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,() 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. () ) ) by ;
now :: thesis: for x being Real st x in Z holds
f | Z is_differentiable_in x
end;
hence A5: f is_differentiable_on Z by A1; :: thesis: for x being Real st x in Z holds
(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. () by ;
A8: (g `| Z) . x = diff (g,x) by ;
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 ; :: thesis: verum