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

theorem Th8:
  (exp_R(x)) #R p = exp_R(p*x)
proof
  exp_R(x) > 0 by SIN_COS:55;
  then consider s being Rational_Sequence such that
A1: s is convergent & lim s = p and
  (exp_R(x)) #Q s is convergent and
A2: lim (exp_R(x)) #Q s = (exp_R(x)) #R p by PREPOWER:def 7;
A3: exp_R is_continuous_in x*p & rng (x(#)s) c= dom (exp_R) by FDIFF_1:24
,SIN_COS:47,65;
A4: now
    let ii be object;
    assume ii in NAT;
    then reconsider i=ii as Element of NAT;
A5: rng (x(#)s) c= dom (exp_R) by SIN_COS:47;
    thus ( (exp_R(x)) #Q s ).ii =(exp_R(x)) #Q (s.i) by PREPOWER:def 6
      .=exp_R((s.i)*x) by Th7
      .=exp_R((x(#)s).i) by SEQ_1:9
      .=exp_R.((x(#)s).i) by SIN_COS:def 23
      .=(exp_R/*(x(#)s)).ii by A5,FUNCT_2:108;
  end;
  x(#)s is convergent & lim (x(#)s) = x*p by A1,SEQ_2:7,8;
  then lim((exp_R/*(x(#)s)) ) = exp_R.(x*p) by A3,FCONT_1:def 1
    .= exp_R(p*x) by SIN_COS:def 23;
  hence thesis by A2,A4,FUNCT_2:12;
end;
