reserve r,p,x for Real;
reserve n for Element of NAT;
reserve A for non empty closed_interval Subset of REAL;
reserve Z for open Subset of REAL;

theorem Th2:
  r<>0 implies (1/r)(#)(exp_R*(AffineMap(r,0)))
  is_differentiable_on REAL & for x holds (((1/r)(#)(exp_R*(AffineMap(r,0))))`|
  REAL).x=(exp_R*(AffineMap(r,0))).x
proof
  assume
A1: r<>0;
A2: [#]REAL = dom (exp_R*(AffineMap(r,0))) by FUNCT_2:def 1;
A3: [#]REAL = dom (AffineMap(r,0)) by FUNCT_2:def 1;
A4: for x st x in REAL holds AffineMap(r,0).x=r*x + 0 by FCONT_1:def 4;
  then
A5: (AffineMap(r,0)) is_differentiable_on REAL by A3,FDIFF_1:23;
  for x st x in REAL holds exp_R*(AffineMap(r,0)) is_differentiable_in x
  proof
    let x;
    assume x in REAL;
    then (AffineMap(r,0)) is_differentiable_in x by A3,A5,FDIFF_1:9;
    hence thesis by TAYLOR_1:19;
  end;
  then
A6: [#]REAL= dom ((1/r)(#)(exp_R*(AffineMap(r,0)))) & exp_R*(AffineMap(r,0))
  is_differentiable_on REAL by A2,FDIFF_1:9,FUNCT_2:def 1;
  hence(1/r)(#)(exp_R*(AffineMap(r,0)))
  is_differentiable_on REAL by FDIFF_1:20;
A7:
  for x st x in REAL holds (((1/r)(#)(exp_R*(AffineMap(r,0))))`|REAL).x =
  (exp_R*(AffineMap(r,0))).x
  proof
    let x;
    assume
A8:x in REAL;
     then
A9: (AffineMap(r,0)) is_differentiable_in x by A3,A5,FDIFF_1:9;
    (((1/r)(#)(exp_R*(AffineMap(r,0))))`|REAL).x = (1/r)*diff(exp_R*(
    AffineMap(r,0)),x) by A6,FDIFF_1:20,A8
      .= (1/r)*(exp_R.((AffineMap(r,0)).x) *diff((AffineMap(r,0)),x)) by A9,
TAYLOR_1:19
      .= (1/r)*(exp_R.((AffineMap(r,0)).x) *(AffineMap(r,0)`|REAL).x) by A5,
FDIFF_1:def 7,A8
      .= (1/r)*(exp_R.((AffineMap(r,0)).x)*r) by A3,A4,FDIFF_1:23,A8
      .= r*(1/r)*(exp_R.((AffineMap(r,0)).x))
      .= r/r*(exp_R.((AffineMap(r,0)).x)) by XCMPLX_1:99
      .= 1*(exp_R.((AffineMap(r,0)).x)) by A1,XCMPLX_1:60
      .= (exp_R*(AffineMap(r,0))).x by A2,FUNCT_1:12,A8;
    hence thesis;
  end;
  let x;
   x in REAL by XREAL_0:def 1;
  hence thesis by A7;
end;
