let x0, y0, r be Real; :: thesis: for z being Element of REAL 2
for f being PartFunc of REAL 2, REAL st z = <*x0,y0*> & f is_partial_differentiable`1_in z holds
( r = partdiff1 f,z iff ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 f,z) & ex L being LINEAR ex R being REST st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 f,z) . x) - ((SVF1 f,z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) ) )
let z be Element of REAL 2; :: thesis: for f being PartFunc of REAL 2, REAL st z = <*x0,y0*> & f is_partial_differentiable`1_in z holds
( r = partdiff1 f,z iff ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 f,z) & ex L being LINEAR ex R being REST st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 f,z) . x) - ((SVF1 f,z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) ) )
let f be PartFunc of REAL 2, REAL ; :: thesis: ( z = <*x0,y0*> & f is_partial_differentiable`1_in z implies ( r = partdiff1 f,z iff ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 f,z) & ex L being LINEAR ex R being REST st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 f,z) . x) - ((SVF1 f,z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) ) ) )
assume AA: ( z = <*x0,y0*> & f is_partial_differentiable`1_in z ) ; :: thesis: ( r = partdiff1 f,z iff ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 f,z) & ex L being LINEAR ex R being REST st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 f,z) . x) - ((SVF1 f,z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) ) )

hereby :: thesis: ( ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 f,z) & ex L being LINEAR ex R being REST st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 f,z) . x) - ((SVF1 f,z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) ) implies r = partdiff1 f,z )

assume r = partdiff1 f,z ; :: thesis: ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 f,z) & ex L being LINEAR ex R being REST st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 f,z) . x) - ((SVF1 f,z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) )
then r = diff (SVF1 f,z),x0 by Th1, AA;
hence ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 f,z) & ex L being LINEAR ex R being REST st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 f,z) . x) - ((SVF1 f,z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) ) by def12, AA; :: thesis: verum

end;

given x1, y1 being Real such that C1: ( z = <*x1,y1*> & ex N being Neighbourhood of x1 st
( N c= dom (SVF1 f,z) & ex L being LINEAR ex R being REST st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 f,z) . x) - ((SVF1 f,z) . x1) = (L . (x - x1)) + (R . (x - x1)) ) ) ) ) ; :: thesis: r = partdiff1 f,z
r = diff (SVF1 f,z),x0 by C1, AA, def12;
hence r = partdiff1 f,z by Th1, AA; :: thesis: verum