let n be non zero Element of NAT ; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL,(REAL n)
for r, p being Element of REAL n st Z c= dom f & ( for x being Real st x in Z holds
f /. x = (x * r) + p ) holds
( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = r ) )

let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,(REAL n)
for r, p being Element of REAL n st Z c= dom f & ( for x being Real st x in Z holds
f /. x = (x * r) + p ) holds
( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = r ) )

let f be PartFunc of REAL,(REAL n); :: thesis: for r, p being Element of REAL n st Z c= dom f & ( for x being Real st x in Z holds
f /. x = (x * r) + p ) holds
( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = r ) )

let r, p be Element of REAL n; :: thesis: ( Z c= dom f & ( for x being Real st x in Z holds
f /. x = (x * r) + p ) implies ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = r ) ) )

assume that
A1: Z c= dom f and
A2: for x being Real st x in Z holds
f /. x = (x * r) + p ; :: thesis: ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = r ) )

reconsider g = f as PartFunc of REAL,() by REAL_NS1:def 4;
reconsider r1 = r, p1 = p as Point of () by REAL_NS1:def 4;
A3: now :: thesis: for x being Real st x in Z holds
g /. x = (x * r1) + p1
let x be Real; :: thesis: ( x in Z implies g /. x = (x * r1) + p1 )
assume x in Z ; :: thesis: g /. x = (x * r1) + p1
then A4: f /. x = (x * r) + p by A2;
A5: f /. x = g /. x by REAL_NS1:def 4;
x * r = x * r1 by REAL_NS1:3;
hence g /. x = (x * r1) + p1 by ; :: thesis: verum
end;
then A6: ( g is_differentiable_on Z & ( for x being Real st x in Z holds
(g `| Z) . x = r1 ) ) by ;
now :: thesis: for x being Real st x in Z holds
f | Z is_differentiable_in x
end;
hence A7: f is_differentiable_on Z by A1; :: thesis: for x being Real st x in Z holds
(f `| Z) . x = r

let x be Real; :: thesis: ( x in Z implies (f `| Z) . x = r )
assume A8: x in Z ; :: thesis: (f `| Z) . x = r
then A9: (g `| Z) . x = diff (g,x) by ;
A10: (f `| Z) . x = diff (f,x) by A8, A7, Def4;
diff (f,x) = diff (g,x) by Th3;
hence (f `| Z) . x = r by ; :: thesis: verum