let X be Subset of COMPLEX ; :: thesis: ( X is open implies for z0 being Complex st z0 in X holds
ex N being Neighbourhood of z0 st N c= X )
assume A1: X is open ; :: thesis: for z0 being Complex st z0 in X holds
ex N being Neighbourhood of z0 st N c= X
let z0 be Complex; :: thesis: ( z0 in X implies ex N being Neighbourhood of z0 st N c= X )
assume that
A2: z0 in X and
A3: for N being Neighbourhood of z0 holds not N c= X ; :: thesis: contradiction

A4: now

let g be Real; :: thesis: ( 0 < g implies ex s being Complex st
( s in { y where y is Complex : |.(y - z0).| < g } & s in X ` ) )
assume A5: 0 < g ; :: thesis: ex s being Complex st
( s in { y where y is Complex : |.(y - z0).| < g } & s in X ` )
set N = { y where y is Complex : |.(y - z0).| < g } ;
{ y where y is Complex : |.(y - z0).| < g } is Neighbourhood of z0 by A5, Th8;
then not { y where y is Complex : |.(y - z0).| < g } c= X by A3;
then consider x being set such that
A6: ( x in { y where y is Complex : |.(y - z0).| < g } & not x in X ) by TARSKI:def 3;
consider s being Complex such that
A7: ( x = s & |.(s - z0).| < g ) by A6;
take s = s; :: thesis: ( s in { y where y is Complex : |.(y - z0).| < g } & s in X ` )
thus s in { y where y is Complex : |.(y - z0).| < g } by A7; :: thesis: s in X `
thus s in X ` by A6, A7, XBOOLE_0:def 5; :: thesis: verum

end;

defpred S₁[ Element of NAT , Complex] means ( $2 in { y where y is Complex : |.(y - z0).| < 1 / ($1 + 1) } & $2 in X ` );
A8: for n being Element of NAT ex s being Complex st S₁[n,s] by A4;
consider s1 being Function of NAT ,COMPLEX such that
A9: for n being Element of NAT holds S₁[n,s1 . n] from FUNCT_2:sch 3(A8);

A10: now

let p be Real; :: thesis: ( 0 < p implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
|.((s1 . m) - z0).| < p )
assume A: 0 < p ; :: thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
|.((s1 . m) - z0).| < p
consider n being Element of NAT such that
A12: p " < n by SEQ_4:10;
(p " ) + 0 < n + 1 by A12, XREAL_1:10;
then A13: 1 / (n + 1) < 1 / (p " ) by A, XREAL_1:78;
take n = n; :: thesis: for m being Element of NAT st n <= m holds
|.((s1 . m) - z0).| < p
let m be Element of NAT ; :: thesis: ( n <= m implies |.((s1 . m) - z0).| < p )
assume n <= m ; :: thesis: |.((s1 . m) - z0).| < p
then A14: n + 1 <= m + 1 by XREAL_1:8;
A15: 1 / (m + 1) <= 1 / (n + 1) by A14, XREAL_1:120;
s1 . m in { y where y is Complex : |.(y - z0).| < 1 / (m + 1) } by A9;
then consider y being Complex such that
A16: ( s1 . m = y & |.(y - z0).| < 1 / (m + 1) ) ;
|.((s1 . m) - z0).| < 1 / (n + 1) by A15, A16, XXREAL_0:2;
hence |.((s1 . m) - z0).| < p by A13, XXREAL_0:2; :: thesis: verum

end;

then A17: s1 is convergent by COMSEQ_2:def 4;
then A18: lim s1 = z0 by A10, COMSEQ_2:def 5;
A19: rng s1 c= X `

proof

let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in rng s1 or y in X ` )
assume y in rng s1 ; :: thesis: y in X `
then consider y1 being set such that
A20: y1 in dom s1 and
A21: s1 . y1 = y by FUNCT_1:def 5;
reconsider y1 = y1 as Element of NAT by A20;
s1 . y1 in X ` by A9;
hence y in X ` by A21; :: thesis: verum

end;

X ` is closed by A1, Def11;
then z0 in COMPLEX \ X by A17, A18, A19, Def10;
hence contradiction by A2, XBOOLE_0:def 5; :: thesis: verum