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 object st x in rng ((f1 /* s) ^\ k) holds
x in (dom f2) \ {(lim (f1,x0))}

let x be object ; :: 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))} ;
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