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 Th41:
  for r being Element of F_Real holds
  FPower(r,n+1) = FPower(r,n)(#)id(REAL)
  proof
    let r be Element of F;
    reconsider f = FPower(r,n) as Function of REAL,REAL;
    reconsider g = f(#)id(REAL) as Function of F,F;
    now
      let y be Element of F;
      reconsider y1 = y as Element of REAL;
      reconsider n1 = n as Element of NAT by ORDINAL1:def 12;
      thus g.y = f.y1*id(REAL).y1 by VALUED_1:5
      .= r*power(y,n)*y by POLYNOM5:def 12
      .= r*((power F).(y,n1)*y)
      .= r*power(y,n+1) by GROUP_1:def 7;
    end;
    hence thesis by POLYNOM5:def 12;
  end;
