let Z be Subset of (REAL 2); :: thesis: for f being PartFunc of (REAL 2),REAL st f is_partial_differentiable`1_on Z holds

( Z c= dom f & ( for z being Element of REAL 2 st z in Z holds

f is_partial_differentiable_in z,1 ) )

let f be PartFunc of (REAL 2),REAL; :: thesis: ( f is_partial_differentiable`1_on Z implies ( Z c= dom f & ( for z being Element of REAL 2 st z in Z holds

f is_partial_differentiable_in z,1 ) ) )

set g = f | Z;

assume A1: f is_partial_differentiable`1_on Z ; :: thesis: ( Z c= dom f & ( for z being Element of REAL 2 st z in Z holds

f is_partial_differentiable_in z,1 ) )

hence Z c= dom f ; :: thesis: for z being Element of REAL 2 st z in Z holds

f is_partial_differentiable_in z,1

let z0 be Element of REAL 2; :: thesis: ( z0 in Z implies f is_partial_differentiable_in z0,1 )

assume z0 in Z ; :: thesis: f is_partial_differentiable_in z0,1

then f | Z is_partial_differentiable_in z0,1 by A1;

then consider x0, y0 being Real such that

A2: z0 = <*x0,y0*> and

A3: ex N being Neighbourhood of x0 st

( N c= dom (SVF1 (1,(f | Z),z0)) & ex L being LinearFunc ex R being RestFunc st

for x being Real st x in N holds

((SVF1 (1,(f | Z),z0)) . x) - ((SVF1 (1,(f | Z),z0)) . x0) = (L . (x - x0)) + (R . (x - x0)) ) by Th9;

consider N being Neighbourhood of x0 such that

A4: N c= dom (SVF1 (1,(f | Z),z0)) and

A5: ex L being LinearFunc ex R being RestFunc st

for x being Real st x in N holds

((SVF1 (1,(f | Z),z0)) . x) - ((SVF1 (1,(f | Z),z0)) . x0) = (L . (x - x0)) + (R . (x - x0)) by A3;

consider L being LinearFunc, R being RestFunc such that

A6: for x being Real st x in N holds

((SVF1 (1,(f | Z),z0)) . x) - ((SVF1 (1,(f | Z),z0)) . x0) = (L . (x - x0)) + (R . (x - x0)) by A5;

A7: for x being Real st x in N holds

((SVF1 (1,f,z0)) . x) - ((SVF1 (1,f,z0)) . x0) = (L . (x - x0)) + (R . (x - x0))

x in dom (SVF1 (1,f,z0))

x in dom (SVF1 (1,f,z0)) ;

then dom (SVF1 (1,(f | Z),z0)) c= dom (SVF1 (1,f,z0)) ;

then N c= dom (SVF1 (1,f,z0)) by A4;

hence f is_partial_differentiable_in z0,1 by A2, A7, Th9; :: thesis: verum

( Z c= dom f & ( for z being Element of REAL 2 st z in Z holds

f is_partial_differentiable_in z,1 ) )

let f be PartFunc of (REAL 2),REAL; :: thesis: ( f is_partial_differentiable`1_on Z implies ( Z c= dom f & ( for z being Element of REAL 2 st z in Z holds

f is_partial_differentiable_in z,1 ) ) )

set g = f | Z;

assume A1: f is_partial_differentiable`1_on Z ; :: thesis: ( Z c= dom f & ( for z being Element of REAL 2 st z in Z holds

f is_partial_differentiable_in z,1 ) )

hence Z c= dom f ; :: thesis: for z being Element of REAL 2 st z in Z holds

f is_partial_differentiable_in z,1

let z0 be Element of REAL 2; :: thesis: ( z0 in Z implies f is_partial_differentiable_in z0,1 )

assume z0 in Z ; :: thesis: f is_partial_differentiable_in z0,1

then f | Z is_partial_differentiable_in z0,1 by A1;

then consider x0, y0 being Real such that

A2: z0 = <*x0,y0*> and

A3: ex N being Neighbourhood of x0 st

( N c= dom (SVF1 (1,(f | Z),z0)) & ex L being LinearFunc ex R being RestFunc st

for x being Real st x in N holds

((SVF1 (1,(f | Z),z0)) . x) - ((SVF1 (1,(f | Z),z0)) . x0) = (L . (x - x0)) + (R . (x - x0)) ) by Th9;

consider N being Neighbourhood of x0 such that

A4: N c= dom (SVF1 (1,(f | Z),z0)) and

A5: ex L being LinearFunc ex R being RestFunc st

for x being Real st x in N holds

((SVF1 (1,(f | Z),z0)) . x) - ((SVF1 (1,(f | Z),z0)) . x0) = (L . (x - x0)) + (R . (x - x0)) by A3;

consider L being LinearFunc, R being RestFunc such that

A6: for x being Real st x in N holds

((SVF1 (1,(f | Z),z0)) . x) - ((SVF1 (1,(f | Z),z0)) . x0) = (L . (x - x0)) + (R . (x - x0)) by A5;

A7: for x being Real st x in N holds

((SVF1 (1,f,z0)) . x) - ((SVF1 (1,f,z0)) . x0) = (L . (x - x0)) + (R . (x - x0))

proof

for x being Real st x in dom (SVF1 (1,(f | Z),z0)) holds
let x be Real; :: thesis: ( x in N implies ((SVF1 (1,f,z0)) . x) - ((SVF1 (1,f,z0)) . x0) = (L . (x - x0)) + (R . (x - x0)) )

A8: for x being Real st x in dom (SVF1 (1,(f | Z),z0)) holds

(SVF1 (1,(f | Z),z0)) . x = (SVF1 (1,f,z0)) . x

assume A13: x in N ; :: thesis: ((SVF1 (1,f,z0)) . x) - ((SVF1 (1,f,z0)) . x0) = (L . (x - x0)) + (R . (x - x0))

then (L . (x - x0)) + (R . (x - x0)) = ((SVF1 (1,(f | Z),z0)) . x) - ((SVF1 (1,(f | Z),z0)) . x0) by A6

.= ((SVF1 (1,f,z0)) . x) - ((SVF1 (1,(f | Z),z0)) . x0) by A4, A13, A8

.= ((SVF1 (1,f,z0)) . x) - ((SVF1 (1,f,z0)) . x0) by A4, A8, A12 ;

hence ((SVF1 (1,f,z0)) . x) - ((SVF1 (1,f,z0)) . x0) = (L . (x - x0)) + (R . (x - x0)) ; :: thesis: verum

end;A8: for x being Real st x in dom (SVF1 (1,(f | Z),z0)) holds

(SVF1 (1,(f | Z),z0)) . x = (SVF1 (1,f,z0)) . x

proof

A12:
x0 in N
by RCOMP_1:16;
let x be Real; :: thesis: ( x in dom (SVF1 (1,(f | Z),z0)) implies (SVF1 (1,(f | Z),z0)) . x = (SVF1 (1,f,z0)) . x )

assume A9: x in dom (SVF1 (1,(f | Z),z0)) ; :: thesis: (SVF1 (1,(f | Z),z0)) . x = (SVF1 (1,f,z0)) . x

then A10: x in dom (reproj (1,z0)) by FUNCT_1:11;

A11: (reproj (1,z0)) . x in dom (f | Z) by A9, FUNCT_1:11;

(SVF1 (1,(f | Z),z0)) . x = (f | Z) . ((reproj (1,z0)) . x) by A9, FUNCT_1:12

.= f . ((reproj (1,z0)) . x) by A11, FUNCT_1:47

.= (SVF1 (1,f,z0)) . x by A10, FUNCT_1:13 ;

hence (SVF1 (1,(f | Z),z0)) . x = (SVF1 (1,f,z0)) . x ; :: thesis: verum

end;assume A9: x in dom (SVF1 (1,(f | Z),z0)) ; :: thesis: (SVF1 (1,(f | Z),z0)) . x = (SVF1 (1,f,z0)) . x

then A10: x in dom (reproj (1,z0)) by FUNCT_1:11;

A11: (reproj (1,z0)) . x in dom (f | Z) by A9, FUNCT_1:11;

(SVF1 (1,(f | Z),z0)) . x = (f | Z) . ((reproj (1,z0)) . x) by A9, FUNCT_1:12

.= f . ((reproj (1,z0)) . x) by A11, FUNCT_1:47

.= (SVF1 (1,f,z0)) . x by A10, FUNCT_1:13 ;

hence (SVF1 (1,(f | Z),z0)) . x = (SVF1 (1,f,z0)) . x ; :: thesis: verum

assume A13: x in N ; :: thesis: ((SVF1 (1,f,z0)) . x) - ((SVF1 (1,f,z0)) . x0) = (L . (x - x0)) + (R . (x - x0))

then (L . (x - x0)) + (R . (x - x0)) = ((SVF1 (1,(f | Z),z0)) . x) - ((SVF1 (1,(f | Z),z0)) . x0) by A6

.= ((SVF1 (1,f,z0)) . x) - ((SVF1 (1,(f | Z),z0)) . x0) by A4, A13, A8

.= ((SVF1 (1,f,z0)) . x) - ((SVF1 (1,f,z0)) . x0) by A4, A8, A12 ;

hence ((SVF1 (1,f,z0)) . x) - ((SVF1 (1,f,z0)) . x0) = (L . (x - x0)) + (R . (x - x0)) ; :: thesis: verum

x in dom (SVF1 (1,f,z0))

proof

then
for x being object st x in dom (SVF1 (1,(f | Z),z0)) holds
let x be Real; :: thesis: ( x in dom (SVF1 (1,(f | Z),z0)) implies x in dom (SVF1 (1,f,z0)) )

dom (f | Z) = (dom f) /\ Z by RELAT_1:61;

then A14: dom (f | Z) c= dom f by XBOOLE_1:17;

assume x in dom (SVF1 (1,(f | Z),z0)) ; :: thesis: x in dom (SVF1 (1,f,z0))

then ( x in dom (reproj (1,z0)) & (reproj (1,z0)) . x in dom (f | Z) ) by FUNCT_1:11;

hence x in dom (SVF1 (1,f,z0)) by A14, FUNCT_1:11; :: thesis: verum

end;dom (f | Z) = (dom f) /\ Z by RELAT_1:61;

then A14: dom (f | Z) c= dom f by XBOOLE_1:17;

assume x in dom (SVF1 (1,(f | Z),z0)) ; :: thesis: x in dom (SVF1 (1,f,z0))

then ( x in dom (reproj (1,z0)) & (reproj (1,z0)) . x in dom (f | Z) ) by FUNCT_1:11;

hence x in dom (SVF1 (1,f,z0)) by A14, FUNCT_1:11; :: thesis: verum

x in dom (SVF1 (1,f,z0)) ;

then dom (SVF1 (1,(f | Z),z0)) c= dom (SVF1 (1,f,z0)) ;

then N c= dom (SVF1 (1,f,z0)) by A4;

hence f is_partial_differentiable_in z0,1 by A2, A7, Th9; :: thesis: verum