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
  f is_continuous_in x0 implies r(#)f is_continuous_in x0
  proof
    assume A1:f is_continuous_in x0;
    reconsider g = f as PartFunc of REAL,REAL-NS n
      by REAL_NS1:def 4;
    g is_continuous_in x0 by A1; then
A2: r(#)g is_continuous_in x0 by NFCONT_3:13;
    r(#)g = r(#)f by Th6;
    hence thesis by A2;
  end;
