let x0 be Real; :: thesis:
let f be PartFunc of REAL ,REAL ; :: thesis:
assume A1: ( f is_divergent_to+infty_in x0 or f is_divergent_to-infty_in x0 ) ; :: thesis:

per cases by A1;

suppose A2: f is_divergent_to+infty_in x0 ; :: thesis:

A3: now

let r1, r2 be Real; :: thesis:
assume ( r1 < x0 & x0 < r2 ) ; :: thesis:
then consider g1, g2 being Real such that
A4: ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) by A2, Def2;
take g1 = g1; :: thesis:
take g2 = g2; :: thesis:
thus ( r1 < g1 & g1 < x0 & g1 in dom (abs f) & g2 < r2 & x0 < g2 & g2 in dom (abs f) ) by A4, VALUED_1:def 11; :: thesis:

end;

now

let seq be Real_Sequence; :: thesis:
assume A5: ( seq is convergent & lim seq = x0 & rng seq c= (dom (abs f)) \ {x0} ) ; :: thesis:
then A6: rng seq c= (dom f) \ {x0} by VALUED_1:def 11;
then f /* seq is divergent_to+infty by A2, A5, Def2;
then A7: abs (f /* seq) is divergent_to+infty by LIMFUNC1:52;
rng seq c= dom f by A6, XBOOLE_1:1;
hence (abs f) /* seq is divergent_to+infty by A7, RFUNCT_2:25; :: thesis:

end;

hence abs f is_divergent_to+infty_in x0 by A3, Def2; :: thesis:

end;

suppose A8: f is_divergent_to-infty_in x0 ; :: thesis:

A9: now

let r1, r2 be Real; :: thesis:
assume ( r1 < x0 & x0 < r2 ) ; :: thesis:
then consider g1, g2 being Real such that
A10: ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) by A8, Def3;
take g1 = g1; :: thesis:
take g2 = g2; :: thesis:
thus ( r1 < g1 & g1 < x0 & g1 in dom (abs f) & g2 < r2 & x0 < g2 & g2 in dom (abs f) ) by A10, VALUED_1:def 11; :: thesis:

end;

now

let seq be Real_Sequence; :: thesis:
assume A11: ( seq is convergent & lim seq = x0 & rng seq c= (dom (abs f)) \ {x0} ) ; :: thesis:
then A12: rng seq c= (dom f) \ {x0} by VALUED_1:def 11;
then f /* seq is divergent_to-infty by A8, A11, Def3;
then A13: abs (f /* seq) is divergent_to+infty by LIMFUNC1:52;
rng seq c= dom f by A12, XBOOLE_1:1;
hence (abs f) /* seq is divergent_to+infty by A13, RFUNCT_2:25; :: thesis:

end;

hence abs f is_divergent_to+infty_in x0 by A9, Def2; :: thesis:

end;

hence abs f is_divergent_to+infty_in x0 ; :: thesis: