reserve n,m,k for Nat;
reserve x,X,X1 for set;
reserve r,p for Real;
reserve s,g,x0,x1,x2 for Real;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of REAL,the carrier of S;
reserve s1,s2 for Real_Sequence;
reserve Y for Subset of 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,Th24;
   hence thesis by RELAT_1:115;
end;
