theorem
  x0 in dom f1 /\ dom f2 & f1 is_continuous_in x0 & f2 is_continuous_in x0
    implies f1+f2 is_continuous_in x0
proof
  assume A1: x0 in dom f1 /\ dom f2 & f1 is_continuous_in x0
     & 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 thesis by A2;
end;
