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, Z be Subset of REAL-NS n st
  Y c= dom f & Z = f.:Y & Y is compact & f|Y is continuous
  holds Z is compact
  proof
    let Y be Subset of REAL, Z be Subset of REAL-NS n;
    assume
A1: Y c= dom f & Z = (f.:Y) & Y is compact & f|Y is continuous;
    reconsider g = f as PartFunc of REAL,REAL-NS n
    by REAL_NS1:def 4;
    g|Y is continuous by A1,Th23;
    hence Z is compact by A1,NFCONT_3:25;
  end;
