reserve n,m for Nat;
reserve x,X,X1 for set;
reserve s,g,r,p for Real;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve s1,s2 for sequence of S;
reserve x0,x1,x2 for Point of S;
reserve Y for Subset of S;

theorem
  Y c= dom f & Y is compact & f is_continuous_on Y implies (f.:Y) is compact
proof
  assume that
A1: Y c= dom f and
A2: Y is compact and
A3: f is_continuous_on Y;
A4: dom (f|Y) = dom f /\ Y by RELAT_1:61
    .= Y by A1,XBOOLE_1:28;
  f|Y is_continuous_on Y
  proof
    thus Y c= dom (f|Y) by A4;
    let r be Point of S;
    assume r in Y;
    then f|Y is_continuous_in r by A3;
    hence thesis by RELAT_1:72;
  end;
  then rng (f|Y) is compact by A2,A4,Th30;
  hence thesis by RELAT_1:115;
end;
