let n be Element of NAT ; :: thesis: for x0 being real number
for f1, f2 being PartFunc of REAL,(REAL n) st x0 in (dom f1) /\ (dom f2) & f1 is_continuous_in x0 & f2 is_continuous_in x0 holds
f1 + f2 is_continuous_in x0
let x0 be real number ; :: thesis: for f1, f2 being PartFunc of REAL,(REAL n) st x0 in (dom f1) /\ (dom f2) & f1 is_continuous_in x0 & f2 is_continuous_in x0 holds
f1 + f2 is_continuous_in x0
let f1, f2 be PartFunc of REAL,(REAL n); :: thesis: ( x0 in (dom f1) /\ (dom f2) & f1 is_continuous_in x0 & f2 is_continuous_in x0 implies f1 + f2 is_continuous_in x0 )
assume AS: ( x0 in (dom f1) /\ (dom f2) & f1 is_continuous_in x0 & f2 is_continuous_in x0 ) ; :: thesis: f1 + f2 is_continuous_in x0
reconsider g1 = f1, g2 = f2 as PartFunc of REAL, the carrier of (REAL-NS n) by REAL_NS1:def 4;
P2: g1 is_continuous_in x0 by AS, Def1Th;
g2 is_continuous_in x0 by AS, Def1Th;
then P3: ( g1 + g2 is_continuous_in x0 & g1 - g2 is_continuous_in x0 ) by AS, P2, NFCONT_3:12;
g1 + g2 = f1 + f2 by LM003A;
hence f1 + f2 is_continuous_in x0 by P3, Def1Th; :: thesis: verum