let x0, r be Real; :: thesis:
let f be PartFunc of REAL ,REAL ; :: thesis:
thus ( f is_divergent_to+infty_in x0 & r > 0 implies r (#) f is_divergent_to+infty_in x0 ) :: thesis:

assume A1: ( f is_divergent_to+infty_in x0 & r > 0 ) ; :: thesis:
thus for r1, r2 being Real st r1 < x0 & x0 < r2 holds
ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) :: according to LIMFUNC3:def 2 :: thesis:

let r1, r2 be Real; :: thesis:
assume ( r1 < x0 & x0 < r2 ) ; :: thesis:
then consider g1, g2 being Real such that
A2: ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) by A1, Def2;
take g1 ; :: thesis:
take g2 ; :: thesis:
thus ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) by A2, VALUED_1:def 5; :: thesis:

end;

let seq be Real_Sequence; :: thesis:
assume A3: ( seq is convergent & lim seq = x0 & rng seq c= (dom (r (#) f)) \ {x0} ) ; :: thesis:
then A4: rng seq c= (dom f) \ {x0} by VALUED_1:def 5;
then f /* seq is divergent_to+infty by A1, A3, Def2;
then r (#) (f /* seq) is divergent_to+infty by A1, LIMFUNC1:40;
hence (r (#) f) /* seq is divergent_to+infty by A4, RFUNCT_2:24, XBOOLE_1:1; :: thesis:

end;

thus ( f is_divergent_to+infty_in x0 & r < 0 implies r (#) f is_divergent_to-infty_in x0 ) :: thesis:

assume A5: ( f is_divergent_to+infty_in x0 & r < 0 ) ; :: thesis:
thus for r1, r2 being Real st r1 < x0 & x0 < r2 holds
ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) :: according to LIMFUNC3:def 3 :: thesis:

let r1, r2 be Real; :: thesis:
assume ( r1 < x0 & x0 < r2 ) ; :: thesis:
then consider g1, g2 being Real such that
A6: ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) by A5, Def2;
take g1 ; :: thesis:
take g2 ; :: thesis:
thus ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) by A6, VALUED_1:def 5; :: thesis:

end;

let seq be Real_Sequence; :: thesis:
assume A7: ( seq is convergent & lim seq = x0 & rng seq c= (dom (r (#) f)) \ {x0} ) ; :: thesis:
then A8: rng seq c= (dom f) \ {x0} by VALUED_1:def 5;
then f /* seq is divergent_to+infty by A5, A7, Def2;
then r (#) (f /* seq) is divergent_to-infty by A5, LIMFUNC1:40;
hence (r (#) f) /* seq is divergent_to-infty by A8, RFUNCT_2:24, XBOOLE_1:1; :: thesis:

end;

thus ( f is_divergent_to-infty_in x0 & r > 0 implies r (#) f is_divergent_to-infty_in x0 ) :: thesis:

assume A9: ( f is_divergent_to-infty_in x0 & r > 0 ) ; :: thesis:
thus for r1, r2 being Real st r1 < x0 & x0 < r2 holds
ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) :: according to LIMFUNC3:def 3 :: thesis:

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 A9, Def3;
take g1 ; :: thesis:
take g2 ; :: thesis:
thus ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) by A10, VALUED_1:def 5; :: thesis:

end;

let seq be Real_Sequence; :: thesis:
assume A11: ( seq is convergent & lim seq = x0 & rng seq c= (dom (r (#) f)) \ {x0} ) ; :: thesis:
then A12: rng seq c= (dom f) \ {x0} by VALUED_1:def 5;
then f /* seq is divergent_to-infty by A9, A11, Def3;
then r (#) (f /* seq) is divergent_to-infty by A9, LIMFUNC1:41;
hence (r (#) f) /* seq is divergent_to-infty by A12, RFUNCT_2:24, XBOOLE_1:1; :: thesis:

end;

assume A13: ( f is_divergent_to-infty_in x0 & r < 0 ) ; :: thesis:
thus for r1, r2 being Real st r1 < x0 & x0 < r2 holds
ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) :: according to LIMFUNC3:def 2 :: thesis:

let r1, r2 be Real; :: thesis:
assume ( r1 < x0 & x0 < r2 ) ; :: thesis:
then consider g1, g2 being Real such that
A14: ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) by A13, Def3;
take g1 ; :: thesis:
take g2 ; :: thesis:
thus ( r1 < g1 & g1 < x0 & g1 in dom (r (#) f) & g2 < r2 & x0 < g2 & g2 in dom (r (#) f) ) by A14, VALUED_1:def 5; :: thesis:

end;

let seq be Real_Sequence; :: thesis:
assume A15: ( seq is convergent & lim seq = x0 & rng seq c= (dom (r (#) f)) \ {x0} ) ; :: thesis:
then A16: rng seq c= (dom f) \ {x0} by VALUED_1:def 5;
then f /* seq is divergent_to-infty by A13, A15, Def3;
then r (#) (f /* seq) is divergent_to+infty by A13, LIMFUNC1:41;
hence (r (#) f) /* seq is divergent_to+infty by A16, RFUNCT_2:24, XBOOLE_1:1; :: thesis: