reserve n,m,i,k for Element of NAT;
reserve x,X,X1 for set;
reserve r,p for Real;
reserve s,x0,x1,x2 for Real;
reserve f,f1,f2 for PartFunc of REAL,REAL n;
reserve h for PartFunc of REAL,REAL-NS n;
reserve W for non empty set;

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;
