reserve F for RealNormSpace;
reserve G for RealNormSpace;
reserve y,X for set;
reserve x,x0,x1,x2,g,g1,g2,r,r1,s,p,p1,p2 for Real;
reserve i,m,k for Element of NAT;
reserve n,k for non zero Element of NAT;
reserve Y for Subset of REAL;
reserve Z for open Subset of REAL;
reserve s1,s3 for Real_Sequence;
reserve seq,seq1 for sequence of G;
reserve f,f1,f2 for PartFunc of REAL,REAL n;
reserve g,g1,g2 for PartFunc of REAL,REAL-NS n;
reserve h for 0-convergent non-zero Real_Sequence;
reserve c for constant Real_Sequence;
reserve GR,R for RestFunc of REAL-NS n;
reserve DFG,L for LinearFunc of REAL-NS n;

theorem Th28:
for g be PartFunc of REAL,REAL-NS n st
 1 <= i & i <= n & g is_differentiable_on X holds
   (Proj(i,n)*g) is_differentiable_on X &
   Proj(i,n)*(g`|X) = (Proj(i,n)*g)`|X
proof
  let g be PartFunc of REAL,REAL-NS n;
  assume
A1: 1 <= i & i <= n & g is_differentiable_on X;
  then
A2: X is open Subset of REAL by NDIFF_3:9,11;
A3: dom(Proj(i,n)) = the carrier of REAL-NS n by FUNCT_2:def 1;
    rng g c= the carrier of REAL-NS n;
    then
    dom(Proj(i,n)*g) = dom g by A3,RELAT_1:27;
    then
A4: X c= dom(Proj(i,n)*g) by A2,A1,NDIFF_3:10;
    now let x;
    assume x in X;
    then g is_differentiable_in x by A2,A1,NDIFF_3:10;
    hence (Proj(i,n)*g) is_differentiable_in x by A1,Th24;
  end;
  hence
A5: (Proj(i,n)*g) is_differentiable_on X by A2,A4,NDIFF_3:10;
  then
A6: dom ((Proj(i,n)*g)`|X) = X &
    for x st x in X holds
    ((Proj(i,n)*g)`|X).x = diff((Proj(i,n)*g),x) by NDIFF_3:def 6;
A7: dom (g`|X) = X &
    for x st x in X holds (g`|X).x = diff(g,x) by A1,NDIFF_3:def 6;
  rng (g`|X) c= the carrier of REAL-NS n;
  then
A8: dom (Proj(i,n)*(g`|X)) = dom (g`|X) by A3,RELAT_1:27;
  now let x be Element of REAL;
    assume A9: x in dom ((Proj(i,n)*g)`|X);
    then
A10: x in X by A5,NDIFF_3:def 6;
    then g is_differentiable_in x by A2,A1,NDIFF_3:10;
    then
A11: Proj(i,n).(diff(g,x)) = diff((Proj(i,n)*g),x) by A1,Th24;
A12: ((Proj(i,n)*g)`|X).x = diff((Proj(i,n)*g),x) by A9,A6;
    (g`|X).x = diff(g,x) by A10,A1,NDIFF_3:def 6;
    hence (Proj(i,n)*(g`|X)).x = ((Proj(i,n)*g)`|X).x
           by A7,A10,A11,A12,FUNCT_1:13;
  end;
  hence thesis by A8,A6,A7,PARTFUN1:5;
end;
