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
  for r,p be Element of REAL n
    st (for x0 st x0 in X holds f/.x0 = x0*r+p)
  holds f|X is continuous
proof
  let r,p be Element of REAL n;
  assume A1: for x0 st x0 in X holds f/.x0 = x0*r+p;
  reconsider g= f as PartFunc of REAL,REAL-NS n
   by REAL_NS1:def 4;
  reconsider r0=r, p0=p as Point of REAL-NS n by REAL_NS1:def 4;
  now let x0;
 assume A2: x0 in X;
A3: x0*r = x0*r0 by REAL_NS1:3;
   thus g/.x0 = f/.x0 by REAL_NS1:def 4
        .= x0*r+p by A2,A1
        .= x0*r0+p0 by A3,REAL_NS1:2;
  end;
  then g|X is continuous by NFCONT_3:33;
  hence thesis by Th23;
end;
