUNCT_2:108 .= x by A14, NAT_1:def_3 ; k <= n + k by NAT_1:12; then ( x0 - g < s . (n + k) & s . (n + k) < x0 + g ) by A8; then s . (n + k) in { g1 where g1 is Real : ( x0 - g < g1 & g1 < x0 + g ) } ; then A16: s . (n + k) in ].(x0 - g),(x0 + g).[ by RCOMP_1:def_2; A17: s . (n + k) in rng s by VALUED_0:28; then not s . (n + k) in {x0} by A7, XBOOLE_0:def_5; then s . (n + k) in ].(x0 - g),(x0 + g).[ \ {x0} by A16, XBOOLE_0:def_5; then s . (n + k) in ].(x0 - g),x0.[ \/ ].x0,(x0 + g).[ by A4, LIMFUNC3:4; then s . (n + k) in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) by A13, A17, XBOOLE_0:def_4; then f1 . (s . (n + k)) <> lim (f1,x0) by A5; then A18: not f1 . (s . (n + k)) in {(lim (f1,x0))} by TARSKI:def_1; f1 . (s . (n + k)) in dom f2 by A9, A17, FUNCT_1:11; hence x in (dom f2) \ {(lim (f1,x0))} by A18, A15, XBOOLE_0:def_5; ::_thesis: verum end; then A19: rng ((f1 /* s) ^\ k) c= (dom f2) \ {(lim (f1,x0))} by TARSKI:def_3; lim (f1 /* s) = lim (f1,x0) by A1, A6, A11, LIMFUNC3:def_4; then lim ((f1 /* s) ^\ k) = lim (f1,x0) by A12, SEQ_4:20; then f2 /* ((f1 /* s) ^\ k) is divergent_to-infty by A2, A12, A19, LIMFUNC3:def_3; hence (f2 * f1) /* s is divergent_to-infty by A10, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is_divergent_to-infty_in x0 by A3, LIMFUNC3:def_3; ::_thesis: verum end; theorem :: LIMFUNC4:41 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_right_divergent_to+infty_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r > lim (f1,x0) ) ) holds f2 * f1 is_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_right_divergent_to+infty_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r > lim (f1,x0) ) ) holds f2 * f1 is_divergent_to+infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_convergent_in x0 & f2 is_right_divergent_to+infty_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r > lim (f1,x0) ) ) implies f2 * f1 is_divergent_to+infty_in x0 ) assume that A1: f1 is_convergent_in x0 and A2: f2 is_right_divergent_to+infty_in lim (f1,x0) and A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) & not f1 . r > lim (f1,x0) ) ) or f2 * f1 is_divergent_to+infty_in x0 ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds lim (f1,x0) < f1 . r ; ::_thesis: f2 * f1 is_divergent_to+infty_in x0 now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f2_*_f1))_\_{x0}_holds_ (f2_*_f1)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) \ {x0} implies (f2 * f1) /* s is divergent_to+infty ) assume that A6: ( s is convergent & lim s = x0 ) and A7: rng s c= (dom (f2 * f1)) \ {x0} ; ::_thesis: (f2 * f1) /* s is divergent_to+infty consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds ( x0 - g < s . n & s . n < x0 + g ) by A4, A6, LIMFUNC3:7; set q = (f1 /* s) ^\ k; A9: rng s c= dom (f2 * f1) by A7, Th2; rng (f1 /* s) c= dom f2 by A7, Th2; then A10: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A9, VALUED_0:31 ; A11: rng s c= (dom f1) \ {x0} by A7, Th2; then A12: f1 /* s is convergent by A1, A2, A6, LIMFUNC3:def_4; A13: rng s c= dom f1 by A7, Th2; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_/\_(right_open_halfline_(lim_(f1,x0))) let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) /\ (right_open_halfline (lim (f1,x0))) ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) /\ (right_open_halfline (lim (f1,x0))) then consider n being Element of NAT such that A14: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A15: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A13, FUNCT_2:108 .= x by A14, NAT_1:def_3 ; k <= n + k by NAT_1:12; then ( x0 - g < s . (n + k) & s . (n + k) < x0 + g ) by A8; then s . (n + k) in { g1 where g1 is Real : ( x0 - g < g1 & g1 < x0 + g ) } ; then A16: s . (n + k) in ].(x0 - g),(x0 + g).[ by RCOMP_1:def_2; A17: s . (n + k) in rng s by VALUED_0:28; then not s . (n + k) in {x0} by A7, XBOOLE_0:def_5; then s . (n + k) in ].(x0 - g),(x0 + g).[ \ {x0} by A16, XBOOLE_0:def_5; then s . (n + k) in ].(x0 - g),x0.[ \/ ].x0,(x0 + g).[ by A4, LIMFUNC3:4; then s . (n + k) in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) by A13, A17, XBOOLE_0:def_4; then f1 . (s . (n + k)) > lim (f1,x0) by A5; then f1 . (s . (n + k)) in { g2 where g2 is Real : lim (f1,x0) < g2 } ; then A18: f1 . (s . (n + k)) in right_open_halfline (lim (f1,x0)) by XXREAL_1:230; f1 . (s . (n + k)) in dom f2 by A9, A17, FUNCT_1:11; hence x in (dom f2) /\ (right_open_halfline (lim (f1,x0))) by A18, A15, XBOOLE_0:def_4; ::_thesis: verum end; then A19: rng ((f1 /* s) ^\ k) c= (dom f2) /\ (right_open_halfline (lim (f1,x0))) by TARSKI:def_3; lim (f1 /* s) = lim (f1,x0) by A1, A6, A11, LIMFUNC3:def_4; then lim ((f1 /* s) ^\ k) = lim (f1,x0) by A12, SEQ_4:20; then f2 /* ((f1 /* s) ^\ k) is divergent_to+infty by A2, A12, A19, LIMFUNC2:def_5; hence (f2 * f1) /* s is divergent_to+infty by A10, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is_divergent_to+infty_in x0 by A3, LIMFUNC3:def_2; ::_thesis: verum end; theorem :: LIMFUNC4:42 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_right_divergent_to-infty_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r > lim (f1,x0) ) ) holds f2 * f1 is_divergent_to-infty_in x0 proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_right_divergent_to-infty_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r > lim (f1,x0) ) ) holds f2 * f1 is_divergent_to-infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_convergent_in x0 & f2 is_right_divergent_to-infty_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r > lim (f1,x0) ) ) implies f2 * f1 is_divergent_to-infty_in x0 ) assume that A1: f1 is_convergent_in x0 and A2: f2 is_right_divergent_to-infty_in lim (f1,x0) and A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) & not f1 . r > lim (f1,x0) ) ) or f2 * f1 is_divergent_to-infty_in x0 ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds lim (f1,x0) < f1 . r ; ::_thesis: f2 * f1 is_divergent_to-infty_in x0 now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f2_*_f1))_\_{x0}_holds_ (f2_*_f1)_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) \ {x0} implies (f2 * f1) /* s is divergent_to-infty ) assume that A6: ( s is convergent & lim s = x0 ) and A7: rng s c= (dom (f2 * f1)) \ {x0} ; ::_thesis: (f2 * f1) /* s is divergent_to-infty consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds ( x0 - g < s . n & s . n < x0 + g ) by A4, A6, LIMFUNC3:7; set q = (f1 /* s) ^\ k; A9: rng s c= dom (f2 * f1) by A7, Th2; rng (f1 /* s) c= dom f2 by A7, Th2; then A10: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A9, VALUED_0:31 ; A11: rng s c= (dom f1) \ {x0} by A7, Th2; then A12: f1 /* s is convergent by A1, A2, A6, LIMFUNC3:def_4; A13: rng s c= dom f1 by A7, Th2; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_/\_(right_open_halfline_(lim_(f1,x0))) let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) /\ (right_open_halfline (lim (f1,x0))) ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) /\ (right_open_halfline (lim (f1,x0))) then consider n being Element of NAT such that A14: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A15: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A13, FUNCT_2:108 .= x by A14, NAT_1:def_3 ; k <= n + k by NAT_1:12; then ( x0 - g < s . (n + k) & s . (n + k) < x0 + g ) by A8; then s . (n + k) in { g1 where g1 is Real : ( x0 - g < g1 & g1 < x0 + g ) } ; then A16: s . (n + k) in ].(x0 - g),(x0 + g).[ by RCOMP_1:def_2; A17: s . (n + k) in rng s by VALUED_0:28; then not s . (n + k) in {x0} by A7, XBOOLE_0:def_5; then s . (n + k) in ].(x0 - g),(x0 + g).[ \ {x0} by A16, XBOOLE_0:def_5; then s . (n + k) in ].(x0 - g),x0.[ \/ ].x0,(x0 + g).[ by A4, LIMFUNC3:4; then s . (n + k) in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) by A13, A17, XBOOLE_0:def_4; then f1 . (s . (n + k)) > lim (f1,x0) by A5; then f1 . (s . (n + k)) in { g2 where g2 is Real : lim (f1,x0) < g2 } ; then A18: f1 . (s . (n + k)) in right_open_halfline (lim (f1,x0)) by XXREAL_1:230; f1 . (s . (n + k)) in dom f2 by A9, A17, FUNCT_1:11; hence x in (dom f2) /\ (right_open_halfline (lim (f1,x0))) by A18, A15, XBOOLE_0:def_4; ::_thesis: verum end; then A19: rng ((f1 /* s) ^\ k) c= (dom f2) /\ (right_open_halfline (lim (f1,x0))) by TARSKI:def_3; lim (f1 /* s) = lim (f1,x0) by A1, A6, A11, LIMFUNC3:def_4; then lim ((f1 /* s) ^\ k) = lim (f1,x0) by A12, SEQ_4:20; then f2 /* ((f1 /* s) ^\ k) is divergent_to-infty by A2, A12, A19, LIMFUNC2:def_6; hence (f2 * f1) /* s is divergent_to-infty by A10, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is_divergent_to-infty_in x0 by A3, LIMFUNC3:def_3; ::_thesis: verum end; theorem :: LIMFUNC4:43 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_divergent_to+infty_in lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds f1 . r <> lim_right (f1,x0) ) ) holds f2 * f1 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_convergent_in x0 & f2 is_divergent_to+infty_in lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds f1 . r <> lim_right (f1,x0) ) ) holds f2 * f1 is_right_divergent_to+infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_right_convergent_in x0 & f2 is_divergent_to+infty_in lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds f1 . r <> lim_right (f1,x0) ) ) implies f2 * f1 is_right_divergent_to+infty_in x0 ) assume that A1: f1 is_right_convergent_in x0 and A2: f2 is_divergent_to+infty_in lim_right (f1,x0) and A3: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ ].x0,(x0 + g).[ & not f1 . r <> lim_right (f1,x0) ) ) or f2 * f1 is_right_divergent_to+infty_in x0 ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds f1 . r <> lim_right (f1,x0) ; ::_thesis: f2 * f1 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_(f2_*_f1))_/\_(right_open_halfline_x0)_holds_ (f2_*_f1)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) implies (f2 * f1) /* s is divergent_to+infty ) assume that A6: ( s is convergent & lim s = x0 ) and A7: rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) ; ::_thesis: (f2 * f1) /* s is divergent_to+infty consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds s . n < x0 + g by A4, A6, Lm1, LIMFUNC2:2; set q = (f1 /* s) ^\ k; A9: rng s c= dom (f2 * f1) by A7, Th1; rng (f1 /* s) c= dom f2 by A7, Th1; then A10: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A9, VALUED_0:31 ; A11: rng s c= (dom f1) /\ (right_open_halfline x0) by A7, Th1; then A12: f1 /* s is convergent by A1, A2, A6, LIMFUNC2:def_8; A13: rng s c= dom f1 by A7, Th1; A14: rng s c= right_open_halfline x0 by A7, Th1; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_\_{(lim_right_(f1,x0))} let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) \ {(lim_right (f1,x0))} ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) \ {(lim_right (f1,x0))} then consider n being Element of NAT such that A15: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A16: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A13, FUNCT_2:108 .= x by A15, NAT_1:def_3 ; A17: s . (n + k) < x0 + g by A8, NAT_1:12; A18: s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in right_open_halfline x0 by A14; then s . (n + k) in { g1 where g1 is Real : x0 < g1 } by XXREAL_1:230; then ex g1 being Real st ( g1 = s . (n + k) & x0 < g1 ) ; then s . (n + k) in { g2 where g2 is Real : ( x0 < g2 & g2 < x0 + g ) } by A17; then s . (n + k) in ].x0,(x0 + g).[ by RCOMP_1:def_2; then s . (n + k) in (dom f1) /\ ].x0,(x0 + g).[ by A13, A18, XBOOLE_0:def_4; then f1 . (s . (n + k)) <> lim_right (f1,x0) by A5; then A19: not f1 . (s . (n + k)) in {(lim_right (f1,x0))} by TARSKI:def_1; f1 . (s . (n + k)) in dom f2 by A9, A18, FUNCT_1:11; hence x in (dom f2) \ {(lim_right (f1,x0))} by A19, A16, XBOOLE_0:def_5; ::_thesis: verum end; then A20: rng ((f1 /* s) ^\ k) c= (dom f2) \ {(lim_right (f1,x0))} by TARSKI:def_3; lim (f1 /* s) = lim_right (f1,x0) by A1, A6, A11, LIMFUNC2:def_8; then lim ((f1 /* s) ^\ k) = lim_right (f1,x0) by A12, SEQ_4:20; then f2 /* ((f1 /* s) ^\ k) is divergent_to+infty by A2, A12, A20, LIMFUNC3:def_2; hence (f2 * f1) /* s is divergent_to+infty by A10, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is_right_divergent_to+infty_in x0 by A3, LIMFUNC2:def_5; ::_thesis: verum end; theorem :: LIMFUNC4:44 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_divergent_to-infty_in lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds f1 . r <> lim_right (f1,x0) ) ) holds f2 * f1 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_convergent_in x0 & f2 is_divergent_to-infty_in lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds f1 . r <> lim_right (f1,x0) ) ) holds f2 * f1 is_right_divergent_to-infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_right_convergent_in x0 & f2 is_divergent_to-infty_in lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds f1 . r <> lim_right (f1,x0) ) ) implies f2 * f1 is_right_divergent_to-infty_in x0 ) assume that A1: f1 is_right_convergent_in x0 and A2: f2 is_divergent_to-infty_in lim_right (f1,x0) and A3: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ ].x0,(x0 + g).[ & not f1 . r <> lim_right (f1,x0) ) ) or f2 * f1 is_right_divergent_to-infty_in x0 ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds f1 . r <> lim_right (f1,x0) ; ::_thesis: f2 * f1 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_(f2_*_f1))_/\_(right_open_halfline_x0)_holds_ (f2_*_f1)_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) implies (f2 * f1) /* s is divergent_to-infty ) assume that A6: ( s is convergent & lim s = x0 ) and A7: rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) ; ::_thesis: (f2 * f1) /* s is divergent_to-infty consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds s . n < x0 + g by A4, A6, Lm1, LIMFUNC2:2; set q = (f1 /* s) ^\ k; A9: rng s c= dom (f2 * f1) by A7, Th1; rng (f1 /* s) c= dom f2 by A7, Th1; then A10: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A9, VALUED_0:31 ; A11: rng s c= (dom f1) /\ (right_open_halfline x0) by A7, Th1; then A12: f1 /* s is convergent by A1, A2, A6, LIMFUNC2:def_8; A13: rng s c= dom f1 by A7, Th1; A14: rng s c= right_open_halfline x0 by A7, Th1; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_\_{(lim_right_(f1,x0))} let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) \ {(lim_right (f1,x0))} ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) \ {(lim_right (f1,x0))} then consider n being Element of NAT such that A15: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A16: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A13, FUNCT_2:108 .= x by A15, NAT_1:def_3 ; A17: s . (n + k) < x0 + g by A8, NAT_1:12; A18: s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in right_open_halfline x0 by A14; then s . (n + k) in { g1 where g1 is Real : x0 < g1 } by XXREAL_1:230; then ex g1 being Real st ( g1 = s . (n + k) & x0 < g1 ) ; then s . (n + k) in { g2 where g2 is Real : ( x0 < g2 & g2 < x0 + g ) } by A17; then s . (n + k) in ].x0,(x0 + g).[ by RCOMP_1:def_2; then s . (n + k) in (dom f1) /\ ].x0,(x0 + g).[ by A13, A18, XBOOLE_0:def_4; then f1 . (s . (n + k)) <> lim_right (f1,x0) by A5; then A19: not f1 . (s . (n + k)) in {(lim_right (f1,x0))} by TARSKI:def_1; f1 . (s . (n + k)) in dom f2 by A9, A18, FUNCT_1:11; hence x in (dom f2) \ {(lim_right (f1,x0))} by A19, A16, XBOOLE_0:def_5; ::_thesis: verum end; then A20: rng ((f1 /* s) ^\ k) c= (dom f2) \ {(lim_right (f1,x0))} by TARSKI:def_3; lim (f1 /* s) = lim_right (f1,x0) by A1, A6, A11, LIMFUNC2:def_8; then lim ((f1 /* s) ^\ k) = lim_right (f1,x0) by A12, SEQ_4:20; then f2 /* ((f1 /* s) ^\ k) is divergent_to-infty by A2, A12, A20, LIMFUNC3:def_3; hence (f2 * f1) /* s is divergent_to-infty by A10, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is_right_divergent_to-infty_in x0 by A3, LIMFUNC2:def_6; ::_thesis: verum end; theorem :: LIMFUNC4:45 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in+infty & f2 is_divergent_to+infty_in lim_in+infty f1 & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (right_open_halfline r) holds f1 . g <> lim_in+infty f1 holds f2 * f1 is divergent_in+infty_to+infty proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in+infty & f2 is_divergent_to+infty_in lim_in+infty f1 & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (right_open_halfline r) holds f1 . g <> lim_in+infty f1 implies f2 * f1 is divergent_in+infty_to+infty ) assume that A1: f1 is convergent_in+infty and A2: f2 is_divergent_to+infty_in lim_in+infty f1 and A3: for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ; ::_thesis: ( for r being Real ex g being Real st ( g in (dom f1) /\ (right_open_halfline r) & not f1 . g <> lim_in+infty f1 ) or f2 * f1 is divergent_in+infty_to+infty ) given r being Real such that A4: for g being Real st g in (dom f1) /\ (right_open_halfline r) holds f1 . g <> lim_in+infty f1 ; ::_thesis: f2 * f1 is divergent_in+infty_to+infty now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to+infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (f2_*_f1)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is divergent_to+infty & rng s c= dom (f2 * f1) implies (f2 * f1) /* s is divergent_to+infty ) assume that A5: s is divergent_to+infty and A6: rng s c= dom (f2 * f1) ; ::_thesis: (f2 * f1) /* s is divergent_to+infty consider k being Element of NAT such that A7: for n being Element of NAT st k <= n holds r < s . n by A5, LIMFUNC1:def_4; A8: rng (f1 /* s) c= dom f2 by A6, Lm2; set q = s ^\ k; A9: s ^\ k is divergent_to+infty by A5, LIMFUNC1:26; A10: rng s c= dom f1 by A6, Lm2; A11: rng (s ^\ k) c= rng s by VALUED_0:21; A12: rng (f1 /* (s ^\ k)) c= (dom f2) \ {(lim_in+infty f1)} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (f1 /* (s ^\ k)) or x in (dom f2) \ {(lim_in+infty f1)} ) assume x in rng (f1 /* (s ^\ k)) ; ::_thesis: x in (dom f2) \ {(lim_in+infty f1)} then consider n being Element of NAT such that A13: (f1 /* (s ^\ k)) . n = x by FUNCT_2:113; A14: x = f1 . ((s ^\ k) . n) by A10, A11, A13, FUNCT_2:108, XBOOLE_1:1 .= f1 . (s . (n + k)) by NAT_1:def_3 ; r < s . (n + k) by A7, NAT_1:12; then s . (n + k) in { r2 where r2 is Real : r < r2 } ; then A15: s . (n + k) in right_open_halfline r by XXREAL_1:230; s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in (dom f1) /\ (right_open_halfline r) by A10, A15, XBOOLE_0:def_4; then f1 . (s . (n + k)) <> lim_in+infty f1 by A4; then A16: not f1 . (s . (n + k)) in {(lim_in+infty f1)} by TARSKI:def_1; (f1 /* s) . (n + k) in rng (f1 /* s) by VALUED_0:28; then (f1 /* s) . (n + k) in dom f2 by A8; then x in dom f2 by A10, A14, FUNCT_2:108; hence x in (dom f2) \ {(lim_in+infty f1)} by A14, A16, XBOOLE_0:def_5; ::_thesis: verum end; rng (s ^\ k) c= dom f1 by A10, A11, XBOOLE_1:1; then ( f1 /* (s ^\ k) is convergent & lim (f1 /* (s ^\ k)) = lim_in+infty f1 ) by A1, A2, A9, LIMFUNC1:def_12; then A17: f2 /* (f1 /* (s ^\ k)) is divergent_to+infty by A2, A12, LIMFUNC3:def_2; f2 /* (f1 /* (s ^\ k)) = f2 /* ((f1 /* s) ^\ k) by A10, VALUED_0:27 .= (f2 /* (f1 /* s)) ^\ k by A8, VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A6, VALUED_0:31 ; hence (f2 * f1) /* s is divergent_to+infty by A17, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is divergent_in+infty_to+infty by A3, LIMFUNC1:def_7; ::_thesis: verum end; theorem :: LIMFUNC4:46 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in+infty & f2 is_divergent_to-infty_in lim_in+infty f1 & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (right_open_halfline r) holds f1 . g <> lim_in+infty f1 holds f2 * f1 is divergent_in+infty_to-infty proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in+infty & f2 is_divergent_to-infty_in lim_in+infty f1 & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (right_open_halfline r) holds f1 . g <> lim_in+infty f1 implies f2 * f1 is divergent_in+infty_to-infty ) assume that A1: f1 is convergent_in+infty and A2: f2 is_divergent_to-infty_in lim_in+infty f1 and A3: for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ; ::_thesis: ( for r being Real ex g being Real st ( g in (dom f1) /\ (right_open_halfline r) & not f1 . g <> lim_in+infty f1 ) or f2 * f1 is divergent_in+infty_to-infty ) given r being Real such that A4: for g being Real st g in (dom f1) /\ (right_open_halfline r) holds f1 . g <> lim_in+infty f1 ; ::_thesis: f2 * f1 is divergent_in+infty_to-infty now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to+infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (f2_*_f1)_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is divergent_to+infty & rng s c= dom (f2 * f1) implies (f2 * f1) /* s is divergent_to-infty ) assume that A5: s is divergent_to+infty and A6: rng s c= dom (f2 * f1) ; ::_thesis: (f2 * f1) /* s is divergent_to-infty consider k being Element of NAT such that A7: for n being Element of NAT st k <= n holds r < s . n by A5, LIMFUNC1:def_4; A8: rng (f1 /* s) c= dom f2 by A6, Lm2; set q = s ^\ k; A9: s ^\ k is divergent_to+infty by A5, LIMFUNC1:26; A10: rng s c= dom f1 by A6, Lm2; A11: rng (s ^\ k) c= rng s by VALUED_0:21; A12: rng (f1 /* (s ^\ k)) c= (dom f2) \ {(lim_in+infty f1)} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (f1 /* (s ^\ k)) or x in (dom f2) \ {(lim_in+infty f1)} ) assume x in rng (f1 /* (s ^\ k)) ; ::_thesis: x in (dom f2) \ {(lim_in+infty f1)} then consider n being Element of NAT such that A13: (f1 /* (s ^\ k)) . n = x by FUNCT_2:113; A14: x = f1 . ((s ^\ k) . n) by A10, A11, A13, FUNCT_2:108, XBOOLE_1:1 .= f1 . (s . (n + k)) by NAT_1:def_3 ; r < s . (n + k) by A7, NAT_1:12; then s . (n + k) in { r2 where r2 is Real : r < r2 } ; then A15: s . (n + k) in right_open_halfline r by XXREAL_1:230; s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in (dom f1) /\ (right_open_halfline r) by A10, A15, XBOOLE_0:def_4; then f1 . (s . (n + k)) <> lim_in+infty f1 by A4; then A16: not f1 . (s . (n + k)) in {(lim_in+infty f1)} by TARSKI:def_1; (f1 /* s) . (n + k) in rng (f1 /* s) by VALUED_0:28; then (f1 /* s) . (n + k) in dom f2 by A8; then x in dom f2 by A10, A14, FUNCT_2:108; hence x in (dom f2) \ {(lim_in+infty f1)} by A14, A16, XBOOLE_0:def_5; ::_thesis: verum end; rng (s ^\ k) c= dom f1 by A10, A11, XBOOLE_1:1; then ( f1 /* (s ^\ k) is convergent & lim (f1 /* (s ^\ k)) = lim_in+infty f1 ) by A1, A2, A9, LIMFUNC1:def_12; then A17: f2 /* (f1 /* (s ^\ k)) is divergent_to-infty by A2, A12, LIMFUNC3:def_3; f2 /* (f1 /* (s ^\ k)) = f2 /* ((f1 /* s) ^\ k) by A10, VALUED_0:27 .= (f2 /* (f1 /* s)) ^\ k by A8, VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A6, VALUED_0:31 ; hence (f2 * f1) /* s is divergent_to-infty by A17, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is divergent_in+infty_to-infty by A3, LIMFUNC1:def_8; ::_thesis: verum end; theorem :: LIMFUNC4:47 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in-infty & f2 is_divergent_to+infty_in lim_in-infty f1 & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (left_open_halfline r) holds f1 . g <> lim_in-infty f1 holds f2 * f1 is divergent_in-infty_to+infty proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in-infty & f2 is_divergent_to+infty_in lim_in-infty f1 & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (left_open_halfline r) holds f1 . g <> lim_in-infty f1 implies f2 * f1 is divergent_in-infty_to+infty ) assume that A1: f1 is convergent_in-infty and A2: f2 is_divergent_to+infty_in lim_in-infty f1 and A3: for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ; ::_thesis: ( for r being Real ex g being Real st ( g in (dom f1) /\ (left_open_halfline r) & not f1 . g <> lim_in-infty f1 ) or f2 * f1 is divergent_in-infty_to+infty ) given r being Real such that A4: for g being Real st g in (dom f1) /\ (left_open_halfline r) holds f1 . g <> lim_in-infty f1 ; ::_thesis: f2 * f1 is divergent_in-infty_to+infty now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to-infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (f2_*_f1)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is divergent_to-infty & rng s c= dom (f2 * f1) implies (f2 * f1) /* s is divergent_to+infty ) assume that A5: s is divergent_to-infty and A6: rng s c= dom (f2 * f1) ; ::_thesis: (f2 * f1) /* s is divergent_to+infty consider k being Element of NAT such that A7: for n being Element of NAT st k <= n holds s . n < r by A5, LIMFUNC1:def_5; A8: rng (f1 /* s) c= dom f2 by A6, Lm2; set q = s ^\ k; A9: s ^\ k is divergent_to-infty by A5, LIMFUNC1:27; A10: rng s c= dom f1 by A6, Lm2; A11: rng (s ^\ k) c= rng s by VALUED_0:21; A12: rng (f1 /* (s ^\ k)) c= (dom f2) \ {(lim_in-infty f1)} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (f1 /* (s ^\ k)) or x in (dom f2) \ {(lim_in-infty f1)} ) assume x in rng (f1 /* (s ^\ k)) ; ::_thesis: x in (dom f2) \ {(lim_in-infty f1)} then consider n being Element of NAT such that A13: (f1 /* (s ^\ k)) . n = x by FUNCT_2:113; A14: x = f1 . ((s ^\ k) . n) by A10, A11, A13, FUNCT_2:108, XBOOLE_1:1 .= f1 . (s . (n + k)) by NAT_1:def_3 ; s . (n + k) < r by A7, NAT_1:12; then s . (n + k) in { r2 where r2 is Real : r2 < r } ; then A15: s . (n + k) in left_open_halfline r by XXREAL_1:229; s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in (dom f1) /\ (left_open_halfline r) by A10, A15, XBOOLE_0:def_4; then f1 . (s . (n + k)) <> lim_in-infty f1 by A4; then A16: not f1 . (s . (n + k)) in {(lim_in-infty f1)} by TARSKI:def_1; (f1 /* s) . (n + k) in rng (f1 /* s) by VALUED_0:28; then (f1 /* s) . (n + k) in dom f2 by A8; then x in dom f2 by A10, A14, FUNCT_2:108; hence x in (dom f2) \ {(lim_in-infty f1)} by A14, A16, XBOOLE_0:def_5; ::_thesis: verum end; rng (s ^\ k) c= dom f1 by A10, A11, XBOOLE_1:1; then ( f1 /* (s ^\ k) is convergent & lim (f1 /* (s ^\ k)) = lim_in-infty f1 ) by A1, A2, A9, LIMFUNC1:def_13; then A17: f2 /* (f1 /* (s ^\ k)) is divergent_to+infty by A2, A12, LIMFUNC3:def_2; f2 /* (f1 /* (s ^\ k)) = f2 /* ((f1 /* s) ^\ k) by A10, VALUED_0:27 .= (f2 /* (f1 /* s)) ^\ k by A8, VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A6, VALUED_0:31 ; hence (f2 * f1) /* s is divergent_to+infty by A17, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is divergent_in-infty_to+infty by A3, LIMFUNC1:def_10; ::_thesis: verum end; theorem :: LIMFUNC4:48 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in-infty & f2 is_divergent_to-infty_in lim_in-infty f1 & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (left_open_halfline r) holds f1 . g <> lim_in-infty f1 holds f2 * f1 is divergent_in-infty_to-infty proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in-infty & f2 is_divergent_to-infty_in lim_in-infty f1 & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (left_open_halfline r) holds f1 . g <> lim_in-infty f1 implies f2 * f1 is divergent_in-infty_to-infty ) assume that A1: f1 is convergent_in-infty and A2: f2 is_divergent_to-infty_in lim_in-infty f1 and A3: for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ; ::_thesis: ( for r being Real ex g being Real st ( g in (dom f1) /\ (left_open_halfline r) & not f1 . g <> lim_in-infty f1 ) or f2 * f1 is divergent_in-infty_to-infty ) given r being Real such that A4: for g being Real st g in (dom f1) /\ (left_open_halfline r) holds f1 . g <> lim_in-infty f1 ; ::_thesis: f2 * f1 is divergent_in-infty_to-infty now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to-infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (f2_*_f1)_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is divergent_to-infty & rng s c= dom (f2 * f1) implies (f2 * f1) /* s is divergent_to-infty ) assume that A5: s is divergent_to-infty and A6: rng s c= dom (f2 * f1) ; ::_thesis: (f2 * f1) /* s is divergent_to-infty consider k being Element of NAT such that A7: for n being Element of NAT st k <= n holds s . n < r by A5, LIMFUNC1:def_5; A8: rng (f1 /* s) c= dom f2 by A6, Lm2; set q = s ^\ k; A9: s ^\ k is divergent_to-infty by A5, LIMFUNC1:27; A10: rng s c= dom f1 by A6, Lm2; A11: rng (s ^\ k) c= rng s by VALUED_0:21; A12: rng (f1 /* (s ^\ k)) c= (dom f2) \ {(lim_in-infty f1)} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (f1 /* (s ^\ k)) or x in (dom f2) \ {(lim_in-infty f1)} ) assume x in rng (f1 /* (s ^\ k)) ; ::_thesis: x in (dom f2) \ {(lim_in-infty f1)} then consider n being Element of NAT such that A13: (f1 /* (s ^\ k)) . n = x by FUNCT_2:113; A14: x = f1 . ((s ^\ k) . n) by A10, A11, A13, FUNCT_2:108, XBOOLE_1:1 .= f1 . (s . (n + k)) by NAT_1:def_3 ; s . (n + k) < r by A7, NAT_1:12; then s . (n + k) in { r2 where r2 is Real : r2 < r } ; then A15: s . (n + k) in left_open_halfline r by XXREAL_1:229; s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in (dom f1) /\ (left_open_halfline r) by A10, A15, XBOOLE_0:def_4; then f1 . (s . (n + k)) <> lim_in-infty f1 by A4; then A16: not f1 . (s . (n + k)) in {(lim_in-infty f1)} by TARSKI:def_1; (f1 /* s) . (n + k) in rng (f1 /* s) by VALUED_0:28; then (f1 /* s) . (n + k) in dom f2 by A8; then x in dom f2 by A10, A14, FUNCT_2:108; hence x in (dom f2) \ {(lim_in-infty f1)} by A14, A16, XBOOLE_0:def_5; ::_thesis: verum end; rng (s ^\ k) c= dom f1 by A10, A11, XBOOLE_1:1; then ( f1 /* (s ^\ k) is convergent & lim (f1 /* (s ^\ k)) = lim_in-infty f1 ) by A1, A2, A9, LIMFUNC1:def_13; then A17: f2 /* (f1 /* (s ^\ k)) is divergent_to-infty by A2, A12, LIMFUNC3:def_3; f2 /* (f1 /* (s ^\ k)) = f2 /* ((f1 /* s) ^\ k) by A10, VALUED_0:27 .= (f2 /* (f1 /* s)) ^\ k by A8, VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A6, VALUED_0:31 ; hence (f2 * f1) /* s is divergent_to-infty by A17, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is divergent_in-infty_to-infty by A3, LIMFUNC1:def_11; ::_thesis: verum end; theorem :: LIMFUNC4:49 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_left_divergent_to+infty_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r < lim (f1,x0) ) ) holds f2 * f1 is_divergent_to+infty_in x0 proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_left_divergent_to+infty_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r < lim (f1,x0) ) ) holds f2 * f1 is_divergent_to+infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_convergent_in x0 & f2 is_left_divergent_to+infty_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r < lim (f1,x0) ) ) implies f2 * f1 is_divergent_to+infty_in x0 ) assume that A1: f1 is_convergent_in x0 and A2: f2 is_left_divergent_to+infty_in lim (f1,x0) and A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) & not f1 . r < lim (f1,x0) ) ) or f2 * f1 is_divergent_to+infty_in x0 ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds lim (f1,x0) > f1 . r ; ::_thesis: f2 * f1 is_divergent_to+infty_in x0 now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f2_*_f1))_\_{x0}_holds_ (f2_*_f1)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) \ {x0} implies (f2 * f1) /* s is divergent_to+infty ) assume that A6: ( s is convergent & lim s = x0 ) and A7: rng s c= (dom (f2 * f1)) \ {x0} ; ::_thesis: (f2 * f1) /* s is divergent_to+infty consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds ( x0 - g < s . n & s . n < x0 + g ) by A4, A6, LIMFUNC3:7; set q = (f1 /* s) ^\ k; A9: rng s c= dom (f2 * f1) by A7, Th2; rng (f1 /* s) c= dom f2 by A7, Th2; then A10: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A9, VALUED_0:31 ; A11: rng s c= (dom f1) \ {x0} by A7, Th2; then A12: f1 /* s is convergent by A1, A2, A6, LIMFUNC3:def_4; A13: rng s c= dom f1 by A7, Th2; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_/\_(left_open_halfline_(lim_(f1,x0))) let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) /\ (left_open_halfline (lim (f1,x0))) ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) /\ (left_open_halfline (lim (f1,x0))) then consider n being Element of NAT such that A14: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A15: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A13, FUNCT_2:108 .= x by A14, NAT_1:def_3 ; k <= n + k by NAT_1:12; then ( x0 - g < s . (n + k) & s . (n + k) < x0 + g ) by A8; then s . (n + k) in { g1 where g1 is Real : ( x0 - g < g1 & g1 < x0 + g ) } ; then A16: s . (n + k) in ].(x0 - g),(x0 + g).[ by RCOMP_1:def_2; A17: s . (n + k) in rng s by VALUED_0:28; then not s . (n + k) in {x0} by A7, XBOOLE_0:def_5; then s . (n + k) in ].(x0 - g),(x0 + g).[ \ {x0} by A16, XBOOLE_0:def_5; then s . (n + k) in ].(x0 - g),x0.[ \/ ].x0,(x0 + g).[ by A4, LIMFUNC3:4; then s . (n + k) in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) by A13, A17, XBOOLE_0:def_4; then f1 . (s . (n + k)) < lim (f1,x0) by A5; then f1 . (s . (n + k)) in { g2 where g2 is Real : g2 < lim (f1,x0) } ; then A18: f1 . (s . (n + k)) in left_open_halfline (lim (f1,x0)) by XXREAL_1:229; f1 . (s . (n + k)) in dom f2 by A9, A17, FUNCT_1:11; hence x in (dom f2) /\ (left_open_halfline (lim (f1,x0))) by A18, A15, XBOOLE_0:def_4; ::_thesis: verum end; then A19: rng ((f1 /* s) ^\ k) c= (dom f2) /\ (left_open_halfline (lim (f1,x0))) by TARSKI:def_3; lim (f1 /* s) = lim (f1,x0) by A1, A6, A11, LIMFUNC3:def_4; then lim ((f1 /* s) ^\ k) = lim (f1,x0) by A12, SEQ_4:20; then f2 /* ((f1 /* s) ^\ k) is divergent_to+infty by A2, A12, A19, LIMFUNC2:def_2; hence (f2 * f1) /* s is divergent_to+infty by A10, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is_divergent_to+infty_in x0 by A3, LIMFUNC3:def_2; ::_thesis: verum end; theorem :: LIMFUNC4:50 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_left_divergent_to-infty_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r < lim (f1,x0) ) ) holds f2 * f1 is_divergent_to-infty_in x0 proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_left_divergent_to-infty_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r < lim (f1,x0) ) ) holds f2 * f1 is_divergent_to-infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_convergent_in x0 & f2 is_left_divergent_to-infty_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r < lim (f1,x0) ) ) implies f2 * f1 is_divergent_to-infty_in x0 ) assume that A1: f1 is_convergent_in x0 and A2: f2 is_left_divergent_to-infty_in lim (f1,x0) and A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) & not f1 . r < lim (f1,x0) ) ) or f2 * f1 is_divergent_to-infty_in x0 ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds lim (f1,x0) > f1 . r ; ::_thesis: f2 * f1 is_divergent_to-infty_in x0 now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f2_*_f1))_\_{x0}_holds_ (f2_*_f1)_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) \ {x0} implies (f2 * f1) /* s is divergent_to-infty ) assume that A6: ( s is convergent & lim s = x0 ) and A7: rng s c= (dom (f2 * f1)) \ {x0} ; ::_thesis: (f2 * f1) /* s is divergent_to-infty consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds ( x0 - g < s . n & s . n < x0 + g ) by A4, A6, LIMFUNC3:7; set q = (f1 /* s) ^\ k; A9: rng s c= dom (f2 * f1) by A7, Th2; rng (f1 /* s) c= dom f2 by A7, Th2; then A10: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A9, VALUED_0:31 ; A11: rng s c= (dom f1) \ {x0} by A7, Th2; then A12: f1 /* s is convergent by A1, A2, A6, LIMFUNC3:def_4; A13: rng s c= dom f1 by A7, Th2; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_/\_(left_open_halfline_(lim_(f1,x0))) let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) /\ (left_open_halfline (lim (f1,x0))) ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) /\ (left_open_halfline (lim (f1,x0))) then consider n being Element of NAT such that A14: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A15: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A13, FUNCT_2:108 .= x by A14, NAT_1:def_3 ; k <= n + k by NAT_1:12; then ( x0 - g < s . (n + k) & s . (n + k) < x0 + g ) by A8; then s . (n + k) in { g1 where g1 is Real : ( x0 - g < g1 & g1 < x0 + g ) } ; then A16: s . (n + k) in ].(x0 - g),(x0 + g).[ by RCOMP_1:def_2; A17: s . (n + k) in rng s by VALUED_0:28; then not s . (n + k) in {x0} by A7, XBOOLE_0:def_5; then s . (n + k) in ].(x0 - g),(x0 + g).[ \ {x0} by A16, XBOOLE_0:def_5; then s . (n + k) in ].(x0 - g),x0.[ \/ ].x0,(x0 + g).[ by A4, LIMFUNC3:4; then s . (n + k) in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) by A13, A17, XBOOLE_0:def_4; then f1 . (s . (n + k)) < lim (f1,x0) by A5; then f1 . (s . (n + k)) in { g2 where g2 is Real : g2 < lim (f1,x0) } ; then A18: f1 . (s . (n + k)) in left_open_halfline (lim (f1,x0)) by XXREAL_1:229; f1 . (s . (n + k)) in dom f2 by A9, A17, FUNCT_1:11; hence x in (dom f2) /\ (left_open_halfline (lim (f1,x0))) by A18, A15, XBOOLE_0:def_4; ::_thesis: verum end; then A19: rng ((f1 /* s) ^\ k) c= (dom f2) /\ (left_open_halfline (lim (f1,x0))) by TARSKI:def_3; lim (f1 /* s) = lim (f1,x0) by A1, A6, A11, LIMFUNC3:def_4; then lim ((f1 /* s) ^\ k) = lim (f1,x0) by A12, SEQ_4:20; then f2 /* ((f1 /* s) ^\ k) is divergent_to-infty by A2, A12, A19, LIMFUNC2:def_3; hence (f2 * f1) /* s is divergent_to-infty by A10, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is_divergent_to-infty_in x0 by A3, LIMFUNC3:def_3; ::_thesis: verum end; theorem :: LIMFUNC4:51 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_divergent_to+infty_in lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds f1 . r <> lim_left (f1,x0) ) ) holds f2 * f1 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_convergent_in x0 & f2 is_divergent_to+infty_in lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds f1 . r <> lim_left (f1,x0) ) ) holds f2 * f1 is_left_divergent_to+infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_convergent_in x0 & f2 is_divergent_to+infty_in lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds f1 . r <> lim_left (f1,x0) ) ) implies f2 * f1 is_left_divergent_to+infty_in x0 ) assume that A1: f1 is_left_convergent_in x0 and A2: f2 is_divergent_to+infty_in lim_left (f1,x0) and A3: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ ].(x0 - g),x0.[ & not f1 . r <> lim_left (f1,x0) ) ) or f2 * f1 is_left_divergent_to+infty_in x0 ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds f1 . r <> lim_left (f1,x0) ; ::_thesis: f2 * f1 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_(f2_*_f1))_/\_(left_open_halfline_x0)_holds_ (f2_*_f1)_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) implies (f2 * f1) /* s is divergent_to+infty ) assume that A6: ( s is convergent & lim s = x0 ) and A7: rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) ; ::_thesis: (f2 * f1) /* s is divergent_to+infty consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds x0 - g < s . n by A4, A6, Lm1, LIMFUNC2:1; set q = (f1 /* s) ^\ k; A9: rng s c= dom (f2 * f1) by A7, Th1; rng (f1 /* s) c= dom f2 by A7, Th1; then A10: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A9, VALUED_0:31 ; A11: rng s c= (dom f1) /\ (left_open_halfline x0) by A7, Th1; then A12: f1 /* s is convergent by A1, A2, A6, LIMFUNC2:def_7; A13: rng s c= dom f1 by A7, Th1; A14: rng s c= left_open_halfline x0 by A7, Th1; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_\_{(lim_left_(f1,x0))} let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) \ {(lim_left (f1,x0))} ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) \ {(lim_left (f1,x0))} then consider n being Element of NAT such that A15: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A16: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A13, FUNCT_2:108 .= x by A15, NAT_1:def_3 ; A17: x0 - g < s . (n + k) by A8, NAT_1:12; A18: s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in left_open_halfline x0 by A14; then s . (n + k) in { g1 where g1 is Real : g1 < x0 } by XXREAL_1:229; then ex g1 being Real st ( g1 = s . (n + k) & g1 < x0 ) ; then s . (n + k) in { g2 where g2 is Real : ( x0 - g < g2 & g2 < x0 ) } by A17; then s . (n + k) in ].(x0 - g),x0.[ by RCOMP_1:def_2; then s . (n + k) in (dom f1) /\ ].(x0 - g),x0.[ by A13, A18, XBOOLE_0:def_4; then f1 . (s . (n + k)) <> lim_left (f1,x0) by A5; then A19: not f1 . (s . (n + k)) in {(lim_left (f1,x0))} by TARSKI:def_1; f1 . (s . (n + k)) in dom f2 by A9, A18, FUNCT_1:11; hence x in (dom f2) \ {(lim_left (f1,x0))} by A19, A16, XBOOLE_0:def_5; ::_thesis: verum end; then A20: rng ((f1 /* s) ^\ k) c= (dom f2) \ {(lim_left (f1,x0))} by TARSKI:def_3; lim (f1 /* s) = lim_left (f1,x0) by A1, A6, A11, LIMFUNC2:def_7; then lim ((f1 /* s) ^\ k) = lim_left (f1,x0) by A12, SEQ_4:20; then f2 /* ((f1 /* s) ^\ k) is divergent_to+infty by A2, A12, A20, LIMFUNC3:def_2; hence (f2 * f1) /* s is divergent_to+infty by A10, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is_left_divergent_to+infty_in x0 by A3, LIMFUNC2:def_2; ::_thesis: verum end; theorem :: LIMFUNC4:52 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_divergent_to-infty_in lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds f1 . r <> lim_left (f1,x0) ) ) holds f2 * f1 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_convergent_in x0 & f2 is_divergent_to-infty_in lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds f1 . r <> lim_left (f1,x0) ) ) holds f2 * f1 is_left_divergent_to-infty_in x0 let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_convergent_in x0 & f2 is_divergent_to-infty_in lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds f1 . r <> lim_left (f1,x0) ) ) implies f2 * f1 is_left_divergent_to-infty_in x0 ) assume that A1: f1 is_left_convergent_in x0 and A2: f2 is_divergent_to-infty_in lim_left (f1,x0) and A3: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ ].(x0 - g),x0.[ & not f1 . r <> lim_left (f1,x0) ) ) or f2 * f1 is_left_divergent_to-infty_in x0 ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds f1 . r <> lim_left (f1,x0) ; ::_thesis: f2 * f1 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_(f2_*_f1))_/\_(left_open_halfline_x0)_holds_ (f2_*_f1)_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) implies (f2 * f1) /* s is divergent_to-infty ) assume that A6: ( s is convergent & lim s = x0 ) and A7: rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) ; ::_thesis: (f2 * f1) /* s is divergent_to-infty consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds x0 - g < s . n by A4, A6, Lm1, LIMFUNC2:1; set q = (f1 /* s) ^\ k; A9: rng s c= dom (f2 * f1) by A7, Th1; rng (f1 /* s) c= dom f2 by A7, Th1; then A10: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A9, VALUED_0:31 ; A11: rng s c= (dom f1) /\ (left_open_halfline x0) by A7, Th1; then A12: f1 /* s is convergent by A1, A2, A6, LIMFUNC2:def_7; A13: rng s c= dom f1 by A7, Th1; A14: rng s c= left_open_halfline x0 by A7, Th1; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_\_{(lim_left_(f1,x0))} let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) \ {(lim_left (f1,x0))} ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) \ {(lim_left (f1,x0))} then consider n being Element of NAT such that A15: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A16: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A13, FUNCT_2:108 .= x by A15, NAT_1:def_3 ; A17: x0 - g < s . (n + k) by A8, NAT_1:12; A18: s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in left_open_halfline x0 by A14; then s . (n + k) in { g1 where g1 is Real : g1 < x0 } by XXREAL_1:229; then ex g1 being Real st ( g1 = s . (n + k) & g1 < x0 ) ; then s . (n + k) in { g2 where g2 is Real : ( x0 - g < g2 & g2 < x0 ) } by A17; then s . (n + k) in ].(x0 - g),x0.[ by RCOMP_1:def_2; then s . (n + k) in (dom f1) /\ ].(x0 - g),x0.[ by A13, A18, XBOOLE_0:def_4; then f1 . (s . (n + k)) <> lim_left (f1,x0) by A5; then A19: not f1 . (s . (n + k)) in {(lim_left (f1,x0))} by TARSKI:def_1; f1 . (s . (n + k)) in dom f2 by A9, A18, FUNCT_1:11; hence x in (dom f2) \ {(lim_left (f1,x0))} by A19, A16, XBOOLE_0:def_5; ::_thesis: verum end; then A20: rng ((f1 /* s) ^\ k) c= (dom f2) \ {(lim_left (f1,x0))} by TARSKI:def_3; lim (f1 /* s) = lim_left (f1,x0) by A1, A6, A11, LIMFUNC2:def_7; then lim ((f1 /* s) ^\ k) = lim_left (f1,x0) by A12, SEQ_4:20; then f2 /* ((f1 /* s) ^\ k) is divergent_to-infty by A2, A12, A20, LIMFUNC3:def_3; hence (f2 * f1) /* s is divergent_to-infty by A10, LIMFUNC1:7; ::_thesis: verum end; hence f2 * f1 is_left_divergent_to-infty_in x0 by A3, LIMFUNC2:def_3; ::_thesis: verum end; theorem :: LIMFUNC4:53 for f1, f2 being PartFunc of REAL,REAL st f1 is divergent_in+infty_to+infty & f2 is convergent_in+infty & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) holds ( f2 * f1 is convergent_in+infty & lim_in+infty (f2 * f1) = lim_in+infty f2 ) proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in+infty_to+infty & f2 is convergent_in+infty & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) implies ( f2 * f1 is convergent_in+infty & lim_in+infty (f2 * f1) = lim_in+infty f2 ) ) assume that A1: f1 is divergent_in+infty_to+infty and A2: f2 is convergent_in+infty and A3: for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ; ::_thesis: ( f2 * f1 is convergent_in+infty & lim_in+infty (f2 * f1) = lim_in+infty f2 ) A4: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to+infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (_(f2_*_f1)_/*_s_is_convergent_&_lim_((f2_*_f1)_/*_s)_=_lim_in+infty_f2_) let s be Real_Sequence; ::_thesis: ( s is divergent_to+infty & rng s c= dom (f2 * f1) implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in+infty f2 ) ) assume that A5: s is divergent_to+infty and A6: rng s c= dom (f2 * f1) ; ::_thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in+infty f2 ) rng s c= dom f1 by A6, Lm2; then A7: f1 /* s is divergent_to+infty by A1, A5, LIMFUNC1:def_7; A8: rng (f1 /* s) c= dom f2 by A6, Lm2; then A9: lim (f2 /* (f1 /* s)) = lim_in+infty f2 by A2, A7, LIMFUNC1:def_12; lim_in+infty f2 = lim_in+infty f2 ; then f2 /* (f1 /* s) is convergent by A2, A8, A7, LIMFUNC1:def_12; hence ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in+infty f2 ) by A6, A9, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is convergent_in+infty by A3, LIMFUNC1:def_6; ::_thesis: lim_in+infty (f2 * f1) = lim_in+infty f2 hence lim_in+infty (f2 * f1) = lim_in+infty f2 by A4, LIMFUNC1:def_12; ::_thesis: verum end; theorem :: LIMFUNC4:54 for f1, f2 being PartFunc of REAL,REAL st f1 is divergent_in+infty_to-infty & f2 is convergent_in-infty & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) holds ( f2 * f1 is convergent_in+infty & lim_in+infty (f2 * f1) = lim_in-infty f2 ) proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in+infty_to-infty & f2 is convergent_in-infty & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) implies ( f2 * f1 is convergent_in+infty & lim_in+infty (f2 * f1) = lim_in-infty f2 ) ) assume that A1: f1 is divergent_in+infty_to-infty and A2: f2 is convergent_in-infty and A3: for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ; ::_thesis: ( f2 * f1 is convergent_in+infty & lim_in+infty (f2 * f1) = lim_in-infty f2 ) A4: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to+infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (_(f2_*_f1)_/*_s_is_convergent_&_lim_((f2_*_f1)_/*_s)_=_lim_in-infty_f2_) let s be Real_Sequence; ::_thesis: ( s is divergent_to+infty & rng s c= dom (f2 * f1) implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in-infty f2 ) ) assume that A5: s is divergent_to+infty and A6: rng s c= dom (f2 * f1) ; ::_thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in-infty f2 ) rng s c= dom f1 by A6, Lm2; then A7: f1 /* s is divergent_to-infty by A1, A5, LIMFUNC1:def_8; A8: rng (f1 /* s) c= dom f2 by A6, Lm2; then A9: lim (f2 /* (f1 /* s)) = lim_in-infty f2 by A2, A7, LIMFUNC1:def_13; lim_in-infty f2 = lim_in-infty f2 ; then f2 /* (f1 /* s) is convergent by A2, A8, A7, LIMFUNC1:def_13; hence ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in-infty f2 ) by A6, A9, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is convergent_in+infty by A3, LIMFUNC1:def_6; ::_thesis: lim_in+infty (f2 * f1) = lim_in-infty f2 hence lim_in+infty (f2 * f1) = lim_in-infty f2 by A4, LIMFUNC1:def_12; ::_thesis: verum end; theorem :: LIMFUNC4:55 for f1, f2 being PartFunc of REAL,REAL st f1 is divergent_in-infty_to+infty & f2 is convergent_in+infty & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) holds ( f2 * f1 is convergent_in-infty & lim_in-infty (f2 * f1) = lim_in+infty f2 ) proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in-infty_to+infty & f2 is convergent_in+infty & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) implies ( f2 * f1 is convergent_in-infty & lim_in-infty (f2 * f1) = lim_in+infty f2 ) ) assume that A1: f1 is divergent_in-infty_to+infty and A2: f2 is convergent_in+infty and A3: for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ; ::_thesis: ( f2 * f1 is convergent_in-infty & lim_in-infty (f2 * f1) = lim_in+infty f2 ) A4: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to-infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (_(f2_*_f1)_/*_s_is_convergent_&_lim_((f2_*_f1)_/*_s)_=_lim_in+infty_f2_) let s be Real_Sequence; ::_thesis: ( s is divergent_to-infty & rng s c= dom (f2 * f1) implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in+infty f2 ) ) assume that A5: s is divergent_to-infty and A6: rng s c= dom (f2 * f1) ; ::_thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in+infty f2 ) rng s c= dom f1 by A6, Lm2; then A7: f1 /* s is divergent_to+infty by A1, A5, LIMFUNC1:def_10; A8: rng (f1 /* s) c= dom f2 by A6, Lm2; then A9: lim (f2 /* (f1 /* s)) = lim_in+infty f2 by A2, A7, LIMFUNC1:def_12; lim_in+infty f2 = lim_in+infty f2 ; then f2 /* (f1 /* s) is convergent by A2, A8, A7, LIMFUNC1:def_12; hence ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in+infty f2 ) by A6, A9, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is convergent_in-infty by A3, LIMFUNC1:def_9; ::_thesis: lim_in-infty (f2 * f1) = lim_in+infty f2 hence lim_in-infty (f2 * f1) = lim_in+infty f2 by A4, LIMFUNC1:def_13; ::_thesis: verum end; theorem :: LIMFUNC4:56 for f1, f2 being PartFunc of REAL,REAL st f1 is divergent_in-infty_to-infty & f2 is convergent_in-infty & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) holds ( f2 * f1 is convergent_in-infty & lim_in-infty (f2 * f1) = lim_in-infty f2 ) proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in-infty_to-infty & f2 is convergent_in-infty & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) implies ( f2 * f1 is convergent_in-infty & lim_in-infty (f2 * f1) = lim_in-infty f2 ) ) assume that A1: f1 is divergent_in-infty_to-infty and A2: f2 is convergent_in-infty and A3: for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ; ::_thesis: ( f2 * f1 is convergent_in-infty & lim_in-infty (f2 * f1) = lim_in-infty f2 ) A4: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to-infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (_(f2_*_f1)_/*_s_is_convergent_&_lim_((f2_*_f1)_/*_s)_=_lim_in-infty_f2_) let s be Real_Sequence; ::_thesis: ( s is divergent_to-infty & rng s c= dom (f2 * f1) implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in-infty f2 ) ) assume that A5: s is divergent_to-infty and A6: rng s c= dom (f2 * f1) ; ::_thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in-infty f2 ) rng s c= dom f1 by A6, Lm2; then A7: f1 /* s is divergent_to-infty by A1, A5, LIMFUNC1:def_11; A8: rng (f1 /* s) c= dom f2 by A6, Lm2; then A9: lim (f2 /* (f1 /* s)) = lim_in-infty f2 by A2, A7, LIMFUNC1:def_13; lim_in-infty f2 = lim_in-infty f2 ; then f2 /* (f1 /* s) is convergent by A2, A8, A7, LIMFUNC1:def_13; hence ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in-infty f2 ) by A6, A9, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is convergent_in-infty by A3, LIMFUNC1:def_9; ::_thesis: lim_in-infty (f2 * f1) = lim_in-infty f2 hence lim_in-infty (f2 * f1) = lim_in-infty f2 by A4, LIMFUNC1:def_13; ::_thesis: verum end; theorem :: LIMFUNC4:57 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_left_divergent_to+infty_in x0 & f2 is convergent_in+infty & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) holds ( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim_in+infty f2 ) 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 convergent_in+infty & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) holds ( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim_in+infty f2 ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_divergent_to+infty_in x0 & f2 is convergent_in+infty & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) implies ( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim_in+infty f2 ) ) assume that A1: f1 is_left_divergent_to+infty_in x0 and A2: f2 is convergent_in+infty and A3: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ; ::_thesis: ( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim_in+infty f2 ) A4: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f2_*_f1))_/\_(left_open_halfline_x0)_holds_ (_(f2_*_f1)_/*_s_is_convergent_&_lim_((f2_*_f1)_/*_s)_=_lim_in+infty_f2_) let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in+infty f2 ) ) assume that A5: ( s is convergent & lim s = x0 ) and A6: rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) ; ::_thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in+infty f2 ) rng s c= (dom f1) /\ (left_open_halfline x0) by A6, Th1; then A7: f1 /* s is divergent_to+infty by A1, A5, LIMFUNC2:def_2; A8: rng (f1 /* s) c= dom f2 by A6, Th1; then A9: lim (f2 /* (f1 /* s)) = lim_in+infty f2 by A2, A7, LIMFUNC1:def_12; A10: rng s c= dom (f2 * f1) by A6, Th1; lim_in+infty f2 = lim_in+infty f2 ; then f2 /* (f1 /* s) is convergent by A2, A8, A7, LIMFUNC1:def_12; hence ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in+infty f2 ) by A10, A9, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is_left_convergent_in x0 by A3, LIMFUNC2:def_1; ::_thesis: lim_left ((f2 * f1),x0) = lim_in+infty f2 hence lim_left ((f2 * f1),x0) = lim_in+infty f2 by A4, LIMFUNC2:def_7; ::_thesis: verum end; theorem :: LIMFUNC4:58 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_left_divergent_to-infty_in x0 & f2 is convergent_in-infty & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) holds ( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim_in-infty f2 ) 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 convergent_in-infty & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) holds ( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim_in-infty f2 ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_divergent_to-infty_in x0 & f2 is convergent_in-infty & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) implies ( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim_in-infty f2 ) ) assume that A1: f1 is_left_divergent_to-infty_in x0 and A2: f2 is convergent_in-infty and A3: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ; ::_thesis: ( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim_in-infty f2 ) A4: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f2_*_f1))_/\_(left_open_halfline_x0)_holds_ (_(f2_*_f1)_/*_s_is_convergent_&_lim_((f2_*_f1)_/*_s)_=_lim_in-infty_f2_) let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in-infty f2 ) ) assume that A5: ( s is convergent & lim s = x0 ) and A6: rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) ; ::_thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in-infty f2 ) rng s c= (dom f1) /\ (left_open_halfline x0) by A6, Th1; then A7: f1 /* s is divergent_to-infty by A1, A5, LIMFUNC2:def_3; A8: rng (f1 /* s) c= dom f2 by A6, Th1; then A9: lim (f2 /* (f1 /* s)) = lim_in-infty f2 by A2, A7, LIMFUNC1:def_13; A10: rng s c= dom (f2 * f1) by A6, Th1; lim_in-infty f2 = lim_in-infty f2 ; then f2 /* (f1 /* s) is convergent by A2, A8, A7, LIMFUNC1:def_13; hence ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in-infty f2 ) by A10, A9, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is_left_convergent_in x0 by A3, LIMFUNC2:def_1; ::_thesis: lim_left ((f2 * f1),x0) = lim_in-infty f2 hence lim_left ((f2 * f1),x0) = lim_in-infty f2 by A4, LIMFUNC2:def_7; ::_thesis: verum end; theorem :: LIMFUNC4:59 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_right_divergent_to+infty_in x0 & f2 is convergent_in+infty & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) holds ( f2 * f1 is_right_convergent_in x0 & lim_right ((f2 * f1),x0) = lim_in+infty f2 ) 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 convergent_in+infty & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) holds ( f2 * f1 is_right_convergent_in x0 & lim_right ((f2 * f1),x0) = lim_in+infty f2 ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_right_divergent_to+infty_in x0 & f2 is convergent_in+infty & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) implies ( f2 * f1 is_right_convergent_in x0 & lim_right ((f2 * f1),x0) = lim_in+infty f2 ) ) assume that A1: f1 is_right_divergent_to+infty_in x0 and A2: f2 is convergent_in+infty and A3: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ; ::_thesis: ( f2 * f1 is_right_convergent_in x0 & lim_right ((f2 * f1),x0) = lim_in+infty f2 ) A4: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f2_*_f1))_/\_(right_open_halfline_x0)_holds_ (_(f2_*_f1)_/*_s_is_convergent_&_lim_((f2_*_f1)_/*_s)_=_lim_in+infty_f2_) let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in+infty f2 ) ) assume that A5: ( s is convergent & lim s = x0 ) and A6: rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) ; ::_thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in+infty f2 ) rng s c= (dom f1) /\ (right_open_halfline x0) by A6, Th1; then A7: f1 /* s is divergent_to+infty by A1, A5, LIMFUNC2:def_5; A8: rng (f1 /* s) c= dom f2 by A6, Th1; then A9: lim (f2 /* (f1 /* s)) = lim_in+infty f2 by A2, A7, LIMFUNC1:def_12; A10: rng s c= dom (f2 * f1) by A6, Th1; lim_in+infty f2 = lim_in+infty f2 ; then f2 /* (f1 /* s) is convergent by A2, A8, A7, LIMFUNC1:def_12; hence ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in+infty f2 ) by A10, A9, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is_right_convergent_in x0 by A3, LIMFUNC2:def_4; ::_thesis: lim_right ((f2 * f1),x0) = lim_in+infty f2 hence lim_right ((f2 * f1),x0) = lim_in+infty f2 by A4, LIMFUNC2:def_8; ::_thesis: verum end; theorem :: LIMFUNC4:60 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_right_divergent_to-infty_in x0 & f2 is convergent_in-infty & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) holds ( f2 * f1 is_right_convergent_in x0 & lim_right ((f2 * f1),x0) = lim_in-infty f2 ) 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 convergent_in-infty & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) holds ( f2 * f1 is_right_convergent_in x0 & lim_right ((f2 * f1),x0) = lim_in-infty f2 ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_right_divergent_to-infty_in x0 & f2 is convergent_in-infty & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) implies ( f2 * f1 is_right_convergent_in x0 & lim_right ((f2 * f1),x0) = lim_in-infty f2 ) ) assume that A1: f1 is_right_divergent_to-infty_in x0 and A2: f2 is convergent_in-infty and A3: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ; ::_thesis: ( f2 * f1 is_right_convergent_in x0 & lim_right ((f2 * f1),x0) = lim_in-infty f2 ) A4: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f2_*_f1))_/\_(right_open_halfline_x0)_holds_ (_(f2_*_f1)_/*_s_is_convergent_&_lim_((f2_*_f1)_/*_s)_=_lim_in-infty_f2_) let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in-infty f2 ) ) assume that A5: ( s is convergent & lim s = x0 ) and A6: rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) ; ::_thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in-infty f2 ) rng s c= (dom f1) /\ (right_open_halfline x0) by A6, Th1; then A7: f1 /* s is divergent_to-infty by A1, A5, LIMFUNC2:def_6; A8: rng (f1 /* s) c= dom f2 by A6, Th1; then A9: lim (f2 /* (f1 /* s)) = lim_in-infty f2 by A2, A7, LIMFUNC1:def_13; A10: rng s c= dom (f2 * f1) by A6, Th1; lim_in-infty f2 = lim_in-infty f2 ; then f2 /* (f1 /* s) is convergent by A2, A8, A7, LIMFUNC1:def_13; hence ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in-infty f2 ) by A10, A9, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is_right_convergent_in x0 by A3, LIMFUNC2:def_4; ::_thesis: lim_right ((f2 * f1),x0) = lim_in-infty f2 hence lim_right ((f2 * f1),x0) = lim_in-infty f2 by A4, LIMFUNC2:def_8; ::_thesis: verum end; theorem :: LIMFUNC4:61 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_convergent_in lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds f1 . r < lim_left (f1,x0) ) ) holds ( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim_left (f2,(lim_left (f1,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 lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds f1 . r < lim_left (f1,x0) ) ) holds ( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim_left (f2,(lim_left (f1,x0))) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_convergent_in x0 & f2 is_left_convergent_in lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds f1 . r < lim_left (f1,x0) ) ) implies ( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim_left (f2,(lim_left (f1,x0))) ) ) assume that A1: f1 is_left_convergent_in x0 and A2: f2 is_left_convergent_in lim_left (f1,x0) and A3: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ ].(x0 - g),x0.[ & not f1 . r < lim_left (f1,x0) ) ) or ( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim_left (f2,(lim_left (f1,x0))) ) ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds f1 . r < lim_left (f1,x0) ; ::_thesis: ( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim_left (f2,(lim_left (f1,x0))) ) A6: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f2_*_f1))_/\_(left_open_halfline_x0)_holds_ (_(f2_*_f1)_/*_s_is_convergent_&_lim_((f2_*_f1)_/*_s)_=_lim_left_(f2,(lim_left_(f1,x0)))_) let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_left (f2,(lim_left (f1,x0))) ) ) assume that A7: ( s is convergent & lim s = x0 ) and A8: rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) ; ::_thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_left (f2,(lim_left (f1,x0))) ) consider k being Element of NAT such that A9: for n being Element of NAT st k <= n holds x0 - g < s . n by A4, A7, Lm1, LIMFUNC2:1; A10: rng s c= dom f1 by A8, Th1; set q = (f1 /* s) ^\ k; A11: rng s c= dom (f2 * f1) by A8, Th1; A12: rng s c= left_open_halfline x0 by A8, Th1; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_/\_(left_open_halfline_(lim_left_(f1,x0))) let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) /\ (left_open_halfline (lim_left (f1,x0))) ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) /\ (left_open_halfline (lim_left (f1,x0))) then consider n being Element of NAT such that A13: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A14: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A10, FUNCT_2:108 .= x by A13, NAT_1:def_3 ; A15: x0 - g < s . (n + k) by A9, NAT_1:12; A16: s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in left_open_halfline x0 by A12; then s . (n + k) in { g1 where g1 is Real : g1 < x0 } by XXREAL_1:229; then ex g1 being Real st ( g1 = s . (n + k) & g1 < x0 ) ; then s . (n + k) in { g2 where g2 is Real : ( x0 - g < g2 & g2 < x0 ) } by A15; then s . (n + k) in ].(x0 - g),x0.[ by RCOMP_1:def_2; then s . (n + k) in (dom f1) /\ ].(x0 - g),x0.[ by A10, A16, XBOOLE_0:def_4; then f1 . (s . (n + k)) < lim_left (f1,x0) by A5; then f1 . (s . (n + k)) in { r1 where r1 is Real : r1 < lim_left (f1,x0) } ; then A17: f1 . (s . (n + k)) in left_open_halfline (lim_left (f1,x0)) by XXREAL_1:229; f1 . (s . (n + k)) in dom f2 by A11, A16, FUNCT_1:11; hence x in (dom f2) /\ (left_open_halfline (lim_left (f1,x0))) by A17, A14, XBOOLE_0:def_4; ::_thesis: verum end; then A18: rng ((f1 /* s) ^\ k) c= (dom f2) /\ (left_open_halfline (lim_left (f1,x0))) by TARSKI:def_3; rng (f1 /* s) c= dom f2 by A8, Th1; then A19: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A11, VALUED_0:31 ; A20: rng s c= (dom f1) /\ (left_open_halfline x0) by A8, Th1; then A21: f1 /* s is convergent by A1, A2, A7, LIMFUNC2:def_7; lim (f1 /* s) = lim_left (f1,x0) by A1, A7, A20, LIMFUNC2:def_7; then A22: lim ((f1 /* s) ^\ k) = lim_left (f1,x0) by A21, SEQ_4:20; lim_left (f2,(lim_left (f1,x0))) = lim_left (f2,(lim_left (f1,x0))) ; then A23: f2 /* ((f1 /* s) ^\ k) is convergent by A2, A21, A22, A18, LIMFUNC2:def_7; hence (f2 * f1) /* s is convergent by A19, SEQ_4:21; ::_thesis: lim ((f2 * f1) /* s) = lim_left (f2,(lim_left (f1,x0))) lim (f2 /* ((f1 /* s) ^\ k)) = lim_left (f2,(lim_left (f1,x0))) by A2, A21, A22, A18, LIMFUNC2:def_7; hence lim ((f2 * f1) /* s) = lim_left (f2,(lim_left (f1,x0))) by A23, A19, SEQ_4:22; ::_thesis: verum end; hence f2 * f1 is_left_convergent_in x0 by A3, LIMFUNC2:def_1; ::_thesis: lim_left ((f2 * f1),x0) = lim_left (f2,(lim_left (f1,x0))) hence lim_left ((f2 * f1),x0) = lim_left (f2,(lim_left (f1,x0))) by A6, LIMFUNC2:def_7; ::_thesis: verum end; theorem :: LIMFUNC4:62 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_right_convergent_in lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds lim_right (f1,x0) < f1 . r ) ) holds ( f2 * f1 is_right_convergent_in x0 & lim_right ((f2 * f1),x0) = lim_right (f2,(lim_right (f1,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 lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds lim_right (f1,x0) < f1 . r ) ) holds ( f2 * f1 is_right_convergent_in x0 & lim_right ((f2 * f1),x0) = lim_right (f2,(lim_right (f1,x0))) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_right_convergent_in x0 & f2 is_right_convergent_in lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds lim_right (f1,x0) < f1 . r ) ) implies ( f2 * f1 is_right_convergent_in x0 & lim_right ((f2 * f1),x0) = lim_right (f2,(lim_right (f1,x0))) ) ) assume that A1: f1 is_right_convergent_in x0 and A2: f2 is_right_convergent_in lim_right (f1,x0) and A3: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ ].x0,(x0 + g).[ & not lim_right (f1,x0) < f1 . r ) ) or ( f2 * f1 is_right_convergent_in x0 & lim_right ((f2 * f1),x0) = lim_right (f2,(lim_right (f1,x0))) ) ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds lim_right (f1,x0) < f1 . r ; ::_thesis: ( f2 * f1 is_right_convergent_in x0 & lim_right ((f2 * f1),x0) = lim_right (f2,(lim_right (f1,x0))) ) A6: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f2_*_f1))_/\_(right_open_halfline_x0)_holds_ (_(f2_*_f1)_/*_s_is_convergent_&_lim_((f2_*_f1)_/*_s)_=_lim_right_(f2,(lim_right_(f1,x0)))_) let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_right (f2,(lim_right (f1,x0))) ) ) assume that A7: ( s is convergent & lim s = x0 ) and A8: rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) ; ::_thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_right (f2,(lim_right (f1,x0))) ) consider k being Element of NAT such that A9: for n being Element of NAT st k <= n holds s . n < x0 + g by A4, A7, Lm1, LIMFUNC2:2; A10: rng s c= dom f1 by A8, Th1; set q = (f1 /* s) ^\ k; A11: rng s c= dom (f2 * f1) by A8, Th1; A12: rng s c= right_open_halfline x0 by A8, Th1; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_/\_(right_open_halfline_(lim_right_(f1,x0))) let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) /\ (right_open_halfline (lim_right (f1,x0))) ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) /\ (right_open_halfline (lim_right (f1,x0))) then consider n being Element of NAT such that A13: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A14: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A10, FUNCT_2:108 .= x by A13, NAT_1:def_3 ; A15: s . (n + k) < x0 + g by A9, NAT_1:12; A16: s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in right_open_halfline x0 by A12; then s . (n + k) in { g1 where g1 is Real : x0 < g1 } by XXREAL_1:230; then ex g1 being Real st ( g1 = s . (n + k) & x0 < g1 ) ; then s . (n + k) in { g2 where g2 is Real : ( x0 < g2 & g2 < x0 + g ) } by A15; then s . (n + k) in ].x0,(x0 + g).[ by RCOMP_1:def_2; then s . (n + k) in (dom f1) /\ ].x0,(x0 + g).[ by A10, A16, XBOOLE_0:def_4; then lim_right (f1,x0) < f1 . (s . (n + k)) by A5; then f1 . (s . (n + k)) in { r1 where r1 is Real : lim_right (f1,x0) < r1 } ; then A17: f1 . (s . (n + k)) in right_open_halfline (lim_right (f1,x0)) by XXREAL_1:230; f1 . (s . (n + k)) in dom f2 by A11, A16, FUNCT_1:11; hence x in (dom f2) /\ (right_open_halfline (lim_right (f1,x0))) by A17, A14, XBOOLE_0:def_4; ::_thesis: verum end; then A18: rng ((f1 /* s) ^\ k) c= (dom f2) /\ (right_open_halfline (lim_right (f1,x0))) by TARSKI:def_3; rng (f1 /* s) c= dom f2 by A8, Th1; then A19: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A11, VALUED_0:31 ; A20: rng s c= (dom f1) /\ (right_open_halfline x0) by A8, Th1; then A21: f1 /* s is convergent by A1, A2, A7, LIMFUNC2:def_8; lim (f1 /* s) = lim_right (f1,x0) by A1, A7, A20, LIMFUNC2:def_8; then A22: lim ((f1 /* s) ^\ k) = lim_right (f1,x0) by A21, SEQ_4:20; lim_right (f2,(lim_right (f1,x0))) = lim_right (f2,(lim_right (f1,x0))) ; then A23: f2 /* ((f1 /* s) ^\ k) is convergent by A2, A21, A22, A18, LIMFUNC2:def_8; hence (f2 * f1) /* s is convergent by A19, SEQ_4:21; ::_thesis: lim ((f2 * f1) /* s) = lim_right (f2,(lim_right (f1,x0))) lim (f2 /* ((f1 /* s) ^\ k)) = lim_right (f2,(lim_right (f1,x0))) by A2, A21, A22, A18, LIMFUNC2:def_8; hence lim ((f2 * f1) /* s) = lim_right (f2,(lim_right (f1,x0))) by A23, A19, SEQ_4:22; ::_thesis: verum end; hence f2 * f1 is_right_convergent_in x0 by A3, LIMFUNC2:def_4; ::_thesis: lim_right ((f2 * f1),x0) = lim_right (f2,(lim_right (f1,x0))) hence lim_right ((f2 * f1),x0) = lim_right (f2,(lim_right (f1,x0))) by A6, LIMFUNC2:def_8; ::_thesis: verum end; theorem :: LIMFUNC4:63 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_right_convergent_in lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds lim_left (f1,x0) < f1 . r ) ) holds ( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim_right (f2,(lim_left (f1,x0))) ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_right_convergent_in lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds lim_left (f1,x0) < f1 . r ) ) holds ( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim_right (f2,(lim_left (f1,x0))) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_convergent_in x0 & f2 is_right_convergent_in lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds lim_left (f1,x0) < f1 . r ) ) implies ( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim_right (f2,(lim_left (f1,x0))) ) ) assume that A1: f1 is_left_convergent_in x0 and A2: f2 is_right_convergent_in lim_left (f1,x0) and A3: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ ].(x0 - g),x0.[ & not lim_left (f1,x0) < f1 . r ) ) or ( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim_right (f2,(lim_left (f1,x0))) ) ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds lim_left (f1,x0) < f1 . r ; ::_thesis: ( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim_right (f2,(lim_left (f1,x0))) ) A6: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f2_*_f1))_/\_(left_open_halfline_x0)_holds_ (_(f2_*_f1)_/*_s_is_convergent_&_lim_((f2_*_f1)_/*_s)_=_lim_right_(f2,(lim_left_(f1,x0)))_) let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_right (f2,(lim_left (f1,x0))) ) ) assume that A7: ( s is convergent & lim s = x0 ) and A8: rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) ; ::_thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_right (f2,(lim_left (f1,x0))) ) consider k being Element of NAT such that A9: for n being Element of NAT st k <= n holds x0 - g < s . n by A4, A7, Lm1, LIMFUNC2:1; A10: rng s c= dom f1 by A8, Th1; set q = (f1 /* s) ^\ k; A11: rng s c= dom (f2 * f1) by A8, Th1; A12: rng s c= left_open_halfline x0 by A8, Th1; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_/\_(right_open_halfline_(lim_left_(f1,x0))) let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) /\ (right_open_halfline (lim_left (f1,x0))) ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) /\ (right_open_halfline (lim_left (f1,x0))) then consider n being Element of NAT such that A13: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A14: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A10, FUNCT_2:108 .= x by A13, NAT_1:def_3 ; A15: x0 - g < s . (n + k) by A9, NAT_1:12; A16: s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in left_open_halfline x0 by A12; then s . (n + k) in { g1 where g1 is Real : g1 < x0 } by XXREAL_1:229; then ex g1 being Real st ( g1 = s . (n + k) & g1 < x0 ) ; then s . (n + k) in { g2 where g2 is Real : ( x0 - g < g2 & g2 < x0 ) } by A15; then s . (n + k) in ].(x0 - g),x0.[ by RCOMP_1:def_2; then s . (n + k) in (dom f1) /\ ].(x0 - g),x0.[ by A10, A16, XBOOLE_0:def_4; then lim_left (f1,x0) < f1 . (s . (n + k)) by A5; then f1 . (s . (n + k)) in { r1 where r1 is Real : lim_left (f1,x0) < r1 } ; then A17: f1 . (s . (n + k)) in right_open_halfline (lim_left (f1,x0)) by XXREAL_1:230; f1 . (s . (n + k)) in dom f2 by A11, A16, FUNCT_1:11; hence x in (dom f2) /\ (right_open_halfline (lim_left (f1,x0))) by A17, A14, XBOOLE_0:def_4; ::_thesis: verum end; then A18: rng ((f1 /* s) ^\ k) c= (dom f2) /\ (right_open_halfline (lim_left (f1,x0))) by TARSKI:def_3; rng (f1 /* s) c= dom f2 by A8, Th1; then A19: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A11, VALUED_0:31 ; A20: rng s c= (dom f1) /\ (left_open_halfline x0) by A8, Th1; then A21: f1 /* s is convergent by A1, A2, A7, LIMFUNC2:def_7; lim (f1 /* s) = lim_left (f1,x0) by A1, A7, A20, LIMFUNC2:def_7; then A22: lim ((f1 /* s) ^\ k) = lim_left (f1,x0) by A21, SEQ_4:20; lim_right (f2,(lim_left (f1,x0))) = lim_right (f2,(lim_left (f1,x0))) ; then A23: f2 /* ((f1 /* s) ^\ k) is convergent by A2, A21, A22, A18, LIMFUNC2:def_8; hence (f2 * f1) /* s is convergent by A19, SEQ_4:21; ::_thesis: lim ((f2 * f1) /* s) = lim_right (f2,(lim_left (f1,x0))) lim (f2 /* ((f1 /* s) ^\ k)) = lim_right (f2,(lim_left (f1,x0))) by A2, A21, A22, A18, LIMFUNC2:def_8; hence lim ((f2 * f1) /* s) = lim_right (f2,(lim_left (f1,x0))) by A23, A19, SEQ_4:22; ::_thesis: verum end; hence f2 * f1 is_left_convergent_in x0 by A3, LIMFUNC2:def_1; ::_thesis: lim_left ((f2 * f1),x0) = lim_right (f2,(lim_left (f1,x0))) hence lim_left ((f2 * f1),x0) = lim_right (f2,(lim_left (f1,x0))) by A6, LIMFUNC2:def_7; ::_thesis: verum end; theorem :: LIMFUNC4:64 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_left_convergent_in lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds f1 . r < lim_right (f1,x0) ) ) holds ( f2 * f1 is_right_convergent_in x0 & lim_right ((f2 * f1),x0) = lim_left (f2,(lim_right (f1,x0))) ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_left_convergent_in lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds f1 . r < lim_right (f1,x0) ) ) holds ( f2 * f1 is_right_convergent_in x0 & lim_right ((f2 * f1),x0) = lim_left (f2,(lim_right (f1,x0))) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_right_convergent_in x0 & f2 is_left_convergent_in lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds f1 . r < lim_right (f1,x0) ) ) implies ( f2 * f1 is_right_convergent_in x0 & lim_right ((f2 * f1),x0) = lim_left (f2,(lim_right (f1,x0))) ) ) assume that A1: f1 is_right_convergent_in x0 and A2: f2 is_left_convergent_in lim_right (f1,x0) and A3: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ ].x0,(x0 + g).[ & not f1 . r < lim_right (f1,x0) ) ) or ( f2 * f1 is_right_convergent_in x0 & lim_right ((f2 * f1),x0) = lim_left (f2,(lim_right (f1,x0))) ) ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds f1 . r < lim_right (f1,x0) ; ::_thesis: ( f2 * f1 is_right_convergent_in x0 & lim_right ((f2 * f1),x0) = lim_left (f2,(lim_right (f1,x0))) ) A6: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f2_*_f1))_/\_(right_open_halfline_x0)_holds_ (_(f2_*_f1)_/*_s_is_convergent_&_lim_((f2_*_f1)_/*_s)_=_lim_left_(f2,(lim_right_(f1,x0)))_) let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_left (f2,(lim_right (f1,x0))) ) ) assume that A7: ( s is convergent & lim s = x0 ) and A8: rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) ; ::_thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_left (f2,(lim_right (f1,x0))) ) consider k being Element of NAT such that A9: for n being Element of NAT st k <= n holds s . n < x0 + g by A4, A7, Lm1, LIMFUNC2:2; A10: rng s c= dom f1 by A8, Th1; set q = (f1 /* s) ^\ k; A11: rng s c= dom (f2 * f1) by A8, Th1; A12: rng s c= right_open_halfline x0 by A8, Th1; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_/\_(left_open_halfline_(lim_right_(f1,x0))) let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) /\ (left_open_halfline (lim_right (f1,x0))) ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) /\ (left_open_halfline (lim_right (f1,x0))) then consider n being Element of NAT such that A13: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A14: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A10, FUNCT_2:108 .= x by A13, NAT_1:def_3 ; A15: s . (n + k) < x0 + g by A9, NAT_1:12; A16: s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in right_open_halfline x0 by A12; then s . (n + k) in { g1 where g1 is Real : x0 < g1 } by XXREAL_1:230; then ex g1 being Real st ( g1 = s . (n + k) & x0 < g1 ) ; then s . (n + k) in { g2 where g2 is Real : ( x0 < g2 & g2 < x0 + g ) } by A15; then s . (n + k) in ].x0,(x0 + g).[ by RCOMP_1:def_2; then s . (n + k) in (dom f1) /\ ].x0,(x0 + g).[ by A10, A16, XBOOLE_0:def_4; then f1 . (s . (n + k)) < lim_right (f1,x0) by A5; then f1 . (s . (n + k)) in { r1 where r1 is Real : r1 < lim_right (f1,x0) } ; then A17: f1 . (s . (n + k)) in left_open_halfline (lim_right (f1,x0)) by XXREAL_1:229; f1 . (s . (n + k)) in dom f2 by A11, A16, FUNCT_1:11; hence x in (dom f2) /\ (left_open_halfline (lim_right (f1,x0))) by A17, A14, XBOOLE_0:def_4; ::_thesis: verum end; then A18: rng ((f1 /* s) ^\ k) c= (dom f2) /\ (left_open_halfline (lim_right (f1,x0))) by TARSKI:def_3; rng (f1 /* s) c= dom f2 by A8, Th1; then A19: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A11, VALUED_0:31 ; A20: rng s c= (dom f1) /\ (right_open_halfline x0) by A8, Th1; then A21: f1 /* s is convergent by A1, A2, A7, LIMFUNC2:def_8; lim (f1 /* s) = lim_right (f1,x0) by A1, A7, A20, LIMFUNC2:def_8; then A22: lim ((f1 /* s) ^\ k) = lim_right (f1,x0) by A21, SEQ_4:20; lim_left (f2,(lim_right (f1,x0))) = lim_left (f2,(lim_right (f1,x0))) ; then A23: f2 /* ((f1 /* s) ^\ k) is convergent by A2, A21, A22, A18, LIMFUNC2:def_7; hence (f2 * f1) /* s is convergent by A19, SEQ_4:21; ::_thesis: lim ((f2 * f1) /* s) = lim_left (f2,(lim_right (f1,x0))) lim (f2 /* ((f1 /* s) ^\ k)) = lim_left (f2,(lim_right (f1,x0))) by A2, A21, A22, A18, LIMFUNC2:def_7; hence lim ((f2 * f1) /* s) = lim_left (f2,(lim_right (f1,x0))) by A23, A19, SEQ_4:22; ::_thesis: verum end; hence f2 * f1 is_right_convergent_in x0 by A3, LIMFUNC2:def_4; ::_thesis: lim_right ((f2 * f1),x0) = lim_left (f2,(lim_right (f1,x0))) hence lim_right ((f2 * f1),x0) = lim_left (f2,(lim_right (f1,x0))) by A6, LIMFUNC2:def_8; ::_thesis: verum end; theorem :: LIMFUNC4:65 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in+infty & f2 is_left_convergent_in lim_in+infty f1 & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (right_open_halfline r) holds f1 . g < lim_in+infty f1 holds ( f2 * f1 is convergent_in+infty & lim_in+infty (f2 * f1) = lim_left (f2,(lim_in+infty f1)) ) proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in+infty & f2 is_left_convergent_in lim_in+infty f1 & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (right_open_halfline r) holds f1 . g < lim_in+infty f1 implies ( f2 * f1 is convergent_in+infty & lim_in+infty (f2 * f1) = lim_left (f2,(lim_in+infty f1)) ) ) assume that A1: f1 is convergent_in+infty and A2: f2 is_left_convergent_in lim_in+infty f1 and A3: for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ; ::_thesis: ( for r being Real ex g being Real st ( g in (dom f1) /\ (right_open_halfline r) & not f1 . g < lim_in+infty f1 ) or ( f2 * f1 is convergent_in+infty & lim_in+infty (f2 * f1) = lim_left (f2,(lim_in+infty f1)) ) ) given r being Real such that A4: for g being Real st g in (dom f1) /\ (right_open_halfline r) holds f1 . g < lim_in+infty f1 ; ::_thesis: ( f2 * f1 is convergent_in+infty & lim_in+infty (f2 * f1) = lim_left (f2,(lim_in+infty f1)) ) A5: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to+infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (_(f2_*_f1)_/*_s_is_convergent_&_lim_((f2_*_f1)_/*_s)_=_lim_left_(f2,(lim_in+infty_f1))_) set L = lim_left (f2,(lim_in+infty f1)); let s be Real_Sequence; ::_thesis: ( s is divergent_to+infty & rng s c= dom (f2 * f1) implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_left (f2,(lim_in+infty f1)) ) ) assume that A6: s is divergent_to+infty and A7: rng s c= dom (f2 * f1) ; ::_thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_left (f2,(lim_in+infty f1)) ) consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds r < s . n by A6, LIMFUNC1:def_4; set q = s ^\ k; A9: s ^\ k is divergent_to+infty by A6, LIMFUNC1:26; A10: rng s c= dom f1 by A7, Lm2; A11: rng (s ^\ k) c= rng s by VALUED_0:21; then rng (s ^\ k) c= dom f1 by A10, XBOOLE_1:1; then A12: ( f1 /* (s ^\ k) is convergent & lim (f1 /* (s ^\ k)) = lim_in+infty f1 ) by A1, A2, A9, LIMFUNC1:def_12; A13: rng (f1 /* s) c= dom f2 by A7, Lm2; A14: rng (f1 /* (s ^\ k)) c= (dom f2) /\ (left_open_halfline (lim_in+infty f1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (f1 /* (s ^\ k)) or x in (dom f2) /\ (left_open_halfline (lim_in+infty f1)) ) assume x in rng (f1 /* (s ^\ k)) ; ::_thesis: x in (dom f2) /\ (left_open_halfline (lim_in+infty f1)) then consider n being Element of NAT such that A15: (f1 /* (s ^\ k)) . n = x by FUNCT_2:113; A16: x = f1 . ((s ^\ k) . n) by A10, A11, A15, FUNCT_2:108, XBOOLE_1:1 .= f1 . (s . (n + k)) by NAT_1:def_3 ; r < s . (n + k) by A8, NAT_1:12; then s . (n + k) in { r2 where r2 is Real : r < r2 } ; then A17: s . (n + k) in right_open_halfline r by XXREAL_1:230; s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in (dom f1) /\ (right_open_halfline r) by A10, A17, XBOOLE_0:def_4; then f1 . (s . (n + k)) < lim_in+infty f1 by A4; then x in { g1 where g1 is Real : g1 < lim_in+infty f1 } by A16; then A18: x in left_open_halfline (lim_in+infty f1) by XXREAL_1:229; (f1 /* s) . (n + k) in rng (f1 /* s) by VALUED_0:28; then (f1 /* s) . (n + k) in dom f2 by A13; then x in dom f2 by A10, A16, FUNCT_2:108; hence x in (dom f2) /\ (left_open_halfline (lim_in+infty f1)) by A18, XBOOLE_0:def_4; ::_thesis: verum end; A19: f2 /* (f1 /* (s ^\ k)) = f2 /* ((f1 /* s) ^\ k) by A10, VALUED_0:27 .= (f2 /* (f1 /* s)) ^\ k by A13, VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A7, VALUED_0:31 ; lim_left (f2,(lim_in+infty f1)) = lim_left (f2,(lim_in+infty f1)) ; then A20: f2 /* (f1 /* (s ^\ k)) is convergent by A2, A12, A14, LIMFUNC2:def_7; hence (f2 * f1) /* s is convergent by A19, SEQ_4:21; ::_thesis: lim ((f2 * f1) /* s) = lim_left (f2,(lim_in+infty f1)) lim (f2 /* (f1 /* (s ^\ k))) = lim_left (f2,(lim_in+infty f1)) by A2, A12, A14, LIMFUNC2:def_7; hence lim ((f2 * f1) /* s) = lim_left (f2,(lim_in+infty f1)) by A20, A19, SEQ_4:22; ::_thesis: verum end; hence f2 * f1 is convergent_in+infty by A3, LIMFUNC1:def_6; ::_thesis: lim_in+infty (f2 * f1) = lim_left (f2,(lim_in+infty f1)) hence lim_in+infty (f2 * f1) = lim_left (f2,(lim_in+infty f1)) by A5, LIMFUNC1:def_12; ::_thesis: verum end; theorem :: LIMFUNC4:66 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in+infty & f2 is_right_convergent_in lim_in+infty f1 & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (right_open_halfline r) holds lim_in+infty f1 < f1 . g holds ( f2 * f1 is convergent_in+infty & lim_in+infty (f2 * f1) = lim_right (f2,(lim_in+infty f1)) ) proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in+infty & f2 is_right_convergent_in lim_in+infty f1 & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (right_open_halfline r) holds lim_in+infty f1 < f1 . g implies ( f2 * f1 is convergent_in+infty & lim_in+infty (f2 * f1) = lim_right (f2,(lim_in+infty f1)) ) ) assume that A1: f1 is convergent_in+infty and A2: f2 is_right_convergent_in lim_in+infty f1 and A3: for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ; ::_thesis: ( for r being Real ex g being Real st ( g in (dom f1) /\ (right_open_halfline r) & not lim_in+infty f1 < f1 . g ) or ( f2 * f1 is convergent_in+infty & lim_in+infty (f2 * f1) = lim_right (f2,(lim_in+infty f1)) ) ) given r being Real such that A4: for g being Real st g in (dom f1) /\ (right_open_halfline r) holds lim_in+infty f1 < f1 . g ; ::_thesis: ( f2 * f1 is convergent_in+infty & lim_in+infty (f2 * f1) = lim_right (f2,(lim_in+infty f1)) ) A5: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to+infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (_(f2_*_f1)_/*_s_is_convergent_&_lim_((f2_*_f1)_/*_s)_=_lim_right_(f2,(lim_in+infty_f1))_) set L = lim_right (f2,(lim_in+infty f1)); let s be Real_Sequence; ::_thesis: ( s is divergent_to+infty & rng s c= dom (f2 * f1) implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_right (f2,(lim_in+infty f1)) ) ) assume that A6: s is divergent_to+infty and A7: rng s c= dom (f2 * f1) ; ::_thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_right (f2,(lim_in+infty f1)) ) consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds r < s . n by A6, LIMFUNC1:def_4; set q = s ^\ k; A9: s ^\ k is divergent_to+infty by A6, LIMFUNC1:26; A10: rng s c= dom f1 by A7, Lm2; A11: rng (s ^\ k) c= rng s by VALUED_0:21; then rng (s ^\ k) c= dom f1 by A10, XBOOLE_1:1; then A12: ( f1 /* (s ^\ k) is convergent & lim (f1 /* (s ^\ k)) = lim_in+infty f1 ) by A1, A2, A9, LIMFUNC1:def_12; A13: rng (f1 /* s) c= dom f2 by A7, Lm2; A14: rng (f1 /* (s ^\ k)) c= (dom f2) /\ (right_open_halfline (lim_in+infty f1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (f1 /* (s ^\ k)) or x in (dom f2) /\ (right_open_halfline (lim_in+infty f1)) ) assume x in rng (f1 /* (s ^\ k)) ; ::_thesis: x in (dom f2) /\ (right_open_halfline (lim_in+infty f1)) then consider n being Element of NAT such that A15: (f1 /* (s ^\ k)) . n = x by FUNCT_2:113; A16: x = f1 . ((s ^\ k) . n) by A10, A11, A15, FUNCT_2:108, XBOOLE_1:1 .= f1 . (s . (n + k)) by NAT_1:def_3 ; r < s . (n + k) by A8, NAT_1:12; then s . (n + k) in { r2 where r2 is Real : r < r2 } ; then A17: s . (n + k) in right_open_halfline r by XXREAL_1:230; s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in (dom f1) /\ (right_open_halfline r) by A10, A17, XBOOLE_0:def_4; then lim_in+infty f1 < f1 . (s . (n + k)) by A4; then x in { g1 where g1 is Real : lim_in+infty f1 < g1 } by A16; then A18: x in right_open_halfline (lim_in+infty f1) by XXREAL_1:230; (f1 /* s) . (n + k) in rng (f1 /* s) by VALUED_0:28; then (f1 /* s) . (n + k) in dom f2 by A13; then x in dom f2 by A10, A16, FUNCT_2:108; hence x in (dom f2) /\ (right_open_halfline (lim_in+infty f1)) by A18, XBOOLE_0:def_4; ::_thesis: verum end; A19: f2 /* (f1 /* (s ^\ k)) = f2 /* ((f1 /* s) ^\ k) by A10, VALUED_0:27 .= (f2 /* (f1 /* s)) ^\ k by A13, VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A7, VALUED_0:31 ; lim_right (f2,(lim_in+infty f1)) = lim_right (f2,(lim_in+infty f1)) ; then A20: f2 /* (f1 /* (s ^\ k)) is convergent by A2, A12, A14, LIMFUNC2:def_8; hence (f2 * f1) /* s is convergent by A19, SEQ_4:21; ::_thesis: lim ((f2 * f1) /* s) = lim_right (f2,(lim_in+infty f1)) lim (f2 /* (f1 /* (s ^\ k))) = lim_right (f2,(lim_in+infty f1)) by A2, A12, A14, LIMFUNC2:def_8; hence lim ((f2 * f1) /* s) = lim_right (f2,(lim_in+infty f1)) by A20, A19, SEQ_4:22; ::_thesis: verum end; hence f2 * f1 is convergent_in+infty by A3, LIMFUNC1:def_6; ::_thesis: lim_in+infty (f2 * f1) = lim_right (f2,(lim_in+infty f1)) hence lim_in+infty (f2 * f1) = lim_right (f2,(lim_in+infty f1)) by A5, LIMFUNC1:def_12; ::_thesis: verum end; theorem :: LIMFUNC4:67 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in-infty & f2 is_left_convergent_in lim_in-infty f1 & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (left_open_halfline r) holds f1 . g < lim_in-infty f1 holds ( f2 * f1 is convergent_in-infty & lim_in-infty (f2 * f1) = lim_left (f2,(lim_in-infty f1)) ) proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in-infty & f2 is_left_convergent_in lim_in-infty f1 & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (left_open_halfline r) holds f1 . g < lim_in-infty f1 implies ( f2 * f1 is convergent_in-infty & lim_in-infty (f2 * f1) = lim_left (f2,(lim_in-infty f1)) ) ) assume that A1: f1 is convergent_in-infty and A2: f2 is_left_convergent_in lim_in-infty f1 and A3: for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ; ::_thesis: ( for r being Real ex g being Real st ( g in (dom f1) /\ (left_open_halfline r) & not f1 . g < lim_in-infty f1 ) or ( f2 * f1 is convergent_in-infty & lim_in-infty (f2 * f1) = lim_left (f2,(lim_in-infty f1)) ) ) given r being Real such that A4: for g being Real st g in (dom f1) /\ (left_open_halfline r) holds f1 . g < lim_in-infty f1 ; ::_thesis: ( f2 * f1 is convergent_in-infty & lim_in-infty (f2 * f1) = lim_left (f2,(lim_in-infty f1)) ) A5: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to-infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (_(f2_*_f1)_/*_s_is_convergent_&_lim_((f2_*_f1)_/*_s)_=_lim_left_(f2,(lim_in-infty_f1))_) set L = lim_left (f2,(lim_in-infty f1)); let s be Real_Sequence; ::_thesis: ( s is divergent_to-infty & rng s c= dom (f2 * f1) implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_left (f2,(lim_in-infty f1)) ) ) assume that A6: s is divergent_to-infty and A7: rng s c= dom (f2 * f1) ; ::_thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_left (f2,(lim_in-infty f1)) ) consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds s . n < r by A6, LIMFUNC1:def_5; set q = s ^\ k; A9: s ^\ k is divergent_to-infty by A6, LIMFUNC1:27; A10: rng s c= dom f1 by A7, Lm2; A11: rng (s ^\ k) c= rng s by VALUED_0:21; then rng (s ^\ k) c= dom f1 by A10, XBOOLE_1:1; then A12: ( f1 /* (s ^\ k) is convergent & lim (f1 /* (s ^\ k)) = lim_in-infty f1 ) by A1, A2, A9, LIMFUNC1:def_13; A13: rng (f1 /* s) c= dom f2 by A7, Lm2; A14: rng (f1 /* (s ^\ k)) c= (dom f2) /\ (left_open_halfline (lim_in-infty f1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (f1 /* (s ^\ k)) or x in (dom f2) /\ (left_open_halfline (lim_in-infty f1)) ) assume x in rng (f1 /* (s ^\ k)) ; ::_thesis: x in (dom f2) /\ (left_open_halfline (lim_in-infty f1)) then consider n being Element of NAT such that A15: (f1 /* (s ^\ k)) . n = x by FUNCT_2:113; A16: x = f1 . ((s ^\ k) . n) by A10, A11, A15, FUNCT_2:108, XBOOLE_1:1 .= f1 . (s . (n + k)) by NAT_1:def_3 ; s . (n + k) < r by A8, NAT_1:12; then s . (n + k) in { r2 where r2 is Real : r2 < r } ; then A17: s . (n + k) in left_open_halfline r by XXREAL_1:229; s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in (dom f1) /\ (left_open_halfline r) by A10, A17, XBOOLE_0:def_4; then f1 . (s . (n + k)) < lim_in-infty f1 by A4; then x in { g1 where g1 is Real : g1 < lim_in-infty f1 } by A16; then A18: x in left_open_halfline (lim_in-infty f1) by XXREAL_1:229; (f1 /* s) . (n + k) in rng (f1 /* s) by VALUED_0:28; then (f1 /* s) . (n + k) in dom f2 by A13; then x in dom f2 by A10, A16, FUNCT_2:108; hence x in (dom f2) /\ (left_open_halfline (lim_in-infty f1)) by A18, XBOOLE_0:def_4; ::_thesis: verum end; A19: f2 /* (f1 /* (s ^\ k)) = f2 /* ((f1 /* s) ^\ k) by A10, VALUED_0:27 .= (f2 /* (f1 /* s)) ^\ k by A13, VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A7, VALUED_0:31 ; lim_left (f2,(lim_in-infty f1)) = lim_left (f2,(lim_in-infty f1)) ; then A20: f2 /* (f1 /* (s ^\ k)) is convergent by A2, A12, A14, LIMFUNC2:def_7; hence (f2 * f1) /* s is convergent by A19, SEQ_4:21; ::_thesis: lim ((f2 * f1) /* s) = lim_left (f2,(lim_in-infty f1)) lim (f2 /* (f1 /* (s ^\ k))) = lim_left (f2,(lim_in-infty f1)) by A2, A12, A14, LIMFUNC2:def_7; hence lim ((f2 * f1) /* s) = lim_left (f2,(lim_in-infty f1)) by A20, A19, SEQ_4:22; ::_thesis: verum end; hence f2 * f1 is convergent_in-infty by A3, LIMFUNC1:def_9; ::_thesis: lim_in-infty (f2 * f1) = lim_left (f2,(lim_in-infty f1)) hence lim_in-infty (f2 * f1) = lim_left (f2,(lim_in-infty f1)) by A5, LIMFUNC1:def_13; ::_thesis: verum end; theorem :: LIMFUNC4:68 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in-infty & f2 is_right_convergent_in lim_in-infty f1 & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (left_open_halfline r) holds lim_in-infty f1 < f1 . g holds ( f2 * f1 is convergent_in-infty & lim_in-infty (f2 * f1) = lim_right (f2,(lim_in-infty f1)) ) proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in-infty & f2 is_right_convergent_in lim_in-infty f1 & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (left_open_halfline r) holds lim_in-infty f1 < f1 . g implies ( f2 * f1 is convergent_in-infty & lim_in-infty (f2 * f1) = lim_right (f2,(lim_in-infty f1)) ) ) assume that A1: f1 is convergent_in-infty and A2: f2 is_right_convergent_in lim_in-infty f1 and A3: for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ; ::_thesis: ( for r being Real ex g being Real st ( g in (dom f1) /\ (left_open_halfline r) & not lim_in-infty f1 < f1 . g ) or ( f2 * f1 is convergent_in-infty & lim_in-infty (f2 * f1) = lim_right (f2,(lim_in-infty f1)) ) ) given r being Real such that A4: for g being Real st g in (dom f1) /\ (left_open_halfline r) holds lim_in-infty f1 < f1 . g ; ::_thesis: ( f2 * f1 is convergent_in-infty & lim_in-infty (f2 * f1) = lim_right (f2,(lim_in-infty f1)) ) A5: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to-infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (_(f2_*_f1)_/*_s_is_convergent_&_lim_((f2_*_f1)_/*_s)_=_lim_right_(f2,(lim_in-infty_f1))_) set L = lim_right (f2,(lim_in-infty f1)); let s be Real_Sequence; ::_thesis: ( s is divergent_to-infty & rng s c= dom (f2 * f1) implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_right (f2,(lim_in-infty f1)) ) ) assume that A6: s is divergent_to-infty and A7: rng s c= dom (f2 * f1) ; ::_thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_right (f2,(lim_in-infty f1)) ) consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds s . n < r by A6, LIMFUNC1:def_5; set q = s ^\ k; A9: s ^\ k is divergent_to-infty by A6, LIMFUNC1:27; A10: rng s c= dom f1 by A7, Lm2; A11: rng (s ^\ k) c= rng s by VALUED_0:21; then rng (s ^\ k) c= dom f1 by A10, XBOOLE_1:1; then A12: ( f1 /* (s ^\ k) is convergent & lim (f1 /* (s ^\ k)) = lim_in-infty f1 ) by A1, A2, A9, LIMFUNC1:def_13; A13: rng (f1 /* s) c= dom f2 by A7, Lm2; A14: rng (f1 /* (s ^\ k)) c= (dom f2) /\ (right_open_halfline (lim_in-infty f1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (f1 /* (s ^\ k)) or x in (dom f2) /\ (right_open_halfline (lim_in-infty f1)) ) assume x in rng (f1 /* (s ^\ k)) ; ::_thesis: x in (dom f2) /\ (right_open_halfline (lim_in-infty f1)) then consider n being Element of NAT such that A15: (f1 /* (s ^\ k)) . n = x by FUNCT_2:113; A16: x = f1 . ((s ^\ k) . n) by A10, A11, A15, FUNCT_2:108, XBOOLE_1:1 .= f1 . (s . (n + k)) by NAT_1:def_3 ; s . (n + k) < r by A8, NAT_1:12; then s . (n + k) in { r2 where r2 is Real : r2 < r } ; then A17: s . (n + k) in left_open_halfline r by XXREAL_1:229; s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in (dom f1) /\ (left_open_halfline r) by A10, A17, XBOOLE_0:def_4; then lim_in-infty f1 < f1 . (s . (n + k)) by A4; then x in { g1 where g1 is Real : lim_in-infty f1 < g1 } by A16; then A18: x in right_open_halfline (lim_in-infty f1) by XXREAL_1:230; (f1 /* s) . (n + k) in rng (f1 /* s) by VALUED_0:28; then (f1 /* s) . (n + k) in dom f2 by A13; then x in dom f2 by A10, A16, FUNCT_2:108; hence x in (dom f2) /\ (right_open_halfline (lim_in-infty f1)) by A18, XBOOLE_0:def_4; ::_thesis: verum end; A19: f2 /* (f1 /* (s ^\ k)) = f2 /* ((f1 /* s) ^\ k) by A10, VALUED_0:27 .= (f2 /* (f1 /* s)) ^\ k by A13, VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A7, VALUED_0:31 ; lim_right (f2,(lim_in-infty f1)) = lim_right (f2,(lim_in-infty f1)) ; then A20: f2 /* (f1 /* (s ^\ k)) is convergent by A2, A12, A14, LIMFUNC2:def_8; hence (f2 * f1) /* s is convergent by A19, SEQ_4:21; ::_thesis: lim ((f2 * f1) /* s) = lim_right (f2,(lim_in-infty f1)) lim (f2 /* (f1 /* (s ^\ k))) = lim_right (f2,(lim_in-infty f1)) by A2, A12, A14, LIMFUNC2:def_8; hence lim ((f2 * f1) /* s) = lim_right (f2,(lim_in-infty f1)) by A20, A19, SEQ_4:22; ::_thesis: verum end; hence f2 * f1 is convergent_in-infty by A3, LIMFUNC1:def_9; ::_thesis: lim_in-infty (f2 * f1) = lim_right (f2,(lim_in-infty f1)) hence lim_in-infty (f2 * f1) = lim_right (f2,(lim_in-infty f1)) by A5, LIMFUNC1:def_13; ::_thesis: verum end; theorem :: LIMFUNC4:69 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_divergent_to+infty_in x0 & f2 is convergent_in+infty & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) holds ( f2 * f1 is_convergent_in x0 & lim ((f2 * f1),x0) = lim_in+infty f2 ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_divergent_to+infty_in x0 & f2 is convergent_in+infty & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) holds ( f2 * f1 is_convergent_in x0 & lim ((f2 * f1),x0) = lim_in+infty f2 ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_divergent_to+infty_in x0 & f2 is convergent_in+infty & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) implies ( f2 * f1 is_convergent_in x0 & lim ((f2 * f1),x0) = lim_in+infty f2 ) ) assume that A1: f1 is_divergent_to+infty_in x0 and A2: f2 is convergent_in+infty and A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ; ::_thesis: ( f2 * f1 is_convergent_in x0 & lim ((f2 * f1),x0) = lim_in+infty f2 ) A4: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f2_*_f1))_\_{x0}_holds_ (_(f2_*_f1)_/*_s_is_convergent_&_lim_((f2_*_f1)_/*_s)_=_lim_in+infty_f2_) let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) \ {x0} implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in+infty f2 ) ) assume that A5: ( s is convergent & lim s = x0 ) and A6: rng s c= (dom (f2 * f1)) \ {x0} ; ::_thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in+infty f2 ) rng s c= (dom f1) \ {x0} by A6, Th2; then A7: f1 /* s is divergent_to+infty by A1, A5, LIMFUNC3:def_2; A8: rng (f1 /* s) c= dom f2 by A6, Th2; then A9: lim (f2 /* (f1 /* s)) = lim_in+infty f2 by A2, A7, LIMFUNC1:def_12; A10: rng s c= dom (f2 * f1) by A6, Th2; lim_in+infty f2 = lim_in+infty f2 ; then f2 /* (f1 /* s) is convergent by A2, A8, A7, LIMFUNC1:def_12; hence ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in+infty f2 ) by A10, A9, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is_convergent_in x0 by A3, LIMFUNC3:def_1; ::_thesis: lim ((f2 * f1),x0) = lim_in+infty f2 hence lim ((f2 * f1),x0) = lim_in+infty f2 by A4, LIMFUNC3:def_4; ::_thesis: verum end; theorem :: LIMFUNC4:70 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_divergent_to-infty_in x0 & f2 is convergent_in-infty & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) holds ( f2 * f1 is_convergent_in x0 & lim ((f2 * f1),x0) = lim_in-infty f2 ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_divergent_to-infty_in x0 & f2 is convergent_in-infty & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) holds ( f2 * f1 is_convergent_in x0 & lim ((f2 * f1),x0) = lim_in-infty f2 ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_divergent_to-infty_in x0 & f2 is convergent_in-infty & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) implies ( f2 * f1 is_convergent_in x0 & lim ((f2 * f1),x0) = lim_in-infty f2 ) ) assume that A1: f1 is_divergent_to-infty_in x0 and A2: f2 is convergent_in-infty and A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ; ::_thesis: ( f2 * f1 is_convergent_in x0 & lim ((f2 * f1),x0) = lim_in-infty f2 ) A4: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f2_*_f1))_\_{x0}_holds_ (_(f2_*_f1)_/*_s_is_convergent_&_lim_((f2_*_f1)_/*_s)_=_lim_in-infty_f2_) let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) \ {x0} implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in-infty f2 ) ) assume that A5: ( s is convergent & lim s = x0 ) and A6: rng s c= (dom (f2 * f1)) \ {x0} ; ::_thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in-infty f2 ) rng s c= (dom f1) \ {x0} by A6, Th2; then A7: f1 /* s is divergent_to-infty by A1, A5, LIMFUNC3:def_3; A8: rng (f1 /* s) c= dom f2 by A6, Th2; then A9: lim (f2 /* (f1 /* s)) = lim_in-infty f2 by A2, A7, LIMFUNC1:def_13; A10: rng s c= dom (f2 * f1) by A6, Th2; lim_in-infty f2 = lim_in-infty f2 ; then f2 /* (f1 /* s) is convergent by A2, A8, A7, LIMFUNC1:def_13; hence ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in-infty f2 ) by A10, A9, VALUED_0:31; ::_thesis: verum end; hence f2 * f1 is_convergent_in x0 by A3, LIMFUNC3:def_1; ::_thesis: lim ((f2 * f1),x0) = lim_in-infty f2 hence lim ((f2 * f1),x0) = lim_in-infty f2 by A4, LIMFUNC3:def_4; ::_thesis: verum end; theorem :: LIMFUNC4:71 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in+infty & f2 is_convergent_in lim_in+infty f1 & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (right_open_halfline r) holds f1 . g <> lim_in+infty f1 holds ( f2 * f1 is convergent_in+infty & lim_in+infty (f2 * f1) = lim (f2,(lim_in+infty f1)) ) proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in+infty & f2 is_convergent_in lim_in+infty f1 & ( for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (right_open_halfline r) holds f1 . g <> lim_in+infty f1 implies ( f2 * f1 is convergent_in+infty & lim_in+infty (f2 * f1) = lim (f2,(lim_in+infty f1)) ) ) assume that A1: f1 is convergent_in+infty and A2: f2 is_convergent_in lim_in+infty f1 and A3: for r being Real ex g being Real st ( r < g & g in dom (f2 * f1) ) ; ::_thesis: ( for r being Real ex g being Real st ( g in (dom f1) /\ (right_open_halfline r) & not f1 . g <> lim_in+infty f1 ) or ( f2 * f1 is convergent_in+infty & lim_in+infty (f2 * f1) = lim (f2,(lim_in+infty f1)) ) ) given r being Real such that A4: for g being Real st g in (dom f1) /\ (right_open_halfline r) holds f1 . g <> lim_in+infty f1 ; ::_thesis: ( f2 * f1 is convergent_in+infty & lim_in+infty (f2 * f1) = lim (f2,(lim_in+infty f1)) ) A5: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to+infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (_(f2_*_f1)_/*_s_is_convergent_&_lim_((f2_*_f1)_/*_s)_=_lim_(f2,(lim_in+infty_f1))_) set L = lim (f2,(lim_in+infty f1)); let s be Real_Sequence; ::_thesis: ( s is divergent_to+infty & rng s c= dom (f2 * f1) implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim (f2,(lim_in+infty f1)) ) ) assume that A6: s is divergent_to+infty and A7: rng s c= dom (f2 * f1) ; ::_thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim (f2,(lim_in+infty f1)) ) consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds r < s . n by A6, LIMFUNC1:def_4; set q = s ^\ k; A9: s ^\ k is divergent_to+infty by A6, LIMFUNC1:26; A10: rng s c= dom f1 by A7, Lm2; A11: rng (s ^\ k) c= rng s by VALUED_0:21; then rng (s ^\ k) c= dom f1 by A10, XBOOLE_1:1; then A12: ( f1 /* (s ^\ k) is convergent & lim (f1 /* (s ^\ k)) = lim_in+infty f1 ) by A1, A2, A9, LIMFUNC1:def_12; A13: rng (f1 /* s) c= dom f2 by A7, Lm2; A14: rng (f1 /* (s ^\ k)) c= (dom f2) \ {(lim_in+infty f1)} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (f1 /* (s ^\ k)) or x in (dom f2) \ {(lim_in+infty f1)} ) assume x in rng (f1 /* (s ^\ k)) ; ::_thesis: x in (dom f2) \ {(lim_in+infty f1)} then consider n being Element of NAT such that A15: (f1 /* (s ^\ k)) . n = x by FUNCT_2:113; A16: x = f1 . ((s ^\ k) . n) by A10, A11, A15, FUNCT_2:108, XBOOLE_1:1 .= f1 . (s . (n + k)) by NAT_1:def_3 ; r < s . (n + k) by A8, NAT_1:12; then s . (n + k) in { r2 where r2 is Real : r < r2 } ; then A17: s . (n + k) in right_open_halfline r by XXREAL_1:230; s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in (dom f1) /\ (right_open_halfline r) by A10, A17, XBOOLE_0:def_4; then f1 . (s . (n + k)) <> lim_in+infty f1 by A4; then A18: not f1 . (s . (n + k)) in {(lim_in+infty f1)} by TARSKI:def_1; (f1 /* s) . (n + k) in rng (f1 /* s) by VALUED_0:28; then (f1 /* s) . (n + k) in dom f2 by A13; then x in dom f2 by A10, A16, FUNCT_2:108; hence x in (dom f2) \ {(lim_in+infty f1)} by A16, A18, XBOOLE_0:def_5; ::_thesis: verum end; A19: f2 /* (f1 /* (s ^\ k)) = f2 /* ((f1 /* s) ^\ k) by A10, VALUED_0:27 .= (f2 /* (f1 /* s)) ^\ k by A13, VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A7, VALUED_0:31 ; lim (f2,(lim_in+infty f1)) = lim (f2,(lim_in+infty f1)) ; then A20: f2 /* (f1 /* (s ^\ k)) is convergent by A2, A12, A14, LIMFUNC3:def_4; hence (f2 * f1) /* s is convergent by A19, SEQ_4:21; ::_thesis: lim ((f2 * f1) /* s) = lim (f2,(lim_in+infty f1)) lim (f2 /* (f1 /* (s ^\ k))) = lim (f2,(lim_in+infty f1)) by A2, A12, A14, LIMFUNC3:def_4; hence lim ((f2 * f1) /* s) = lim (f2,(lim_in+infty f1)) by A20, A19, SEQ_4:22; ::_thesis: verum end; hence f2 * f1 is convergent_in+infty by A3, LIMFUNC1:def_6; ::_thesis: lim_in+infty (f2 * f1) = lim (f2,(lim_in+infty f1)) hence lim_in+infty (f2 * f1) = lim (f2,(lim_in+infty f1)) by A5, LIMFUNC1:def_12; ::_thesis: verum end; theorem :: LIMFUNC4:72 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in-infty & f2 is_convergent_in lim_in-infty f1 & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (left_open_halfline r) holds f1 . g <> lim_in-infty f1 holds ( f2 * f1 is convergent_in-infty & lim_in-infty (f2 * f1) = lim (f2,(lim_in-infty f1)) ) proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in-infty & f2 is_convergent_in lim_in-infty f1 & ( for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ) & ex r being Real st for g being Real st g in (dom f1) /\ (left_open_halfline r) holds f1 . g <> lim_in-infty f1 implies ( f2 * f1 is convergent_in-infty & lim_in-infty (f2 * f1) = lim (f2,(lim_in-infty f1)) ) ) assume that A1: f1 is convergent_in-infty and A2: f2 is_convergent_in lim_in-infty f1 and A3: for r being Real ex g being Real st ( g < r & g in dom (f2 * f1) ) ; ::_thesis: ( for r being Real ex g being Real st ( g in (dom f1) /\ (left_open_halfline r) & not f1 . g <> lim_in-infty f1 ) or ( f2 * f1 is convergent_in-infty & lim_in-infty (f2 * f1) = lim (f2,(lim_in-infty f1)) ) ) given r being Real such that A4: for g being Real st g in (dom f1) /\ (left_open_halfline r) holds f1 . g <> lim_in-infty f1 ; ::_thesis: ( f2 * f1 is convergent_in-infty & lim_in-infty (f2 * f1) = lim (f2,(lim_in-infty f1)) ) A5: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to-infty_&_rng_s_c=_dom_(f2_*_f1)_holds_ (_(f2_*_f1)_/*_s_is_convergent_&_lim_((f2_*_f1)_/*_s)_=_lim_(f2,(lim_in-infty_f1))_) set L = lim (f2,(lim_in-infty f1)); let s be Real_Sequence; ::_thesis: ( s is divergent_to-infty & rng s c= dom (f2 * f1) implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim (f2,(lim_in-infty f1)) ) ) assume that A6: s is divergent_to-infty and A7: rng s c= dom (f2 * f1) ; ::_thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim (f2,(lim_in-infty f1)) ) consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds s . n < r by A6, LIMFUNC1:def_5; set q = s ^\ k; A9: s ^\ k is divergent_to-infty by A6, LIMFUNC1:27; A10: rng s c= dom f1 by A7, Lm2; A11: rng (s ^\ k) c= rng s by VALUED_0:21; then rng (s ^\ k) c= dom f1 by A10, XBOOLE_1:1; then A12: ( f1 /* (s ^\ k) is convergent & lim (f1 /* (s ^\ k)) = lim_in-infty f1 ) by A1, A2, A9, LIMFUNC1:def_13; A13: rng (f1 /* s) c= dom f2 by A7, Lm2; A14: rng (f1 /* (s ^\ k)) c= (dom f2) \ {(lim_in-infty f1)} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (f1 /* (s ^\ k)) or x in (dom f2) \ {(lim_in-infty f1)} ) assume x in rng (f1 /* (s ^\ k)) ; ::_thesis: x in (dom f2) \ {(lim_in-infty f1)} then consider n being Element of NAT such that A15: (f1 /* (s ^\ k)) . n = x by FUNCT_2:113; A16: x = f1 . ((s ^\ k) . n) by A10, A11, A15, FUNCT_2:108, XBOOLE_1:1 .= f1 . (s . (n + k)) by NAT_1:def_3 ; s . (n + k) < r by A8, NAT_1:12; then s . (n + k) in { r2 where r2 is Real : r2 < r } ; then A17: s . (n + k) in left_open_halfline r by XXREAL_1:229; s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in (dom f1) /\ (left_open_halfline r) by A10, A17, XBOOLE_0:def_4; then f1 . (s . (n + k)) <> lim_in-infty f1 by A4; then A18: not f1 . (s . (n + k)) in {(lim_in-infty f1)} by TARSKI:def_1; (f1 /* s) . (n + k) in rng (f1 /* s) by VALUED_0:28; then (f1 /* s) . (n + k) in dom f2 by A13; then x in dom f2 by A10, A16, FUNCT_2:108; hence x in (dom f2) \ {(lim_in-infty f1)} by A16, A18, XBOOLE_0:def_5; ::_thesis: verum end; A19: f2 /* (f1 /* (s ^\ k)) = f2 /* ((f1 /* s) ^\ k) by A10, VALUED_0:27 .= (f2 /* (f1 /* s)) ^\ k by A13, VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A7, VALUED_0:31 ; lim (f2,(lim_in-infty f1)) = lim (f2,(lim_in-infty f1)) ; then A20: f2 /* (f1 /* (s ^\ k)) is convergent by A2, A12, A14, LIMFUNC3:def_4; hence (f2 * f1) /* s is convergent by A19, SEQ_4:21; ::_thesis: lim ((f2 * f1) /* s) = lim (f2,(lim_in-infty f1)) lim (f2 /* (f1 /* (s ^\ k))) = lim (f2,(lim_in-infty f1)) by A2, A12, A14, LIMFUNC3:def_4; hence lim ((f2 * f1) /* s) = lim (f2,(lim_in-infty f1)) by A20, A19, SEQ_4:22; ::_thesis: verum end; hence f2 * f1 is convergent_in-infty by A3, LIMFUNC1:def_9; ::_thesis: lim_in-infty (f2 * f1) = lim (f2,(lim_in-infty f1)) hence lim_in-infty (f2 * f1) = lim (f2,(lim_in-infty f1)) by A5, LIMFUNC1:def_13; ::_thesis: verum end; theorem :: LIMFUNC4:73 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_left_convergent_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r < lim (f1,x0) ) ) holds ( f2 * f1 is_convergent_in x0 & lim ((f2 * f1),x0) = lim_left (f2,(lim (f1,x0))) ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_left_convergent_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r < lim (f1,x0) ) ) holds ( f2 * f1 is_convergent_in x0 & lim ((f2 * f1),x0) = lim_left (f2,(lim (f1,x0))) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_convergent_in x0 & f2 is_left_convergent_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r < lim (f1,x0) ) ) implies ( f2 * f1 is_convergent_in x0 & lim ((f2 * f1),x0) = lim_left (f2,(lim (f1,x0))) ) ) assume that A1: f1 is_convergent_in x0 and A2: f2 is_left_convergent_in lim (f1,x0) and A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) & not f1 . r < lim (f1,x0) ) ) or ( f2 * f1 is_convergent_in x0 & lim ((f2 * f1),x0) = lim_left (f2,(lim (f1,x0))) ) ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r < lim (f1,x0) ; ::_thesis: ( f2 * f1 is_convergent_in x0 & lim ((f2 * f1),x0) = lim_left (f2,(lim (f1,x0))) ) A6: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f2_*_f1))_\_{x0}_holds_ (_(f2_*_f1)_/*_s_is_convergent_&_lim_((f2_*_f1)_/*_s)_=_lim_left_(f2,(lim_(f1,x0)))_) let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) \ {x0} implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_left (f2,(lim (f1,x0))) ) ) assume that A7: ( s is convergent & lim s = x0 ) and A8: rng s c= (dom (f2 * f1)) \ {x0} ; ::_thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_left (f2,(lim (f1,x0))) ) consider k being Element of NAT such that A9: for n being Element of NAT st k <= n holds ( x0 - g < s . n & s . n < x0 + g ) by A4, A7, LIMFUNC3:7; set q = (f1 /* s) ^\ k; A10: rng s c= dom (f2 * f1) by A8, Th2; A11: rng s c= dom f1 by A8, Th2; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_/\_(left_open_halfline_(lim_(f1,x0))) let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) /\ (left_open_halfline (lim (f1,x0))) ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) /\ (left_open_halfline (lim (f1,x0))) then consider n being Element of NAT such that A12: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A13: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A11, FUNCT_2:108 .= x by A12, NAT_1:def_3 ; k <= n + k by NAT_1:12; then ( x0 - g < s . (n + k) & s . (n + k) < x0 + g ) by A9; then s . (n + k) in { g1 where g1 is Real : ( x0 - g < g1 & g1 < x0 + g ) } ; then A14: s . (n + k) in ].(x0 - g),(x0 + g).[ by RCOMP_1:def_2; A15: s . (n + k) in rng s by VALUED_0:28; then not s . (n + k) in {x0} by A8, XBOOLE_0:def_5; then s . (n + k) in ].(x0 - g),(x0 + g).[ \ {x0} by A14, XBOOLE_0:def_5; then s . (n + k) in ].(x0 - g),x0.[ \/ ].x0,(x0 + g).[ by A4, LIMFUNC3:4; then s . (n + k) in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) by A11, A15, XBOOLE_0:def_4; then f1 . (s . (n + k)) < lim (f1,x0) by A5; then f1 . (s . (n + k)) in { g2 where g2 is Real : g2 < lim (f1,x0) } ; then A16: f1 . (s . (n + k)) in left_open_halfline (lim (f1,x0)) by XXREAL_1:229; f1 . (s . (n + k)) in dom f2 by A10, A15, FUNCT_1:11; hence x in (dom f2) /\ (left_open_halfline (lim (f1,x0))) by A16, A13, XBOOLE_0:def_4; ::_thesis: verum end; then A17: rng ((f1 /* s) ^\ k) c= (dom f2) /\ (left_open_halfline (lim (f1,x0))) by TARSKI:def_3; rng (f1 /* s) c= dom f2 by A8, Th2; then A18: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A10, VALUED_0:31 ; A19: rng s c= (dom f1) \ {x0} by A8, Th2; then A20: f1 /* s is convergent by A1, A2, A7, LIMFUNC3:def_4; lim (f1 /* s) = lim (f1,x0) by A1, A7, A19, LIMFUNC3:def_4; then A21: lim ((f1 /* s) ^\ k) = lim (f1,x0) by A20, SEQ_4:20; lim_left (f2,(lim (f1,x0))) = lim_left (f2,(lim (f1,x0))) ; then A22: f2 /* ((f1 /* s) ^\ k) is convergent by A2, A20, A21, A17, LIMFUNC2:def_7; hence (f2 * f1) /* s is convergent by A18, SEQ_4:21; ::_thesis: lim ((f2 * f1) /* s) = lim_left (f2,(lim (f1,x0))) lim (f2 /* ((f1 /* s) ^\ k)) = lim_left (f2,(lim (f1,x0))) by A2, A20, A21, A17, LIMFUNC2:def_7; hence lim ((f2 * f1) /* s) = lim_left (f2,(lim (f1,x0))) by A22, A18, SEQ_4:22; ::_thesis: verum end; hence f2 * f1 is_convergent_in x0 by A3, LIMFUNC3:def_1; ::_thesis: lim ((f2 * f1),x0) = lim_left (f2,(lim (f1,x0))) hence lim ((f2 * f1),x0) = lim_left (f2,(lim (f1,x0))) by A6, LIMFUNC3:def_4; ::_thesis: verum end; theorem :: LIMFUNC4:74 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_convergent_in lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds f1 . r <> lim_left (f1,x0) ) ) holds ( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim (f2,(lim_left (f1,x0))) ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_convergent_in lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds f1 . r <> lim_left (f1,x0) ) ) holds ( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim (f2,(lim_left (f1,x0))) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_left_convergent_in x0 & f2 is_convergent_in lim_left (f1,x0) & ( for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds f1 . r <> lim_left (f1,x0) ) ) implies ( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim (f2,(lim_left (f1,x0))) ) ) assume that A1: f1 is_left_convergent_in x0 and A2: f2 is_convergent_in lim_left (f1,x0) and A3: for r being Real st r < x0 holds ex g being Real st ( r < g & g < x0 & g in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ ].(x0 - g),x0.[ & not f1 . r <> lim_left (f1,x0) ) ) or ( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim (f2,(lim_left (f1,x0))) ) ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds f1 . r <> lim_left (f1,x0) ; ::_thesis: ( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim (f2,(lim_left (f1,x0))) ) A6: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f2_*_f1))_/\_(left_open_halfline_x0)_holds_ (_(f2_*_f1)_/*_s_is_convergent_&_lim_((f2_*_f1)_/*_s)_=_lim_(f2,(lim_left_(f1,x0)))_) let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim (f2,(lim_left (f1,x0))) ) ) assume that A7: ( s is convergent & lim s = x0 ) and A8: rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) ; ::_thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim (f2,(lim_left (f1,x0))) ) consider k being Element of NAT such that A9: for n being Element of NAT st k <= n holds x0 - g < s . n by A4, A7, Lm1, LIMFUNC2:1; A10: rng s c= dom f1 by A8, Th1; set q = (f1 /* s) ^\ k; A11: rng s c= dom (f2 * f1) by A8, Th1; A12: rng s c= left_open_halfline x0 by A8, Th1; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_\_{(lim_left_(f1,x0))} let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) \ {(lim_left (f1,x0))} ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) \ {(lim_left (f1,x0))} then consider n being Element of NAT such that A13: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A14: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A10, FUNCT_2:108 .= x by A13, NAT_1:def_3 ; A15: x0 - g < s . (n + k) by A9, NAT_1:12; A16: s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in left_open_halfline x0 by A12; then s . (n + k) in { g1 where g1 is Real : g1 < x0 } by XXREAL_1:229; then ex g1 being Real st ( g1 = s . (n + k) & g1 < x0 ) ; then s . (n + k) in { g2 where g2 is Real : ( x0 - g < g2 & g2 < x0 ) } by A15; then s . (n + k) in ].(x0 - g),x0.[ by RCOMP_1:def_2; then s . (n + k) in (dom f1) /\ ].(x0 - g),x0.[ by A10, A16, XBOOLE_0:def_4; then f1 . (s . (n + k)) <> lim_left (f1,x0) by A5; then A17: not f1 . (s . (n + k)) in {(lim_left (f1,x0))} by TARSKI:def_1; f1 . (s . (n + k)) in dom f2 by A11, A16, FUNCT_1:11; hence x in (dom f2) \ {(lim_left (f1,x0))} by A17, A14, XBOOLE_0:def_5; ::_thesis: verum end; then A18: rng ((f1 /* s) ^\ k) c= (dom f2) \ {(lim_left (f1,x0))} by TARSKI:def_3; rng (f1 /* s) c= dom f2 by A8, Th1; then A19: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A11, VALUED_0:31 ; A20: rng s c= (dom f1) /\ (left_open_halfline x0) by A8, Th1; then A21: f1 /* s is convergent by A1, A2, A7, LIMFUNC2:def_7; lim (f1 /* s) = lim_left (f1,x0) by A1, A7, A20, LIMFUNC2:def_7; then A22: lim ((f1 /* s) ^\ k) = lim_left (f1,x0) by A21, SEQ_4:20; lim (f2,(lim_left (f1,x0))) = lim (f2,(lim_left (f1,x0))) ; then A23: f2 /* ((f1 /* s) ^\ k) is convergent by A2, A21, A22, A18, LIMFUNC3:def_4; hence (f2 * f1) /* s is convergent by A19, SEQ_4:21; ::_thesis: lim ((f2 * f1) /* s) = lim (f2,(lim_left (f1,x0))) lim (f2 /* ((f1 /* s) ^\ k)) = lim (f2,(lim_left (f1,x0))) by A2, A21, A22, A18, LIMFUNC3:def_4; hence lim ((f2 * f1) /* s) = lim (f2,(lim_left (f1,x0))) by A23, A19, SEQ_4:22; ::_thesis: verum end; hence f2 * f1 is_left_convergent_in x0 by A3, LIMFUNC2:def_1; ::_thesis: lim_left ((f2 * f1),x0) = lim (f2,(lim_left (f1,x0))) hence lim_left ((f2 * f1),x0) = lim (f2,(lim_left (f1,x0))) by A6, LIMFUNC2:def_7; ::_thesis: verum end; theorem :: LIMFUNC4:75 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_right_convergent_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds lim (f1,x0) < f1 . r ) ) holds ( f2 * f1 is_convergent_in x0 & lim ((f2 * f1),x0) = lim_right (f2,(lim (f1,x0))) ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_right_convergent_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds lim (f1,x0) < f1 . r ) ) holds ( f2 * f1 is_convergent_in x0 & lim ((f2 * f1),x0) = lim_right (f2,(lim (f1,x0))) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_convergent_in x0 & f2 is_right_convergent_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds lim (f1,x0) < f1 . r ) ) implies ( f2 * f1 is_convergent_in x0 & lim ((f2 * f1),x0) = lim_right (f2,(lim (f1,x0))) ) ) assume that A1: f1 is_convergent_in x0 and A2: f2 is_right_convergent_in lim (f1,x0) and A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) & not lim (f1,x0) < f1 . r ) ) or ( f2 * f1 is_convergent_in x0 & lim ((f2 * f1),x0) = lim_right (f2,(lim (f1,x0))) ) ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds lim (f1,x0) < f1 . r ; ::_thesis: ( f2 * f1 is_convergent_in x0 & lim ((f2 * f1),x0) = lim_right (f2,(lim (f1,x0))) ) A6: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f2_*_f1))_\_{x0}_holds_ (_(f2_*_f1)_/*_s_is_convergent_&_lim_((f2_*_f1)_/*_s)_=_lim_right_(f2,(lim_(f1,x0)))_) let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) \ {x0} implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_right (f2,(lim (f1,x0))) ) ) assume that A7: ( s is convergent & lim s = x0 ) and A8: rng s c= (dom (f2 * f1)) \ {x0} ; ::_thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_right (f2,(lim (f1,x0))) ) consider k being Element of NAT such that A9: for n being Element of NAT st k <= n holds ( x0 - g < s . n & s . n < x0 + g ) by A4, A7, LIMFUNC3:7; set q = (f1 /* s) ^\ k; A10: rng s c= dom (f2 * f1) by A8, Th2; A11: rng s c= dom f1 by A8, Th2; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_/\_(right_open_halfline_(lim_(f1,x0))) let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) /\ (right_open_halfline (lim (f1,x0))) ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) /\ (right_open_halfline (lim (f1,x0))) then consider n being Element of NAT such that A12: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A13: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A11, FUNCT_2:108 .= x by A12, NAT_1:def_3 ; k <= n + k by NAT_1:12; then ( x0 - g < s . (n + k) & s . (n + k) < x0 + g ) by A9; then s . (n + k) in { g1 where g1 is Real : ( x0 - g < g1 & g1 < x0 + g ) } ; then A14: s . (n + k) in ].(x0 - g),(x0 + g).[ by RCOMP_1:def_2; A15: s . (n + k) in rng s by VALUED_0:28; then not s . (n + k) in {x0} by A8, XBOOLE_0:def_5; then s . (n + k) in ].(x0 - g),(x0 + g).[ \ {x0} by A14, XBOOLE_0:def_5; then s . (n + k) in ].(x0 - g),x0.[ \/ ].x0,(x0 + g).[ by A4, LIMFUNC3:4; then s . (n + k) in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) by A11, A15, XBOOLE_0:def_4; then f1 . (s . (n + k)) > lim (f1,x0) by A5; then f1 . (s . (n + k)) in { g2 where g2 is Real : lim (f1,x0) < g2 } ; then A16: f1 . (s . (n + k)) in right_open_halfline (lim (f1,x0)) by XXREAL_1:230; f1 . (s . (n + k)) in dom f2 by A10, A15, FUNCT_1:11; hence x in (dom f2) /\ (right_open_halfline (lim (f1,x0))) by A16, A13, XBOOLE_0:def_4; ::_thesis: verum end; then A17: rng ((f1 /* s) ^\ k) c= (dom f2) /\ (right_open_halfline (lim (f1,x0))) by TARSKI:def_3; rng (f1 /* s) c= dom f2 by A8, Th2; then A18: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A10, VALUED_0:31 ; A19: rng s c= (dom f1) \ {x0} by A8, Th2; then A20: f1 /* s is convergent by A1, A2, A7, LIMFUNC3:def_4; lim (f1 /* s) = lim (f1,x0) by A1, A7, A19, LIMFUNC3:def_4; then A21: lim ((f1 /* s) ^\ k) = lim (f1,x0) by A20, SEQ_4:20; lim_right (f2,(lim (f1,x0))) = lim_right (f2,(lim (f1,x0))) ; then A22: f2 /* ((f1 /* s) ^\ k) is convergent by A2, A20, A21, A17, LIMFUNC2:def_8; hence (f2 * f1) /* s is convergent by A18, SEQ_4:21; ::_thesis: lim ((f2 * f1) /* s) = lim_right (f2,(lim (f1,x0))) lim (f2 /* ((f1 /* s) ^\ k)) = lim_right (f2,(lim (f1,x0))) by A2, A20, A21, A17, LIMFUNC2:def_8; hence lim ((f2 * f1) /* s) = lim_right (f2,(lim (f1,x0))) by A22, A18, SEQ_4:22; ::_thesis: verum end; hence f2 * f1 is_convergent_in x0 by A3, LIMFUNC3:def_1; ::_thesis: lim ((f2 * f1),x0) = lim_right (f2,(lim (f1,x0))) hence lim ((f2 * f1),x0) = lim_right (f2,(lim (f1,x0))) by A6, LIMFUNC3:def_4; ::_thesis: verum end; theorem :: LIMFUNC4:76 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_convergent_in lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds f1 . r <> lim_right (f1,x0) ) ) holds ( f2 * f1 is_right_convergent_in x0 & lim_right ((f2 * f1),x0) = lim (f2,(lim_right (f1,x0))) ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_right_convergent_in x0 & f2 is_convergent_in lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds f1 . r <> lim_right (f1,x0) ) ) holds ( f2 * f1 is_right_convergent_in x0 & lim_right ((f2 * f1),x0) = lim (f2,(lim_right (f1,x0))) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_right_convergent_in x0 & f2 is_convergent_in lim_right (f1,x0) & ( for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds f1 . r <> lim_right (f1,x0) ) ) implies ( f2 * f1 is_right_convergent_in x0 & lim_right ((f2 * f1),x0) = lim (f2,(lim_right (f1,x0))) ) ) assume that A1: f1 is_right_convergent_in x0 and A2: f2 is_convergent_in lim_right (f1,x0) and A3: for r being Real st x0 < r holds ex g being Real st ( g < r & x0 < g & g in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ ].x0,(x0 + g).[ & not f1 . r <> lim_right (f1,x0) ) ) or ( f2 * f1 is_right_convergent_in x0 & lim_right ((f2 * f1),x0) = lim (f2,(lim_right (f1,x0))) ) ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ ].x0,(x0 + g).[ holds f1 . r <> lim_right (f1,x0) ; ::_thesis: ( f2 * f1 is_right_convergent_in x0 & lim_right ((f2 * f1),x0) = lim (f2,(lim_right (f1,x0))) ) A6: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f2_*_f1))_/\_(right_open_halfline_x0)_holds_ (_(f2_*_f1)_/*_s_is_convergent_&_lim_((f2_*_f1)_/*_s)_=_lim_(f2,(lim_right_(f1,x0)))_) let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim (f2,(lim_right (f1,x0))) ) ) assume that A7: ( s is convergent & lim s = x0 ) and A8: rng s c= (dom (f2 * f1)) /\ (right_open_halfline x0) ; ::_thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim (f2,(lim_right (f1,x0))) ) consider k being Element of NAT such that A9: for n being Element of NAT st k <= n holds s . n < x0 + g by A4, A7, Lm1, LIMFUNC2:2; A10: rng s c= dom f1 by A8, Th1; set q = (f1 /* s) ^\ k; A11: rng s c= dom (f2 * f1) by A8, Th1; A12: rng s c= right_open_halfline x0 by A8, Th1; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_\_{(lim_right_(f1,x0))} let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) \ {(lim_right (f1,x0))} ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) \ {(lim_right (f1,x0))} then consider n being Element of NAT such that A13: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A14: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A10, FUNCT_2:108 .= x by A13, NAT_1:def_3 ; A15: s . (n + k) < x0 + g by A9, NAT_1:12; A16: s . (n + k) in rng s by VALUED_0:28; then s . (n + k) in right_open_halfline x0 by A12; then s . (n + k) in { g1 where g1 is Real : x0 < g1 } by XXREAL_1:230; then ex g1 being Real st ( g1 = s . (n + k) & x0 < g1 ) ; then s . (n + k) in { g2 where g2 is Real : ( x0 < g2 & g2 < x0 + g ) } by A15; then s . (n + k) in ].x0,(x0 + g).[ by RCOMP_1:def_2; then s . (n + k) in (dom f1) /\ ].x0,(x0 + g).[ by A10, A16, XBOOLE_0:def_4; then f1 . (s . (n + k)) <> lim_right (f1,x0) by A5; then A17: not f1 . (s . (n + k)) in {(lim_right (f1,x0))} by TARSKI:def_1; f1 . (s . (n + k)) in dom f2 by A11, A16, FUNCT_1:11; hence x in (dom f2) \ {(lim_right (f1,x0))} by A17, A14, XBOOLE_0:def_5; ::_thesis: verum end; then A18: rng ((f1 /* s) ^\ k) c= (dom f2) \ {(lim_right (f1,x0))} by TARSKI:def_3; rng (f1 /* s) c= dom f2 by A8, Th1; then A19: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A11, VALUED_0:31 ; A20: rng s c= (dom f1) /\ (right_open_halfline x0) by A8, Th1; then A21: f1 /* s is convergent by A1, A2, A7, LIMFUNC2:def_8; lim (f1 /* s) = lim_right (f1,x0) by A1, A7, A20, LIMFUNC2:def_8; then A22: lim ((f1 /* s) ^\ k) = lim_right (f1,x0) by A21, SEQ_4:20; lim (f2,(lim_right (f1,x0))) = lim (f2,(lim_right (f1,x0))) ; then A23: f2 /* ((f1 /* s) ^\ k) is convergent by A2, A21, A22, A18, LIMFUNC3:def_4; hence (f2 * f1) /* s is convergent by A19, SEQ_4:21; ::_thesis: lim ((f2 * f1) /* s) = lim (f2,(lim_right (f1,x0))) lim (f2 /* ((f1 /* s) ^\ k)) = lim (f2,(lim_right (f1,x0))) by A2, A21, A22, A18, LIMFUNC3:def_4; hence lim ((f2 * f1) /* s) = lim (f2,(lim_right (f1,x0))) by A23, A19, SEQ_4:22; ::_thesis: verum end; hence f2 * f1 is_right_convergent_in x0 by A3, LIMFUNC2:def_4; ::_thesis: lim_right ((f2 * f1),x0) = lim (f2,(lim_right (f1,x0))) hence lim_right ((f2 * f1),x0) = lim (f2,(lim_right (f1,x0))) by A6, LIMFUNC2:def_8; ::_thesis: verum end; theorem :: LIMFUNC4:77 for x0 being Real for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_convergent_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r <> lim (f1,x0) ) ) holds ( f2 * f1 is_convergent_in x0 & lim ((f2 * f1),x0) = lim (f2,(lim (f1,x0))) ) proof let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_convergent_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r <> lim (f1,x0) ) ) holds ( f2 * f1 is_convergent_in x0 & lim ((f2 * f1),x0) = lim (f2,(lim (f1,x0))) ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_convergent_in x0 & f2 is_convergent_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st ( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r <> lim (f1,x0) ) ) implies ( f2 * f1 is_convergent_in x0 & lim ((f2 * f1),x0) = lim (f2,(lim (f1,x0))) ) ) assume that A1: f1 is_convergent_in x0 and A2: f2 is_convergent_in lim (f1,x0) and A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds ex g1, g2 being Real st ( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ; ::_thesis: ( for g being Real holds ( not 0 < g or ex r being Real st ( r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) & not f1 . r <> lim (f1,x0) ) ) or ( f2 * f1 is_convergent_in x0 & lim ((f2 * f1),x0) = lim (f2,(lim (f1,x0))) ) ) given g being Real such that A4: 0 < g and A5: for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds f1 . r <> lim (f1,x0) ; ::_thesis: ( f2 * f1 is_convergent_in x0 & lim ((f2 * f1),x0) = lim (f2,(lim (f1,x0))) ) A6: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_convergent_&_lim_s_=_x0_&_rng_s_c=_(dom_(f2_*_f1))_\_{x0}_holds_ (_(f2_*_f1)_/*_s_is_convergent_&_lim_((f2_*_f1)_/*_s)_=_lim_(f2,(lim_(f1,x0)))_) let s be Real_Sequence; ::_thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) \ {x0} implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim (f2,(lim (f1,x0))) ) ) assume that A7: ( s is convergent & lim s = x0 ) and A8: rng s c= (dom (f2 * f1)) \ {x0} ; ::_thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim (f2,(lim (f1,x0))) ) consider k being Element of NAT such that A9: for n being Element of NAT st k <= n holds ( x0 - g < s . n & s . n < x0 + g ) by A4, A7, LIMFUNC3:7; set q = (f1 /* s) ^\ k; A10: rng s c= dom (f2 * f1) by A8, Th2; A11: rng s c= dom f1 by A8, Th2; now__::_thesis:_for_x_being_set_st_x_in_rng_((f1_/*_s)_^\_k)_holds_ x_in_(dom_f2)_\_{(lim_(f1,x0))} let x be set ; ::_thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) \ {(lim (f1,x0))} ) assume x in rng ((f1 /* s) ^\ k) ; ::_thesis: x in (dom f2) \ {(lim (f1,x0))} then consider n being Element of NAT such that A12: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113; A13: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A11, FUNCT_2:108 .= x by A12, NAT_1:def_3 ; k <= n + k by NAT_1:12; then ( x0 - g < s . (n + k) & s . (n + k) < x0 + g ) by A9; then s . (n + k) in { g1 where g1 is Real : ( x0 - g < g1 & g1 < x0 + g ) } ; then A14: s . (n + k) in ].(x0 - g),(x0 + g).[ by RCOMP_1:def_2; A15: s . (n + k) in rng s by VALUED_0:28; then not s . (n + k) in {x0} by A8, XBOOLE_0:def_5; then s . (n + k) in ].(x0 - g),(x0 + g).[ \ {x0} by A14, XBOOLE_0:def_5; then s . (n + k) in ].(x0 - g),x0.[ \/ ].x0,(x0 + g).[ by A4, LIMFUNC3:4; then s . (n + k) in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) by A11, A15, XBOOLE_0:def_4; then f1 . (s . (n + k)) <> lim (f1,x0) by A5; then A16: not f1 . (s . (n + k)) in {(lim (f1,x0))} by TARSKI:def_1; f1 . (s . (n + k)) in dom f2 by A10, A15, FUNCT_1:11; hence x in (dom f2) \ {(lim (f1,x0))} by A16, A13, XBOOLE_0:def_5; ::_thesis: verum end; then A17: rng ((f1 /* s) ^\ k) c= (dom f2) \ {(lim (f1,x0))} by TARSKI:def_3; rng (f1 /* s) c= dom f2 by A8, Th2; then A18: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27 .= ((f2 * f1) /* s) ^\ k by A10, VALUED_0:31 ; A19: rng s c= (dom f1) \ {x0} by A8, Th2; then A20: f1 /* s is convergent by A1, A2, A7, LIMFUNC3:def_4; lim (f1 /* s) = lim (f1,x0) by A1, A7, A19, LIMFUNC3:def_4; then A21: lim ((f1 /* s) ^\ k) = lim (f1,x0) by A20, SEQ_4:20; lim (f2,(lim (f1,x0))) = lim (f2,(lim (f1,x0))) ; then A22: f2 /* ((f1 /* s) ^\ k) is convergent by A2, A20, A21, A17, LIMFUNC3:def_4; hence (f2 * f1) /* s is convergent by A18, SEQ_4:21; ::_thesis: lim ((f2 * f1) /* s) = lim (f2,(lim (f1,x0))) lim (f2 /* ((f1 /* s) ^\ k)) = lim (f2,(lim (f1,x0))) by A2, A20, A21, A17, LIMFUNC3:def_4; hence lim ((f2 * f1) /* s) = lim (f2,(lim (f1,x0))) by A22, A18, SEQ_4:22; ::_thesis: verum end; hence f2 * f1 is_convergent_in x0 by A3, LIMFUNC3:def_1; ::_thesis: lim ((f2 * f1),x0) = lim (f2,(lim (f1,x0))) hence lim ((f2 * f1),x0) = lim (f2,(lim (f1,x0))) by A6, LIMFUNC3:def_4; ::_thesis: verum end;