let x0 be Real; :: thesis:
let f be PartFunc of REAL ,REAL ; :: thesis:
assume A1: ( f is_convergent_in x0 & lim f,x0 <> 0 & ( 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:
A2: (dom f) \ (f " {0 }) = dom (f ^ ) by RFUNCT_1:def 8;

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 & f . g1 <> 0 & f . g2 <> 0 ) by A1;
take g1 = g1; :: thesis:
take g2 = g2; :: thesis:
( not f . g1 in {0 } & not f . g2 in {0 } ) by A4, TARSKI:def 1;
then ( not g1 in f " {0 } & not g2 in f " {0 } ) by FUNCT_1:def 13;
hence ( r1 < g1 & g1 < x0 & g1 in dom (f ^ ) & g2 < r2 & x0 < g2 & g2 in dom (f ^ ) ) by A2, A4, XBOOLE_0:def 5; :: thesis:

end;

A5: now

let seq be Real_Sequence; :: thesis:
assume A6: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f ^ )) \ {x0} ) ; :: thesis:
then rng seq c= dom (f ^ ) by XBOOLE_1:1;
then A7: rng seq c= dom f by A2, XBOOLE_1:1;

let x be set ; :: thesis:
assume A8: x in rng seq ; :: thesis:
then not x in {x0} by A6, XBOOLE_0:def 5;
hence x in (dom f) \ {x0} by A7, A8, XBOOLE_0:def 5; :: thesis:

end;

then rng seq c= (dom f) \ {x0} by TARSKI:def 3;
then A9: ( f /* seq is convergent & lim (f /* seq) = lim f,x0 ) by A1, A6, Def4;
A10: f /* seq is non-zero by A6, RFUNCT_2:26, XBOOLE_1:1;
A11: (f /* seq) " = (f ^ ) /* seq by A6, RFUNCT_2:27, XBOOLE_1:1;
hence (f ^ ) /* seq is convergent by A1, A9, A10, SEQ_2:35; :: thesis:
thus lim ((f ^ ) /* seq) = (lim f,x0) " by A1, A9, A10, A11, SEQ_2:36; :: thesis:

end;

hence f ^ is_convergent_in x0 by A3, Def1; :: thesis:
hence lim (f ^ ),x0 = (lim f,x0) " by A5, Def4; :: thesis: