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 & ( 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 & ( 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) ) ; :: thesis: ( 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; :: thesis: 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)) ; :: thesis: verum