let Y be Subset of COMPLEX ; :: thesis: for f being PartFunc of COMPLEX ,COMPLEX st Y c= dom f & Y is compact & f is_continuous_on Y holds
f .: Y is compact
let f be PartFunc of COMPLEX ,COMPLEX ; :: thesis: ( Y c= dom f & Y is compact & f is_continuous_on Y implies f .: Y is compact )
assume that
A1: Y c= dom f and
A2: Y is compact and
A3: f is_continuous_on Y ; :: thesis: f .: Y is compact
A4: dom (f | Y) = (dom f) /\ Y by RELAT_1:90
.= Y by A1, XBOOLE_1:28 ;
f | Y is_continuous_on Y

proof

thus Y c= dom (f | Y) by A4; :: according to CFCONT_1:def 5 :: thesis: for x0 being Element of COMPLEX st x0 in Y holds
(f | Y) | Y is_continuous_in x0
let g be Element of COMPLEX ; :: thesis: ( g in Y implies (f | Y) | Y is_continuous_in g )
assume g in Y ; :: thesis: (f | Y) | Y is_continuous_in g
then f | Y is_continuous_in g by A3, Def5;
hence (f | Y) | Y is_continuous_in g by RELAT_1:101; :: thesis: verum

end;

then rng (f | Y) is compact by A2, A4, Th73;
hence f .: Y is compact by RELAT_1:148; :: thesis: verum