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

hereby :: thesis: ( ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 2,f,z) & ex L being LINEAR ex R being REST st
( r = L . 1 & ( for y being Real st y in N holds
((SVF1 2,f,z) . y) - ((SVF1 2,f,z) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) implies r = partdiff f,z,2 )

assume r = partdiff f,z,2 ; :: thesis: ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 2,f,z) & ex L being LINEAR ex R being REST st
( r = L . 1 & ( for y being Real st y in N holds
((SVF1 2,f,z) . y) - ((SVF1 2,f,z) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) )
then r = diff (SVF1 2,f,z),y0 by Th2, AA;
hence ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 2,f,z) & ex L being LINEAR ex R being REST st
( r = L . 1 & ( for y being Real st y in N holds
((SVF1 2,f,z) . y) - ((SVF1 2,f,z) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) by def13, AA; :: thesis: verum

end;

given x1, y1 being Real such that C1: ( z = <*x1,y1*> & ex N being Neighbourhood of y1 st
( N c= dom (SVF1 2,f,z) & ex L being LINEAR ex R being REST st
( r = L . 1 & ( for y being Real st y in N holds
((SVF1 2,f,z) . y) - ((SVF1 2,f,z) . y1) = (L . (y - y1)) + (R . (y - y1)) ) ) ) ) ; :: thesis: r = partdiff f,z,2
r = diff (SVF1 2,f,z),y0 by C1, AA, def13;
hence r = partdiff f,z,2 by Th2, AA; :: thesis: verum