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

theorem Th34:
for r be Real, f,g be PartFunc of REAL m,REAL n
  st f is_continuous_on Z holds r(#)f is_continuous_on Z
proof
   let r be Real, f,g be PartFunc of REAL m,REAL n;
   assume A1: f 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 as PartFunc of REAL-NS m,REAL-NS n;
   f1 is_continuous_on Z by A1,PDIFF_7:37; then
A3:r(#)f1 is_continuous_on Z by NFCONT_1:27;
   r(#)f1=r(#)f by A2,NFCONT_4:6;
   hence r(#)f is_continuous_on Z by A3,PDIFF_7:37;
end;
