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

let f be PartFunc of (REAL 2),REAL; :: thesis: ( f is_partial_differentiable`2_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,2 ) ) )

set g = f | Z;
assume A1: f is_partial_differentiable`2_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,2 ) )

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

let z0 be Element of REAL 2; :: thesis: ( z0 in Z implies f is_partial_differentiable_in z0,2 )
assume z0 in Z ; :: thesis:
then f | Z is_partial_differentiable_in z0,2 by A1;
then consider x0, y0 being Real such that
A2: z0 = <*x0,y0*> and
A3: ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(f | Z),z0)) & ex L being LinearFunc ex R being RestFunc st
for y being Real st y in N holds
((SVF1 (2,(f | Z),z0)) . y) - ((SVF1 (2,(f | Z),z0)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) by Th10;
consider N being Neighbourhood of y0 such that
A4: N c= dom (SVF1 (2,(f | Z),z0)) and
A5: ex L being LinearFunc ex R being RestFunc st
for y being Real st y in N holds
((SVF1 (2,(f | Z),z0)) . y) - ((SVF1 (2,(f | Z),z0)) . y0) = (L . (y - y0)) + (R . (y - y0)) by A3;
consider L being LinearFunc, R being RestFunc such that
A6: for y being Real st y in N holds
((SVF1 (2,(f | Z),z0)) . y) - ((SVF1 (2,(f | Z),z0)) . y0) = (L . (y - y0)) + (R . (y - y0)) by A5;
A7: for y being Real st y in N holds
((SVF1 (2,f,z0)) . y) - ((SVF1 (2,f,z0)) . y0) = (L . (y - y0)) + (R . (y - y0))
proof
let y be Real; :: thesis: ( y in N implies ((SVF1 (2,f,z0)) . y) - ((SVF1 (2,f,z0)) . y0) = (L . (y - y0)) + (R . (y - y0)) )
A8: for y being Real st y in dom (SVF1 (2,(f | Z),z0)) holds
(SVF1 (2,(f | Z),z0)) . y = (SVF1 (2,f,z0)) . y
proof
let y be Real; :: thesis: ( y in dom (SVF1 (2,(f | Z),z0)) implies (SVF1 (2,(f | Z),z0)) . y = (SVF1 (2,f,z0)) . y )
assume A9: y in dom (SVF1 (2,(f | Z),z0)) ; :: thesis: (SVF1 (2,(f | Z),z0)) . y = (SVF1 (2,f,z0)) . y
then A10: y in dom (reproj (2,z0)) by FUNCT_1:11;
A11: (reproj (2,z0)) . y in dom (f | Z) by ;
(SVF1 (2,(f | Z),z0)) . y = (f | Z) . ((reproj (2,z0)) . y) by
.= f . ((reproj (2,z0)) . y) by
.= (SVF1 (2,f,z0)) . y by ;
hence (SVF1 (2,(f | Z),z0)) . y = (SVF1 (2,f,z0)) . y ; :: thesis: verum
end;
A12: y0 in N by RCOMP_1:16;
assume A13: y in N ; :: thesis: ((SVF1 (2,f,z0)) . y) - ((SVF1 (2,f,z0)) . y0) = (L . (y - y0)) + (R . (y - y0))
then (L . (y - y0)) + (R . (y - y0)) = ((SVF1 (2,(f | Z),z0)) . y) - ((SVF1 (2,(f | Z),z0)) . y0) by A6
.= ((SVF1 (2,f,z0)) . y) - ((SVF1 (2,(f | Z),z0)) . y0) by A4, A13, A8
.= ((SVF1 (2,f,z0)) . y) - ((SVF1 (2,f,z0)) . y0) by A4, A8, A12 ;
hence ((SVF1 (2,f,z0)) . y) - ((SVF1 (2,f,z0)) . y0) = (L . (y - y0)) + (R . (y - y0)) ; :: thesis: verum
end;
for y being Real st y in dom (SVF1 (2,(f | Z),z0)) holds
y in dom (SVF1 (2,f,z0))
proof
let y be Real; :: thesis: ( y in dom (SVF1 (2,(f | Z),z0)) implies y in dom (SVF1 (2,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 y in dom (SVF1 (2,(f | Z),z0)) ; :: thesis: y in dom (SVF1 (2,f,z0))
then ( y in dom (reproj (2,z0)) & (reproj (2,z0)) . y in dom (f | Z) ) by FUNCT_1:11;
hence y in dom (SVF1 (2,f,z0)) by ; :: thesis: verum
end;
then for y being object st y in dom (SVF1 (2,(f | Z),z0)) holds
y in dom (SVF1 (2,f,z0)) ;
then dom (SVF1 (2,(f | Z),z0)) c= dom (SVF1 (2,f,z0)) ;
then N c= dom (SVF1 (2,f,z0)) by A4;
hence f is_partial_differentiable_in z0,2 by A2, A7, Th10; :: thesis: verum