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
( r < g & 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 A1: ( f1 is convergent_in+infty & f2 is convergent_in+infty & lim_in+infty f2 <> 0 & ( for r being Real ex g being Real st
( r < g & 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 A2: dom (f1 / f2) = (dom f1) /\ (dom (f2 ^ )) by RFUNCT_1:def 8;
A3: ( (dom f1) /\ (dom (f2 ^ )) c= dom f1 & (dom f1) /\ (dom (f2 ^ )) c= dom (f2 ^ ) ) by XBOOLE_1:17;

A4: now

let r be Real; :: thesis: ex g being Real st
( r < g & g in dom (f1 (#) (f2 ^ )) )
consider g being Real such that
A5: ( r < g & g in dom (f1 / f2) ) by A1;
take g = g; :: thesis: ( r < g & g in dom (f1 (#) (f2 ^ )) )
thus ( r < g & g in dom (f1 (#) (f2 ^ )) ) by A2, A5, VALUED_1:def 4; :: thesis: verum

end;

now

let r be Real; :: thesis: ex g being Real st
( r < g & g in dom f2 & f2 . g <> 0 )
consider g being Real such that
A6: ( r < g & g in dom (f1 / f2) ) by A1;
take g = g; :: thesis: ( r < g & g in dom f2 & f2 . g <> 0 )
g in dom (f2 ^ ) by A2, A3, A6;
then A7: 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 ( r < g & g in dom f2 & f2 . g <> 0 ) by A6, A7, TARSKI:def 1, XBOOLE_0:def 5; :: thesis: verum

end;

then A8: ( f2 ^ is convergent_in+infty & lim_in+infty (f2 ^ ) = (lim_in+infty f2) " ) by A1, Th121;
then f1 (#) (f2 ^ ) is convergent_in+infty by A1, A4, Th122;
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, A4, A8, Th122
.= (lim_in+infty f1) / (lim_in+infty f2) by XCMPLX_0:def 9 ; :: thesis: verum