reserve n,m,k for Element of NAT;
reserve x, X,X1,Z,Z1 for set;
reserve s,g,r,p,x0,x1,x2 for Real;
reserve s1,s2,q1 for Real_Sequence;
reserve Y for Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem
  Y c= dom f & Y is compact & f|Y is continuous implies (f.:Y) is compact
proof
  assume that
A1: Y c= dom f and
A2: Y is compact and
A3: f|Y is continuous;
A4: f|Y|Y is continuous by A3;
  dom (f|Y) = dom f /\ Y by RELAT_1:61
    .= Y by A1,XBOOLE_1:28;
  then rng (f|Y) is compact by A2,A4,Th28;
  hence thesis by RELAT_1:115;
end;
