ex r being Real st
( r > 0 & [.(x0 - r),x0.] c= dom f ) by A1;
then consider h being non-zero 0 -convergent Real_Sequence, c being constant Real_Sequence such that
A11: ( rng c = {x0} & rng (h + c) c= dom f & ( for n being Nat holds h . n < 0 ) ) by Th1;
let g1, g2 be Real; :: thesis: ( ( for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {x0} & rng (h + c) c= dom f & ( for n being Nat holds h . n < 0 ) holds
g1 = lim ((h ") (#) ((f /* (h + c)) - (f /* c))) ) & ( for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {x0} & rng (h + c) c= dom f & ( for n being Nat holds h . n < 0 ) holds
g2 = lim ((h ") (#) ((f /* (h + c)) - (f /* c))) ) implies g1 = g2 )

assume that
A12: for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {x0} & rng (h + c) c= dom f & ( for n being Nat holds h . n < 0 ) holds
g1 = lim ((h ") (#) ((f /* (h + c)) - (f /* c))) and
A13: for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {x0} & rng (h + c) c= dom f & ( for n being Nat holds h . n < 0 ) holds
g2 = lim ((h ") (#) ((f /* (h + c)) - (f /* c))) ; :: thesis: g1 = g2
g1 = lim ((h ") (#) ((f /* (h + c)) - (f /* c))) by ;
hence g1 = g2 by ; :: thesis: verum