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
   st J=proj(1,1)
 holds J is_differentiable_in x0 & diff(J,x0) = J
proof
  let J be Function of REAL-NS 1,REAL,
      x0 be Point of REAL-NS 1;
  assume A1: J=proj(1,1);
  reconsider J0=J as Function of REAL 1,REAL by Lm1;
  reconsider y0=x0 as Element of REAL 1 by REAL_NS1:def 4;
  J0 is_differentiable_in y0 & diff(J0,y0) = J by A1,Th38;
  hence J is_differentiable_in x0;
  ex g be PartFunc of REAL 1,REAL, y be Element of REAL 1 st
  J=g & x0=y & diff(J,x0) = diff(g,y) by Def7;
  hence thesis by A1,Th38;
end;
