let D be set ; :: thesis: for f being PartFunc of (REAL 3),REAL st f is_partial_differentiable`1_on D holds
( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f is_partial_differentiable_in u,1 ) )
let f be PartFunc of (REAL 3),REAL; :: thesis: ( f is_partial_differentiable`1_on D implies ( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f is_partial_differentiable_in u,1 ) ) )
assume A1: f is_partial_differentiable`1_on D ; :: thesis: ( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f is_partial_differentiable_in u,1 ) )
hence D c= dom f ; :: thesis: for u being Element of REAL 3 st u in D holds
f is_partial_differentiable_in u,1
set g = f | D;
let u0 be Element of REAL 3; :: thesis: ( u0 in D implies f is_partial_differentiable_in u0,1 )
assume u0 in D ; :: thesis: f is_partial_differentiable_in u0,1
then f | D is_partial_differentiable_in u0,1 by A1;
then consider x0, y0, z0 being Real such that
A2: ( u0 = <*x0,y0,z0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 (1,(f | D),u0)) & ex L being LinearFunc ex R being RestFunc st
for x being Real st x in N holds
((SVF1 (1,(f | D),u0)) . x) - ((SVF1 (1,(f | D),u0)) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) by Th13;
consider N being Neighbourhood of x0 such that
A3: ( N c= dom (SVF1 (1,(f | D),u0)) & ex L being LinearFunc ex R being RestFunc st
for x being Real st x in N holds
((SVF1 (1,(f | D),u0)) . x) - ((SVF1 (1,(f | D),u0)) . x0) = (L . (x - x0)) + (R . (x - x0)) ) by A2;
for x being Real st x in dom (SVF1 (1,(f | D),u0)) holds
x in dom (SVF1 (1,f,u0))

proof

let x be Real; :: thesis: ( x in dom (SVF1 (1,(f | D),u0)) implies x in dom (SVF1 (1,f,u0)) )
assume x in dom (SVF1 (1,(f | D),u0)) ; :: thesis: x in dom (SVF1 (1,f,u0))
then A4: ( x in dom (reproj (1,u0)) & (reproj (1,u0)) . x in dom (f | D) ) by FUNCT_1:11;
dom (f | D) = (dom f) /\ D by RELAT_1:61;
then dom (f | D) c= dom f by XBOOLE_1:17;
hence x in dom (SVF1 (1,f,u0)) by A4, FUNCT_1:11; :: thesis: verum

end;

then for x being object st x in dom (SVF1 (1,(f | D),u0)) holds
x in dom (SVF1 (1,f,u0)) ;
then dom (SVF1 (1,(f | D),u0)) c= dom (SVF1 (1,f,u0)) ;
then A5: N c= dom (SVF1 (1,f,u0)) by A3;
consider L being LinearFunc, R being RestFunc such that
A6: for x being Real st x in N holds
((SVF1 (1,(f | D),u0)) . x) - ((SVF1 (1,(f | D),u0)) . x0) = (L . (x - x0)) + (R . (x - x0)) by A3;
for x being Real st x in N holds
((SVF1 (1,f,u0)) . x) - ((SVF1 (1,f,u0)) . x0) = (L . (x - x0)) + (R . (x - x0))

proof

let x be Real; :: thesis: ( x in N implies ((SVF1 (1,f,u0)) . x) - ((SVF1 (1,f,u0)) . x0) = (L . (x - x0)) + (R . (x - x0)) )
assume A7: x in N ; :: thesis: ((SVF1 (1,f,u0)) . x) - ((SVF1 (1,f,u0)) . x0) = (L . (x - x0)) + (R . (x - x0))
A8: for x being Real st x in dom (SVF1 (1,(f | D),u0)) holds
(SVF1 (1,(f | D),u0)) . x = (SVF1 (1,f,u0)) . x

proof

let x be Real; :: thesis: ( x in dom (SVF1 (1,(f | D),u0)) implies (SVF1 (1,(f | D),u0)) . x = (SVF1 (1,f,u0)) . x )
assume A9: x in dom (SVF1 (1,(f | D),u0)) ; :: thesis: (SVF1 (1,(f | D),u0)) . x = (SVF1 (1,f,u0)) . x
then A10: ( x in dom (reproj (1,u0)) & (reproj (1,u0)) . x in dom (f | D) ) by FUNCT_1:11;
(SVF1 (1,(f | D),u0)) . x = (f | D) . ((reproj (1,u0)) . x) by A9, FUNCT_1:12
.= f . ((reproj (1,u0)) . x) by A10, FUNCT_1:47
.= (SVF1 (1,f,u0)) . x by A10, FUNCT_1:13 ;
hence (SVF1 (1,(f | D),u0)) . x = (SVF1 (1,f,u0)) . x ; :: thesis: verum

end;

A11: x0 in N by RCOMP_1:16;
(L . (x - x0)) + (R . (x - x0)) = ((SVF1 (1,(f | D),u0)) . x) - ((SVF1 (1,(f | D),u0)) . x0) by A6, A7
.= ((SVF1 (1,f,u0)) . x) - ((SVF1 (1,(f | D),u0)) . x0) by A3, A7, A8
.= ((SVF1 (1,f,u0)) . x) - ((SVF1 (1,f,u0)) . x0) by A3, A8, A11 ;
hence ((SVF1 (1,f,u0)) . x) - ((SVF1 (1,f,u0)) . x0) = (L . (x - x0)) + (R . (x - x0)) ; :: thesis: verum

end;

hence f is_partial_differentiable_in u0,1 by A2, A5, Th13; :: thesis: verum