let x0 be Real; :: thesis:
let f be PartFunc of REAL,REAL; :: thesis:
A1: (dom f) \ (f " {0}) = dom (f ^) by RFUNCT_1:def 2;
assume A2: ( f is_divergent_to+infty_in x0 or f is_divergent_to-infty_in x0 ) ; :: thesis:

A3: now :: thesis:

let seq be Real_Sequence; :: thesis:
assume that
A4: seq is convergent and
A5: lim seq = x0 and
A6: rng seq c= (dom (f ^)) \ {x0} ; :: thesis:
rng seq c= dom (f ^) by A6, XBOOLE_1:1;
then A7: rng seq c= dom f by A1, XBOOLE_1:1;
A8: rng seq c= (dom f) \ {x0}

let x be object ; :: according to TARSKI:def 3 :: thesis:
assume A9: x in rng seq ; :: thesis:
then not x in {x0} by A6, XBOOLE_0:def 5;
hence x in (dom f) \ {x0} by A7, A9, XBOOLE_0:def 5; :: thesis:

end;

per cases by A2;

suppose f is_divergent_to+infty_in x0 ; :: thesis:

then A10: f /* seq is divergent_to+infty by A4, A5, A8;
then A11: lim ((f /* seq) ") = 0 by LIMFUNC1:34;
(f /* seq) " is convergent by A10, LIMFUNC1:34;
hence ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) by A6, A11, RFUNCT_2:12, XBOOLE_1:1; :: thesis:

end;

suppose f is_divergent_to-infty_in x0 ; :: thesis:

then A12: f /* seq is divergent_to-infty by A4, A5, A8;
then A13: lim ((f /* seq) ") = 0 by LIMFUNC1:34;
(f /* seq) " is convergent by A12, LIMFUNC1:34;
hence ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) by A6, A13, RFUNCT_2:12, XBOOLE_1:1; :: thesis:

end;

end;

end;

hence ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) ; :: thesis:

end;

assume A14: for r1, r2 being Real st r1 < x0 & x0 < r2 holds
ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f & f . g1 <> 0 & f . g2 <> 0 ) ; :: thesis:

let r1, r2 be Real; :: thesis:
assume that
A15: r1 < x0 and
A16: x0 < r2 ; :: thesis:
consider g1, g2 being Real such that
A17: r1 < g1 and
A18: g1 < x0 and
A19: g1 in dom f and
A20: g2 < r2 and
A21: x0 < g2 and
A22: g2 in dom f and
A23: f . g1 <> 0 and
A24: f . g2 <> 0 by A14, A15, A16;
take g1 = g1; :: thesis:
take g2 = g2; :: thesis:
not f . g2 in {0} by A24, TARSKI:def 1;
then A25: not g2 in f " {0} by FUNCT_1:def 7;
not f . g1 in {0} by A23, TARSKI:def 1;
then not g1 in f " {0} by FUNCT_1:def 7;
hence ( r1 < g1 & g1 < x0 & g1 in dom (f ^) & g2 < r2 & x0 < g2 & g2 in dom (f ^) ) by A1, A17, A18, A19, A20, A21, A22, A25, XBOOLE_0:def 5; :: thesis:

end;

hence f ^ is_convergent_in x0 by A3; :: thesis:
hence lim ((f ^),x0) = 0 by A3, Def4; :: thesis: