reserve m,n for non zero Element of NAT;
reserve i,j,k for Element of NAT;
reserve Z for set;

theorem Th33:
for Z be set, f,g be PartFunc of REAL m,REAL n
  st f is_continuous_on Z & g is_continuous_on Z
  holds f+g is_continuous_on Z & f-g is_continuous_on Z
proof
   let Z be set, f,g be PartFunc of REAL m,REAL n;
   assume A1: f is_continuous_on Z & g is_continuous_on Z;
A2:the carrier of REAL-NS m = REAL m &
   the carrier of REAL-NS n = REAL n by REAL_NS1:def 4; then
   reconsider f1=f, g1=g as PartFunc of REAL-NS m,REAL-NS n;
   f1 is_continuous_on Z & g1 is_continuous_on Z by A1,PDIFF_7:37; then
A3:f1+g1 is_continuous_on Z & f1-g1 is_continuous_on Z by NFCONT_1:25;
   f+g = f1+g1 & f-g=f1-g1 by A2,NFCONT_4:5,10;
   hence thesis by A3,PDIFF_7:37;
end;
