let f be PartFunc of REAL , REAL ; :: thesis: for x0 being Real st f is_left_differentiable_in x0 & f . x0 <> 0 holds
( f ^ is_left_differentiable_in x0 & Ldiff (f ^ ),x0 = - ((Ldiff f,x0) / ((f . x0) ^2 )) )
let x0 be Real; :: thesis: ( f is_left_differentiable_in x0 & f . x0 <> 0 implies ( f ^ is_left_differentiable_in x0 & Ldiff (f ^ ),x0 = - ((Ldiff f,x0) / ((f . x0) ^2 )) ) )
assume A1: ( f is_left_differentiable_in x0 & f . x0 <> 0 ) ; :: thesis: ( f ^ is_left_differentiable_in x0 & Ldiff (f ^ ),x0 = - ((Ldiff f,x0) / ((f . x0) ^2 )) )
then A2: f is_Lcontinuous_in x0 by Th5;
ex r being Real st
( r > 0 & [.(x0 - r),x0.] c= dom f ) by A1, Def4;
then consider r1 being Real such that
A3: ( r1 > 0 & [.(x0 - r1),x0.] c= dom f & ( for g being Real st g in [.(x0 - r1),x0.] holds
f . g <> 0 ) ) by A1, A2, Th6;

now

take r1 = r1; :: thesis: ( r1 > 0 & ( for g being Real st g in dom f & g in [.(x0 - r1),x0.] holds
f . g <> 0 ) )
thus r1 > 0 by A3; :: thesis: for g being Real st g in dom f & g in [.(x0 - r1),x0.] holds
f . g <> 0
let g be Real; :: thesis: ( g in dom f & g in [.(x0 - r1),x0.] implies f . g <> 0 )
assume ( g in dom f & g in [.(x0 - r1),x0.] ) ; :: thesis: f . g <> 0
hence f . g <> 0 by A3; :: thesis: verum

end;

hence ( f ^ is_left_differentiable_in x0 & Ldiff (f ^ ),x0 = - ((Ldiff f,x0) / ((f . x0) ^2 )) ) by A1, Lm2; :: thesis: verum