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
  -(exp_R*(AffineMap(-1,0))) is_differentiable_on REAL & for x holds ((-
  (exp_R*(AffineMap(-1,0))))`|REAL).x = exp_R(-x)
proof
A1: [#]REAL = dom (exp_R*(AffineMap(-1,0))) by FUNCT_2:def 1;
A2: [#]REAL = dom (AffineMap(-jj,0)) by FUNCT_2:def 1;
A3: for x st x in REAL holds AffineMap(-1,0).x=(-1)*x + 0 by FCONT_1:def 4;
  then
A4: (AffineMap(-jj,0)) is_differentiable_on REAL by A2,FDIFF_1:23;
  for x st x in REAL holds
exp_R*(AffineMap(-1,0)) qua PartFunc of REAL, REAL
is_differentiable_in x
  proof
    let x;
    assume x in REAL;
    then (AffineMap(-jj,0)) is_differentiable_in x by A2,A4,FDIFF_1:9;
    hence thesis by TAYLOR_1:19;
  end;
  then
A5:  exp_R*(AffineMap(-1,0)) is_differentiable_on REAL by A1,FDIFF_1:9;
  hence -(exp_R*(AffineMap(-1,0))) is_differentiable_on REAL by Lm1,FDIFF_1:20;
A6:
for x st x in REAL holds ((-(exp_R*(AffineMap(-1,0))))`|REAL).x = exp_R( -x)
  proof
    let x;
    assume
A7:  x in REAL;
    then
A8: (AffineMap(-1,0)) is_differentiable_in x by A2,A4,FDIFF_1:9;
   ((-(exp_R*(AffineMap(-1,0))))`|REAL).x = (-1)*diff(exp_R*(AffineMap(-
    1,0)),x) by A5,Lm1,FDIFF_1:20,A7
      .= (-1)*(exp_R.((AffineMap(-1,0)).x) *diff((AffineMap(-1,0)),x)) by A8,
TAYLOR_1:19
      .= (-1)*(exp_R.((AffineMap(-1,0)).x) *((AffineMap(-1,0))`|REAL).x) by A4,
FDIFF_1:def 7,A7
      .= (-1)*(exp_R.((AffineMap(-1,0)).x)*(-1)) by A2,A3,FDIFF_1:23,A7
      .= (-1)*(exp_R.((-1)*x + 0)*(-1)) by FCONT_1:def 4
      .= exp_R(-x);
    hence thesis;
  end;
  let x;
   x in REAL by XREAL_0:def 1;
  hence ((-
  (exp_R*(AffineMap(-1,0))))`|REAL).x = exp_R(-x) by A6;
end;
