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, Th52;
hence f1 - f2 is_left_convergent_in x0 by A1, A3, Th53; :: 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, Th53
.= (lim_left f1,x0) + (- (lim_left f2,x0)) by A2, Th52
.= (lim_left f1,x0) - (lim_left f2,x0) ; :: thesis: verum