let X be open Subset of REAL; :: thesis:
let r be Real; :: thesis:
assume that
A1: r in X and
A2: for N being Neighbourhood of r holds not N c= X ; :: thesis:
defpred S₁[ Nat, Real] means ( $2 in ].(r - (1 / ($1 + 1))),(r + (1 / ($1 + 1))).[ & $2 in X ` );

let N be Neighbourhood of r; :: thesis:
consider g being Real such that
0 < g and
A4: N = ].(r - g),(r + g).[ by Def6;
not N c= X by A2;
then consider x being object such that
A5: x in N and
A6: not x in X ;
consider s being Real such that
A7: x = s and
r - g < s and
s < r + g by A5, A4;
reconsider s = s as Element of REAL by XREAL_0:def 1;
take s = s; :: thesis:
thus s in N by A5, A7; :: thesis:
thus s in X ` by A6, A7, XBOOLE_0:def 5; :: thesis:

end;

A8: for n being Element of NAT ex s being Element of REAL st S₁[n,s]

let n be Element of NAT ; :: thesis:
0 < 1 * ((n + 1) ") ;
then 0 < 1 / (n + 1) by XCMPLX_0:def 9;
then reconsider N = ].(r - (1 / (n + 1))),(r + (1 / (n + 1))).[ as Neighbourhood of r by Def6;
ex s being Element of REAL st
( s in N & s in X ` ) by A3;
hence ex s being Element of REAL st S₁[n,s] ; :: thesis:

end;

consider s3 being Real_Sequence such that
A9: for n being Element of NAT holds S₁[n,s3 . n] from FUNCT_2:sch 3(A8);
deffunc H₁( Nat) -> set = r + (1 / ($1 + 1));
deffunc H₂( Nat) -> set = r - (1 / ($1 + 1));
consider s1 being Real_Sequence such that
A10: for n being Nat holds s1 . n = H₂(n) from SEQ_1:sch 1();
consider s2 being Real_Sequence such that
A11: for n being Nat holds s2 . n = H₁(n) from SEQ_1:sch 1();
A12: for n being Nat holds
( s1 . n <= s3 . n & s3 . n <= s2 . n )

let n be Nat; :: thesis:
n in NAT by ORDINAL1:def 12;
then s3 . n in ].(r - (1 / (n + 1))),(r + (1 / (n + 1))).[ by A9;
then A13: ex s being Real st
( s = s3 . n & r - (1 / (n + 1)) < s & s < r + (1 / (n + 1)) ) ;
hence s1 . n <= s3 . n by A10; :: thesis:
thus s3 . n <= s2 . n by A11, A13; :: thesis:

end;

let x be object ; :: according to TARSKI:def 3 :: thesis:
assume x in rng s3 ; :: thesis:
then consider y being object such that
A15: y in dom s3 and
A16: s3 . y = x by FUNCT_1:def 3;
reconsider y = y as Element of NAT by A15, FUNCT_2:def 1;
s3 . y in X ` by A9;
hence x in X ` by A16; :: thesis:

end;

let s be Real; :: thesis:
assume A19: 0 < s ; :: thesis:
consider n being Nat such that
A20: s " < n by SEQ_4:3;
take n = n; :: thesis:
let m be Nat; :: thesis:
assume n <= m ; :: thesis:
then n + 1 <= m + 1 by XREAL_1:6;
then A21: 1 / (m + 1) <= 1 / (n + 1) by XREAL_1:118;
(s ") + 0 < n + 1 by A20, XREAL_1:8;
then 1 / (n + 1) < 1 / (s ") by A19, XREAL_1:76;
then A22: 1 / (n + 1) < s by XCMPLX_1:216;
|.((s2 . m) - r).| = |.((r + (1 / (m + 1))) - r).| by A11
.= 1 / (m + 1) by ABSVALUE:def 1 ;
hence |.((s2 . m) - r).| < s by A22, A21, XXREAL_0:2; :: thesis:

end;

let s be Real; :: thesis:
assume A26: 0 < s ; :: thesis:
consider n being Nat such that
A27: s " < n by SEQ_4:3;
take n = n; :: thesis:
let m be Nat; :: thesis:
assume n <= m ; :: thesis:
then n + 1 <= m + 1 by XREAL_1:6;
then A28: 1 / (m + 1) <= 1 / (n + 1) by XREAL_1:118;
(s ") + 0 < n + 1 by A27, XREAL_1:8;
then 1 / (n + 1) < 1 / (s ") by A26, XREAL_1:76;
then A29: 1 / (n + 1) < s by XCMPLX_1:216;
|.((s1 . m) - r).| = |.((r - (1 / (m + 1))) - r).| by A10
.= |.(- (1 / (m + 1))).|
.= |.(1 / (m + 1)).| by COMPLEX1:52
.= 1 / (m + 1) by ABSVALUE:def 1 ;
hence |.((s1 . m) - r).| < s by A29, A28, XXREAL_0:2; :: thesis:

end;