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
for f be PartFunc of REAL,REAL n,
    x0 be Real st
 1 <= i & i <= n & f is_differentiable_in x0 holds
   (Proj(i,n)*f) is_differentiable_in x0 &
   Proj(i,n).(diff(f,x0)) = diff((Proj(i,n)*f),x0)
proof
  let f be PartFunc of REAL,REAL n,
      x0 be Real;
  assume
A1: 1 <= i & i <= n & f is_differentiable_in x0;
  then
  consider g be PartFunc of REAL,REAL-NS n such that
A2: f=g & g is_differentiable_in x0;
  diff(f,x0) = diff(g,x0) by A2,Th3;
  hence thesis by A2,A1,Th24;
end;
