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 g;
    assume g in Y;
    then f|Y is_continuous_in g by A3;
    hence thesis by RELAT_1:72;
  end;
  then rng (f|Y) is compact by A2,A4,Th51;
  hence thesis by RELAT_1:115;
end;
