let f1, f2 be PartFunc of REAL ,REAL ; :: thesis: ( f1 is convergent_in-infty & f2 is convergent_in-infty & lim_in-infty f2 <> 0 & ( for r being Real ex g being Real st
( g < r & g in dom (f1 / f2) ) ) implies ( f1 / f2 is convergent_in-infty & lim_in-infty (f1 / f2) = (lim_in-infty f1) / (lim_in-infty f2) ) )
assume that
A1: f1 is convergent_in-infty and
A2: ( f2 is convergent_in-infty & lim_in-infty f2 <> 0 ) and
A3: for r being Real ex g being Real st
( g < r & g in dom (f1 / f2) ) ; :: thesis: ( f1 / f2 is convergent_in-infty & lim_in-infty (f1 / f2) = (lim_in-infty f1) / (lim_in-infty f2) )
dom (f1 / f2) = (dom f1) /\ ((dom f2) \ (f2 " {0 })) by RFUNCT_1:def 4;
then A4: dom (f1 / f2) = (dom f1) /\ (dom (f2 ^ )) by RFUNCT_1:def 8;
A5: (dom f1) /\ (dom (f2 ^ )) c= dom (f2 ^ ) by XBOOLE_1:17;

A6: now

let r be Real; :: thesis: ex g being Real st
( g < r & g in dom f2 & f2 . g <> 0 )
consider g being Real such that
A7: g < r and
A8: g in dom (f1 / f2) by A3;
take g = g; :: thesis: ( g < r & g in dom f2 & f2 . g <> 0 )
g in dom (f2 ^ ) by A4, A5, A8;
then A9: g in (dom f2) \ (f2 " {0 }) by RFUNCT_1:def 8;
then ( g in dom f2 & not g in f2 " {0 } ) by XBOOLE_0:def 5;
then not f2 . g in {0 } by FUNCT_1:def 13;
hence ( g < r & g in dom f2 & f2 . g <> 0 ) by A7, A9, TARSKI:def 1, XBOOLE_0:def 5; :: thesis: verum

end;

then A10: f2 ^ is convergent_in-infty by A2, Th130;
A11: lim_in-infty (f2 ^ ) = (lim_in-infty f2) " by A2, A6, Th130;

A12: now

let r be Real; :: thesis: ex g being Real st
( g < r & g in dom (f1 (#) (f2 ^ )) )
consider g being Real such that
A13: ( g < r & g in dom (f1 / f2) ) by A3;
take g = g; :: thesis: ( g < r & g in dom (f1 (#) (f2 ^ )) )
thus ( g < r & g in dom (f1 (#) (f2 ^ )) ) by A4, A13, VALUED_1:def 4; :: thesis: verum

end;

then f1 (#) (f2 ^ ) is convergent_in-infty by A1, A10, Th131;
hence f1 / f2 is convergent_in-infty by RFUNCT_1:47; :: thesis: lim_in-infty (f1 / f2) = (lim_in-infty f1) / (lim_in-infty f2)
thus lim_in-infty (f1 / f2) = lim_in-infty (f1 (#) (f2 ^ )) by RFUNCT_1:47
.= (lim_in-infty f1) * ((lim_in-infty f2) " ) by A1, A12, A10, A11, Th131
.= (lim_in-infty f1) / (lim_in-infty f2) by XCMPLX_0:def 9 ; :: thesis: verum