let x0, y0, z0 be Real; :: thesis: for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_hpartial_differentiable`12_in u holds
SVF1 2,(pdiff1 f,1),u is_differentiable_in y0
let u be Element of REAL 3; :: thesis: for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_hpartial_differentiable`12_in u holds
SVF1 2,(pdiff1 f,1),u is_differentiable_in y0
let f be PartFunc of (REAL 3),REAL ; :: thesis: ( u = <*x0,y0,z0*> & f is_hpartial_differentiable`12_in u implies SVF1 2,(pdiff1 f,1),u is_differentiable_in y0 )
assume that
A1: u = <*x0,y0,z0*> and
A2: f is_hpartial_differentiable`12_in u ; :: thesis: SVF1 2,(pdiff1 f,1),u is_differentiable_in y0
consider x1, y1, z1 being Real such that
A4: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of y1 st
( N c= dom (SVF1 2,(pdiff1 f,1),u) & ex L being LINEAR ex R being REST st
for y being Real st y in N holds
((SVF1 2,(pdiff1 f,1),u) . y) - ((SVF1 2,(pdiff1 f,1),u) . y1) = (L . (y - y1)) + (R . (y - y1)) ) ) by A2, Def4;
y0 = y1 by A1, A4, FINSEQ_1:99;
hence SVF1 2,(pdiff1 f,1),u is_differentiable_in y0 by A4, FDIFF_1:def 5; :: thesis: verum