let x0, y0 be Real; :: thesis: for z being Element of REAL 2
for f being PartFunc of (REAL 2),REAL st z = <*x0,y0*> & f is_hpartial_differentiable`12_in z holds
SVF1 (2,(pdiff1 (f,1)),z) is_differentiable_in y0
let z be Element of REAL 2; :: thesis: for f being PartFunc of (REAL 2),REAL st z = <*x0,y0*> & f is_hpartial_differentiable`12_in z holds
SVF1 (2,(pdiff1 (f,1)),z) is_differentiable_in y0
let f be PartFunc of (REAL 2),REAL; :: thesis: ( z = <*x0,y0*> & f is_hpartial_differentiable`12_in z implies SVF1 (2,(pdiff1 (f,1)),z) is_differentiable_in y0 )
assume that
A1: z = <*x0,y0*> and
A2: f is_hpartial_differentiable`12_in z ; :: thesis: SVF1 (2,(pdiff1 (f,1)),z) is_differentiable_in y0
consider x1, y1 being Real such that
A3: z = <*x1,y1*> and
A4: ex N being Neighbourhood of y1 st
( N c= dom (SVF1 (2,(pdiff1 (f,1)),z)) & ex L being LinearFunc ex R being RestFunc st
for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,1)),z)) . y) - ((SVF1 (2,(pdiff1 (f,1)),z)) . y1) = (L . (y - y1)) + (R . (y - y1)) ) by A2;
y0 = y1 by A1, A3, FINSEQ_1:77;
hence SVF1 (2,(pdiff1 (f,1)),z) is_differentiable_in y0 by A4, FDIFF_1:def 4; :: thesis: verum