reserve n for Nat,
  i for Integer,
  p, x, x0, y for Real,
  q for Rational,
  f for PartFunc of REAL,REAL;

theorem
  (exp_R.1) #R x =exp_R.x & (exp_R.1) to_power x = exp_R.x & number_e
  to_power x = exp_R.x & number_e #R x =exp_R.x
proof
  thus (exp_R.1) #R x =(exp_R(1)) #R x by SIN_COS:def 23
    .=exp_R(x) by Th9
    .=exp_R.x by SIN_COS:def 23;
  thus (exp_R.1) to_power x=(exp_R(1)) to_power x by SIN_COS:def 23
    .=exp_R(x) by Th9
    .=exp_R.x by SIN_COS:def 23;
  thus number_e to_power x = exp_R(x) by Th9
    .=exp_R.x by SIN_COS:def 23;
  thus number_e #R x =exp_R(x) by Th9
    .=exp_R.x by SIN_COS:def 23;
end;
