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 J be Function of REAL-NS 1,REAL,
      x0 be Point of REAL-NS 1,
      y0 be Element of REAL,
      g be PartFunc of REAL,REAL-NS n,
      f be PartFunc of REAL-NS 1,REAL-NS n
   st J=proj(1,1) & x0 in dom f & y0 in dom g
   & x0=<*y0*> & f = g*J & g is_differentiable_in y0
  holds
   f is_differentiable_in x0
   & diff(g,y0)=diff(f,x0).<*1*>
   & for r be Element of REAL holds diff(f,x0).<*r*> = r*diff(g,y0)
proof
  let J be Function of REAL-NS 1,REAL,
      x0 be Point of REAL-NS 1,
      y0 be Element of REAL,
      g be PartFunc of REAL,REAL-NS n,
      f be PartFunc of REAL-NS 1,REAL-NS n;
  assume A1: J=proj(1,1) & x0 in dom f & y0 in dom g & x0=<*y0*> & f = g*J
   & g is_differentiable_in y0;
  hence
A2: f is_differentiable_in x0 by Th44;
  reconsider I= (proj(1,1) qua Function") as Function of REAL,REAL-NS 1
                       by PDIFF_1:2,REAL_NS1:def 4;
A3: J*I = id REAL by A1,Lm2,FUNCT_1:39;
   f*I = g*(id REAL) by A1,A3,RELAT_1:36
   .= g by FUNCT_2:17;
  hence thesis by A1,Th42,A2;
end;
