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;
