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 Y be Subset of REAL-NS n st
  dom f is compact & f| (dom f) is continuous & Y = rng f
  holds Y is compact
proof
  let Y be Subset of REAL-NS n;
  assume A1:dom f is compact & f| (dom f) is continuous & Y = (rng f);
  reconsider g= f as PartFunc of REAL,REAL-NS n
    by REAL_NS1:def 4;
  g| (dom g) is continuous by A1,Th23;
  hence Y is compact by A1,NFCONT_3:24;
end;
