let x0 be Real; :: thesis: for f1, f2 being PartFunc of REAL ,REAL st f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & lim_left f2,x0 <> 0 & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom (f1 / f2) ) ) holds
( f1 / f2 is_left_convergent_in x0 & lim_left (f1 / f2),x0 = (lim_left f1,x0) / (lim_left f2,x0) )
let f1, f2 be PartFunc of REAL ,REAL ; :: thesis: ( f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & lim_left f2,x0 <> 0 & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom (f1 / f2) ) ) implies ( f1 / f2 is_left_convergent_in x0 & lim_left (f1 / f2),x0 = (lim_left f1,x0) / (lim_left f2,x0) ) )
assume that
A1: f1 is_left_convergent_in x0 and
A2: f2 is_left_convergent_in x0 and
A3: lim_left f2,x0 <> 0 and
A4: for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom (f1 / f2) ) ; :: thesis: ( f1 / f2 is_left_convergent_in x0 & lim_left (f1 / f2),x0 = (lim_left f1,x0) / (lim_left f2,x0) )

A5: now

let r be Real; :: thesis: ( r < x0 implies ex g being Real st
( r < g & g < x0 & g in dom f2 & f2 . g <> 0 ) )
assume r < x0 ; :: thesis: ex g being Real st
( r < g & g < x0 & g in dom f2 & f2 . g <> 0 )
then consider g being Real such that
A6: r < g and
A7: g < x0 and
A8: g in dom (f1 / f2) by A4;
take g = g; :: thesis: ( r < g & g < x0 & g in dom f2 & f2 . g <> 0 )
thus ( r < g & g < x0 ) by A6, A7; :: thesis: ( g in dom f2 & f2 . g <> 0 )
dom (f1 / f2) = (dom f1) /\ ((dom f2) \ (f2 " {0 })) by RFUNCT_1:def 4;
then A9: g in (dom f2) \ (f2 " {0 }) by A8, XBOOLE_0:def 4;
then A10: not g in f2 " {0 } by XBOOLE_0:def 5;
g in dom f2 by A9, XBOOLE_0:def 5;
then not f2 . g in {0 } by A10, FUNCT_1:def 13;
hence ( g in dom f2 & f2 . g <> 0 ) by A9, TARSKI:def 1, XBOOLE_0:def 5; :: thesis: verum

end;

then A11: f2 ^ is_left_convergent_in x0 by A2, A3, Th57;
A12: f1 / f2 = f1 (#) (f2 ^ ) by RFUNCT_1:47;
hence f1 / f2 is_left_convergent_in x0 by A1, A4, A11, Th58; :: thesis: lim_left (f1 / f2),x0 = (lim_left f1,x0) / (lim_left f2,x0)
lim_left (f2 ^ ),x0 = (lim_left f2,x0) " by A2, A3, A5, Th57;
hence lim_left (f1 / f2),x0 = (lim_left f1,x0) * ((lim_left f2,x0) " ) by A1, A4, A12, A11, Th58
.= (lim_left f1,x0) / (lim_left f2,x0) by XCMPLX_0:def 9 ;
:: thesis: verum