let CNS1, CNS2 be ComplexNormSpace; :: thesis: for f being PartFunc of CNS1,CNS2 st dom f is compact & f is_continuous_on dom f holds
rng f is compact
let f be PartFunc of CNS1,CNS2; :: thesis: ( dom f is compact & f is_continuous_on dom f implies rng f is compact )
assume that
A1: dom f is compact and
A2: f is_continuous_on dom f ; :: thesis: rng f is compact

now

let s1 be sequence of CNS2; :: thesis: ( rng s1 c= rng f implies ex q2 being Element of K21(K22(NAT ,the carrier of CNS2)) st
( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f ) )
assume A3: rng s1 c= rng f ; :: thesis: ex q2 being Element of K21(K22(NAT ,the carrier of CNS2)) st
( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f )
defpred S₁[ set , set ] means ( $2 in dom f & f /. $2 = s1 . $1 );
A4: for n being Element of NAT ex p being Point of CNS1 st S₁[n,p]

proof

let n be Element of NAT ; :: thesis: ex p being Point of CNS1 st S₁[n,p]
dom s1 = NAT by FUNCT_2:def 1;
then s1 . n in rng s1 by FUNCT_1:12;
then consider p being Point of CNS1 such that
A5: ( p in dom f & s1 . n = f . p ) by A3, PARTFUN1:26;
take p ; :: thesis: S₁[n,p]
thus S₁[n,p] by A5, PARTFUN1:def 8; :: thesis: verum

end;

consider q1 being sequence of CNS1 such that
A6: for n being Element of NAT holds S₁[n,q1 . n] from FUNCT_2:sch 3(A4);

now

let x be set ; :: thesis: ( x in rng q1 implies x in dom f )
assume x in rng q1 ; :: thesis: x in dom f
then ex n being Element of NAT st x = q1 . n by Th8;
hence x in dom f by A6; :: thesis: verum

end;

then A7: rng q1 c= dom f by TARSKI:def 3;

now

let n be Element of NAT ; :: thesis: (f /* q1) . n = s1 . n
( q1 . n in dom f & f /. (q1 . n) = s1 . n ) by A6;
hence (f /* q1) . n = s1 . n by A7, FUNCT_2:186; :: thesis: verum

end;

then A8: f /* q1 = s1 by FUNCT_2:113;
consider s2 being sequence of CNS1 such that
A9: ( s2 is subsequence of q1 & s2 is convergent & lim s2 in dom f ) by A1, A7, Def6;
f | (dom f) is_continuous_in lim s2 by A2, A9, Def21;
then A10: f is_continuous_in lim s2 by RELAT_1:97;
rng s2 c= rng q1 by A9, VALUED_0:21;
then rng s2 c= dom f by A7, XBOOLE_1:1;
then A11: ( f /* s2 is convergent & f /. (lim s2) = lim (f /* s2) ) by A9, A10, Def15;
take q2 = f /* s2; :: thesis: ( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f )
thus ( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f ) by A7, A8, A9, A11, PARTFUN2:4, VALUED_0:22; :: thesis: verum

end;

hence rng f is compact by Def6; :: thesis: verum