let n be Element of NAT ; :: thesis: for x0 being Real
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; :: 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 A1: ( 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,(REAL-NS n) by REAL_NS1:def 4;
A2: ( g1 + g2 is_continuous_in x0 & g1 - g2 is_continuous_in x0 ) by A1, NFCONT_3:12;
g1 + g2 = f1 + f2 by Th5;
hence f1 + f2 is_continuous_in x0 by A2; :: thesis: verum