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 holds
f is_differentiable_in x0 iff
(for i be Element of NAT st 1<=i & i <=n holds
      Proj(i,n)*f is_differentiable_in x0)
proof
  let f be PartFunc of REAL,REAL n, x0 be Real;
  thus f is_differentiable_in x0 implies
   (for i be Element of NAT st 1<=i & i <=n holds
      Proj(i,n)*f is_differentiable_in x0)
  by Th25;
  assume
A1: for i be Element of NAT st 1<=i & i <=n holds
      Proj(i,n)*f is_differentiable_in x0;
    reconsider g=f as PartFunc of REAL,REAL-NS n by REAL_NS1:def 4;
    for i be Element of NAT st 1<=i & i <=n holds
                  Proj(i,n)*g is_differentiable_in x0 by A1;
    then g is_differentiable_in x0 by Th25;
    hence thesis;
end;
