theorem
  X c= dom f & f|X is continuous implies (r(#)f) | X is continuous
  proof
  assume A1:X c= dom f & f|X is continuous;
  reconsider g= f as PartFunc of REAL,REAL-NS n
   by REAL_NS1:def 4;
  g|X is continuous PartFunc of REAL,REAL-NS n by A1,Th23;
  then
A2: (r(#)g) | X is continuous by A1,NFCONT_3:21;
  r(#)g = r(#)f by Th6;
  hence thesis by A2,Th23;
  end;
