let x0 be Real; :: thesis: for f being PartFunc of REAL,REAL st f is_left_convergent_in x0 holds
( - f is_left_convergent_in x0 & lim_left ((- f),x0) = - (lim_left (f,x0)) )
let f be PartFunc of REAL,REAL; :: thesis: ( f is_left_convergent_in x0 implies ( - f is_left_convergent_in x0 & lim_left ((- f),x0) = - (lim_left (f,x0)) ) )
assume A1: f is_left_convergent_in x0 ; :: thesis: ( - f is_left_convergent_in x0 & lim_left ((- f),x0) = - (lim_left (f,x0)) )
(- 1) (#) f = - f ;
hence - f is_left_convergent_in x0 by A1, Th51; :: thesis: lim_left ((- f),x0) = - (lim_left (f,x0))
thus lim_left ((- f),x0) = (- 1) * (lim_left (f,x0)) by A1, Th51
.= - (lim_left (f,x0)) ; :: thesis: verum