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_partial_differentiable_in u,2 holds
ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,f,u)) & ex L being LinearFunc ex R being RestFunc st
for y being Real st y in N holds
((SVF1 (2,f,u)) . y) - ((SVF1 (2,f,u)) . y0) = (L . (y - y0)) + (R . (y - y0)) )
let u be Element of REAL 3; :: thesis: for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_partial_differentiable_in u,2 holds
ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,f,u)) & ex L being LinearFunc ex R being RestFunc st
for y being Real st y in N holds
((SVF1 (2,f,u)) . y) - ((SVF1 (2,f,u)) . y0) = (L . (y - y0)) + (R . (y - y0)) )
let f be PartFunc of (REAL 3),REAL; :: thesis: ( u = <*x0,y0,z0*> & f is_partial_differentiable_in u,2 implies ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,f,u)) & ex L being LinearFunc ex R being RestFunc st
for y being Real st y in N holds
((SVF1 (2,f,u)) . y) - ((SVF1 (2,f,u)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) )
assume that
A1: u = <*x0,y0,z0*> and
A2: f is_partial_differentiable_in u,2 ; :: thesis: ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,f,u)) & ex L being LinearFunc ex R being RestFunc st
for y being Real st y in N holds
((SVF1 (2,f,u)) . y) - ((SVF1 (2,f,u)) . y0) = (L . (y - y0)) + (R . (y - y0)) )
consider x1, y1, z1 being Real such that
A3: ( u = <*x1,y1,z1*> & SVF1 (2,f,u) is_differentiable_in y1 ) by A2, Th8;
SVF1 (2,f,u) is_differentiable_in y0 by A1, A3, FINSEQ_1:78;
hence ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,f,u)) & ex L being LinearFunc ex R being RestFunc st
for y being Real st y in N holds
((SVF1 (2,f,u)) . y) - ((SVF1 (2,f,u)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) by FDIFF_1:def 4; :: thesis: verum