let x0, g be Real; for f being PartFunc of REAL,REAL holds
( f is_differentiable_in x0 & diff (f,x0) = g iff ( ex N being Neighbourhood of x0 st N c= dom f & ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g ) ) ) )
let f be PartFunc of REAL,REAL; ( f is_differentiable_in x0 & diff (f,x0) = g iff ( ex N being Neighbourhood of x0 st N c= dom f & ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g ) ) ) )
thus
( f is_differentiable_in x0 & diff (f,x0) = g implies ( ex N being Neighbourhood of x0 st N c= dom f & ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g ) ) ) )
( ex N being Neighbourhood of x0 st N c= dom f & ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g ) ) implies ( f is_differentiable_in x0 & diff (f,x0) = g ) )
assume that
A4:
ex N being Neighbourhood of x0 st N c= dom f
and
A5:
for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g )
; ( f is_differentiable_in x0 & diff (f,x0) = g )
A6:
for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
(h ") (#) ((f /* (h + c)) - (f /* c)) is convergent
by A5;
hence
f is_differentiable_in x0
by A4, Lm1; diff (f,x0) = g
consider h being non-zero 0 -convergent Real_Sequence, c being V8() Real_Sequence such that
A7:
rng c = {x0}
and
A8:
rng (h + c) c= dom f
and
{x0} c= dom f
by A4, Th8;
lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g
by A5, A7, A8;
hence
diff (f,x0) = g
by A4, A6, A7, A8, Lm1; verum