reserve c for Complex;
reserve r for Real;
reserve m,n for Nat;
reserve f for complex-valued Function;
reserve f,g for differentiable Function of REAL,REAL;
reserve L for non empty ZeroStr;
reserve x for Element of L;
reserve p,q for Polynomial of F_Real;

theorem Th40:
  #Z n = FPower(1.F_Real,n)
  proof
    reconsider f = FPower(1.F,n) as Function of REAL,REAL;
    #Z n = f
    proof
      let r be Element of REAL;
      thus ( #Z n).r = r #Z n by TAYLOR_1:def 1
      .= r |^ n by PREPOWER:36
      .= 1.F * power(In(r,F),n) by Th39
      .= f.r by POLYNOM5:def 12;
    end;
    hence thesis;
  end;
