let x0 be Real; for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & ( 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; ( f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & ( 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:
for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom (f1 - f2) )
; ( f1 - f2 is_left_convergent_in x0 & lim_left ((f1 - f2),x0) = (lim_left (f1,x0)) - (lim_left (f2,x0)) )
A4:
- f2 is_left_convergent_in x0
by A2, Th44;
hence
f1 - f2 is_left_convergent_in x0
by A1, A3, Th45; lim_left ((f1 - f2),x0) = (lim_left (f1,x0)) - (lim_left (f2,x0))
thus lim_left ((f1 - f2),x0) =
(lim_left (f1,x0)) + (lim_left ((- f2),x0))
by A1, A3, A4, Th45
.=
(lim_left (f1,x0)) + (- (lim_left (f2,x0)))
by A2, Th44
.=
(lim_left (f1,x0)) - (lim_left (f2,x0))
; verum