ex r being Real st
( r > 0 & [.(x0 - r),x0.] c= dom f ) by A1, Def4;
then consider h1 being convergent_to_0 Real_Sequence, c1 being V8() Real_Sequence such that
A2: ( rng c1 = {x0} & rng (h1 + c1) c= dom f & ( for n being Element of NAT holds h1 . n < 0 ) ) by Th1;
take lim ((h1 " ) (#) ((f /* (h1 + c1)) - (f /* c1))) ; :: thesis: for h being convergent_to_0 Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f & ( for n being Element of NAT holds h . n < 0 ) holds
lim ((h1 " ) (#) ((f /* (h1 + c1)) - (f /* c1))) = lim ((h " ) (#) ((f /* (h + c)) - (f /* c)))
A3: {x0} c= dom f

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in {x0} or x in dom f )
assume x in {x0} ; :: thesis: x in dom f
then A4: x = x0 by TARSKI:def 1;
consider r being Real such that
A5: ( r > 0 & [.(x0 - r),x0.] c= dom f ) by A1, Def4;
x0 - r <= x0 by A5, XREAL_1:46;
then x0 in [.(x0 - r),x0.] by XXREAL_1:1;
hence x in dom f by A4, A5; :: thesis: verum

end;

let h be convergent_to_0 Real_Sequence; :: thesis: for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f & ( for n being Element of NAT holds h . n < 0 ) holds
lim ((h1 " ) (#) ((f /* (h1 + c1)) - (f /* c1))) = lim ((h " ) (#) ((f /* (h + c)) - (f /* c)))
let c be V8() Real_Sequence; :: thesis: ( rng c = {x0} & rng (h + c) c= dom f & ( for n being Element of NAT holds h . n < 0 ) implies lim ((h1 " ) (#) ((f /* (h1 + c1)) - (f /* c1))) = lim ((h " ) (#) ((f /* (h + c)) - (f /* c))) )
assume A6: ( rng c = {x0} & rng (h + c) c= dom f & ( for n being Element of NAT holds h . n < 0 ) ) ; :: thesis: lim ((h1 " ) (#) ((f /* (h1 + c1)) - (f /* c1))) = lim ((h " ) (#) ((f /* (h + c)) - (f /* c)))
A7: for h being convergent_to_0 Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f & ( for n being Element of NAT holds h . n < 0 ) holds
(h " ) (#) ((f /* (h + c)) - (f /* c)) is convergent by A1, Def4;
c1 = c by A2, A6, FDIFF_2:5;
hence lim ((h1 " ) (#) ((f /* (h1 + c1)) - (f /* c1))) = lim ((h " ) (#) ((f /* (h + c)) - (f /* c))) by A2, A3, A6, A7, Th3; :: thesis: verum