let n be non zero Element of NAT ; :: thesis: for Z being open Subset of REAL

for f being PartFunc of REAL,(REAL n) st Z c= dom f & f | Z is constant holds

( f is_differentiable_on Z & ( for x being Real st x in Z holds

(f `| Z) . x = 0* n ) )

let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,(REAL n) st Z c= dom f & f | Z is constant holds

( f is_differentiable_on Z & ( for x being Real st x in Z holds

(f `| Z) . x = 0* n ) )

let f be PartFunc of REAL,(REAL n); :: thesis: ( Z c= dom f & f | Z is constant implies ( f is_differentiable_on Z & ( for x being Real st x in Z holds

(f `| Z) . x = 0* n ) ) )

assume that

A1: Z c= dom f and

A2: f | Z is constant ; :: thesis: ( f is_differentiable_on Z & ( for x being Real st x in Z holds

(f `| Z) . x = 0* n ) )

reconsider g = f as PartFunc of REAL,(REAL-NS n) by REAL_NS1:def 4;

A3: g | Z is constant by A2;

then A4: ( g is_differentiable_on Z & ( for x being Real st x in Z holds

(g `| Z) . x = 0. (REAL-NS n) ) ) by A1, NDIFF_3:20;

(f `| Z) . x = 0* n

let x be Real; :: thesis: ( x in Z implies (f `| Z) . x = 0* n )

assume A6: x in Z ; :: thesis: (f `| Z) . x = 0* n

then A7: (g `| Z) . x = 0. (REAL-NS n) by A3, A1, NDIFF_3:20;

A8: (g `| Z) . x = diff (g,x) by A6, A4, NDIFF_3:def 6;

A9: (f `| Z) . x = diff (f,x) by A6, A5, Def4;

diff (f,x) = diff (g,x) by Th3;

hence (f `| Z) . x = 0* n by A7, A8, A9, REAL_NS1:def 4; :: thesis: verum

for f being PartFunc of REAL,(REAL n) st Z c= dom f & f | Z is constant holds

( f is_differentiable_on Z & ( for x being Real st x in Z holds

(f `| Z) . x = 0* n ) )

let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,(REAL n) st Z c= dom f & f | Z is constant holds

( f is_differentiable_on Z & ( for x being Real st x in Z holds

(f `| Z) . x = 0* n ) )

let f be PartFunc of REAL,(REAL n); :: thesis: ( Z c= dom f & f | Z is constant implies ( f is_differentiable_on Z & ( for x being Real st x in Z holds

(f `| Z) . x = 0* n ) ) )

assume that

A1: Z c= dom f and

A2: f | Z is constant ; :: thesis: ( f is_differentiable_on Z & ( for x being Real st x in Z holds

(f `| Z) . x = 0* n ) )

reconsider g = f as PartFunc of REAL,(REAL-NS n) by REAL_NS1:def 4;

A3: g | Z is constant by A2;

then A4: ( g is_differentiable_on Z & ( for x being Real st x in Z holds

(g `| Z) . x = 0. (REAL-NS n) ) ) by A1, NDIFF_3:20;

now :: thesis: for x being Real st x in Z holds

f | Z is_differentiable_in x

hence A5:
f is_differentiable_on Z
by A1; :: thesis: for x being Real st x in Z holds f | Z is_differentiable_in x

let x be Real; :: thesis: ( x in Z implies f | Z is_differentiable_in x )

assume x in Z ; :: thesis: f | Z is_differentiable_in x

then g | Z is_differentiable_in x by A4, NDIFF_3:def 5;

hence f | Z is_differentiable_in x ; :: thesis: verum

end;assume x in Z ; :: thesis: f | Z is_differentiable_in x

then g | Z is_differentiable_in x by A4, NDIFF_3:def 5;

hence f | Z is_differentiable_in x ; :: thesis: verum

(f `| Z) . x = 0* n

let x be Real; :: thesis: ( x in Z implies (f `| Z) . x = 0* n )

assume A6: x in Z ; :: thesis: (f `| Z) . x = 0* n

then A7: (g `| Z) . x = 0. (REAL-NS n) by A3, A1, NDIFF_3:20;

A8: (g `| Z) . x = diff (g,x) by A6, A4, NDIFF_3:def 6;

A9: (f `| Z) . x = diff (f,x) by A6, A5, Def4;

diff (f,x) = diff (g,x) by Th3;

hence (f `| Z) . x = 0* n by A7, A8, A9, REAL_NS1:def 4; :: thesis: verum