let x0 be Real; :: thesis: for f1, f2 being PartFunc of REAL ,REAL st f1 is_convergent_in x0 & f2 is_convergent_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds
ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom (f1 - f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 - f2) ) ) holds
( f1 - f2 is_convergent_in x0 & lim (f1 - f2),x0 = (lim f1,x0) - (lim f2,x0) )
let f1, f2 be PartFunc of REAL ,REAL ; :: thesis: ( f1 is_convergent_in x0 & f2 is_convergent_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds
ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom (f1 - f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 - f2) ) ) implies ( f1 - f2 is_convergent_in x0 & lim (f1 - f2),x0 = (lim f1,x0) - (lim f2,x0) ) )
assume that
A1: f1 is_convergent_in x0 and
A2: f2 is_convergent_in x0 and
A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds
ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom (f1 - f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 - f2) ) ; :: thesis: ( f1 - f2 is_convergent_in x0 & lim (f1 - f2),x0 = (lim f1,x0) - (lim f2,x0) )
A4: - f2 is_convergent_in x0 by A2, Th36;
hence f1 - f2 is_convergent_in x0 by A1, A3, Th37; :: thesis: lim (f1 - f2),x0 = (lim f1,x0) - (lim f2,x0)
thus lim (f1 - f2),x0 = (lim f1,x0) + (lim (- f2),x0) by A1, A3, A4, Th37
.= (lim f1,x0) + (- (lim f2,x0)) by A2, Th36
.= (lim f1,x0) - (lim f2,x0) ; :: thesis: verum