 reserve j for set;
 reserve p,r for Real;
 reserve S,T,F for RealNormSpace;
 reserve x0 for Point of S;
 reserve g for PartFunc of S,T;
 reserve c for constant sequence of S;
 reserve R for RestFunc of S,T;
 reserve G for RealNormSpace-Sequence;
 reserve i for Element of dom G;
 reserve f for PartFunc of product G,F;
 reserve x for Element of product G;
reserve G for RealNormSpace-Sequence;
reserve F for RealNormSpace;
reserve i for Element of dom G;
reserve f,f1,f2 for PartFunc of product G, F;
reserve x for Point of product G;
reserve X for set;

theorem Th53:
for G be RealNormSpace-Sequence,
    S be RealNormSpace,
    f be PartFunc of product G,S,
    X be Subset of product G,
    x,y,z be Point of product G,
    i be set,
    p,q be Point of G.(In(i,dom G)),
    d,r be Real
 st i in dom G & X is open & x in X &
    ||. y-x .|| < d & ||. z-x .|| < d & X c= dom f &
    (for x be Point of product G st x in X holds
      f is_partial_differentiable_in x,i) &
    (for z be Point of product G st ||. z-x .|| < d holds z in X) &
    (for z be Point of product G st ||. z-x .|| < d
        holds ||. partdiff(f,z,i) - partdiff(f,x,i).|| <=r) &
    z = reproj(In(i,dom G),y).p & q = proj(In(i,dom G)).y
holds
   ||. f/.z - f/.y - ((partdiff(f,x,i)).(p-q)) .|| <= ||. p-q .||*r
proof
   let G be RealNormSpace-Sequence,
       S be RealNormSpace,
       f be PartFunc of product G,S,
       X be Subset of product G,
       x,y,z be Point of product G,
       i0 be set,
       p,q be Point of G.(In(i0,dom G)),
       d,r be Real;
   assume
A1: i0 in dom G & X is open & x in X &
    ||. y-x .|| < d & ||. z-x .|| < d & X c= dom f &
    (for x be Point of product G st x in X holds
       f is_partial_differentiable_in x,i0) &
    (for z be Point of product G st ||. z-x .|| < d holds z in X) &
    (for z be Point of product G st ||. z-x .|| < d
       holds ||. partdiff(f,z,i0) - partdiff(f,x,i0).|| <=r) &
    z = reproj(In(i0,dom G),y).p & q = proj(In(i0,dom G)).y;
   set i=In(i0,dom G);
A2:y = reproj(i,y).q by A1,Th47;
A3:now let h be Point of G.i;
    assume h in [. q,p .]; then
    ||. reproj(i,y).h - x .|| < d by A1,Th52;
    hence reproj(i,y).h in dom f by A1;
   end;
A4:now let h be Point of G.i;
    assume h in [. q,p .]; then
    ||. reproj(i,y).h - x .|| < d by A1,Th52;
    hence f is_partial_differentiable_in (reproj(i,y).h),i0 by A1;
   end;
   for h be Point of G.i st h in ]. q,p .[ holds
     ||. partdiff(f,reproj(i,y).h,i0) - partdiff(f,x,i0) .|| <=r
   proof
    let h be Point of G.i;
    assume A5: h in ]. q,p .[;
    ].q,p.[ c= [. q,p .] by XBOOLE_1:36; then
    ||. reproj(i,y).h - x .|| < d by A1,A5,Th52;
    hence ||. partdiff(f,reproj(i,y).h,i0) - partdiff(f,x,i0).|| <=r by A1;
   end;
   hence thesis by A2,A1,Th51,A3,A4;
end;
