let f be PartFunc of REAL,REAL; :: thesis: for x0, g1 being Real holds

( f is_right_differentiable_in x0 & Rdiff (f,x0) = g1 iff ( ex r being Real st

( r > 0 & [.x0,(x0 + r).] c= dom f ) & ( 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

( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g1 ) ) ) )

let x0, g1 be Real; :: thesis: ( f is_right_differentiable_in x0 & Rdiff (f,x0) = g1 iff ( ex r being Real st

( r > 0 & [.x0,(x0 + r).] c= dom f ) & ( 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

( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g1 ) ) ) )

thus ( f is_right_differentiable_in x0 & Rdiff (f,x0) = g1 implies ( ex r being Real st

( r > 0 & [.x0,(x0 + r).] c= dom f ) & ( 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

( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g1 ) ) ) ) by Def6; :: thesis: ( ex r being Real st

( r > 0 & [.x0,(x0 + r).] c= dom f ) & ( 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

( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g1 ) ) implies ( f is_right_differentiable_in x0 & Rdiff (f,x0) = g1 ) )

assume that

A1: ex r being Real st

( r > 0 & [.x0,(x0 + r).] c= dom f ) and

A2: 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

( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g1 ) ; :: thesis: ( f is_right_differentiable_in x0 & Rdiff (f,x0) = g1 )

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

(h ") (#) ((f /* (h + c)) - (f /* c)) is convergent by A2;

hence A3: f is_right_differentiable_in x0 by A1; :: thesis: Rdiff (f,x0) = g1

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

lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g1 by A2;

hence Rdiff (f,x0) = g1 by A3, Def6; :: thesis: verum

( f is_right_differentiable_in x0 & Rdiff (f,x0) = g1 iff ( ex r being Real st

( r > 0 & [.x0,(x0 + r).] c= dom f ) & ( 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

( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g1 ) ) ) )

let x0, g1 be Real; :: thesis: ( f is_right_differentiable_in x0 & Rdiff (f,x0) = g1 iff ( ex r being Real st

( r > 0 & [.x0,(x0 + r).] c= dom f ) & ( 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

( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g1 ) ) ) )

thus ( f is_right_differentiable_in x0 & Rdiff (f,x0) = g1 implies ( ex r being Real st

( r > 0 & [.x0,(x0 + r).] c= dom f ) & ( 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

( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g1 ) ) ) ) by Def6; :: thesis: ( ex r being Real st

( r > 0 & [.x0,(x0 + r).] c= dom f ) & ( 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

( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g1 ) ) implies ( f is_right_differentiable_in x0 & Rdiff (f,x0) = g1 ) )

assume that

A1: ex r being Real st

( r > 0 & [.x0,(x0 + r).] c= dom f ) and

A2: 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

( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g1 ) ; :: thesis: ( f is_right_differentiable_in x0 & Rdiff (f,x0) = g1 )

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

(h ") (#) ((f /* (h + c)) - (f /* c)) is convergent by A2;

hence A3: f is_right_differentiable_in x0 by A1; :: thesis: Rdiff (f,x0) = g1

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

lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g1 by A2;

hence Rdiff (f,x0) = g1 by A3, Def6; :: thesis: verum