reserve n,m for Element of NAT;
reserve r,s for Real;
reserve z for Complex;
reserve CNS,CNS1,CNS2 for ComplexNormSpace;
reserve RNS for RealNormSpace;
reserve X,X1 for set;

theorem
  for Y be Subset of CNS1, f be PartFunc of CNS1,CNS2 st Y c= dom f & Y
  is compact & f is_continuous_on Y holds (f.:Y) is compact
proof
  let Y be Subset of CNS1;
  let f be PartFunc of CNS1,CNS2;
  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 CNS1;
    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,Th77;
  hence thesis by RELAT_1:115;
end;
