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 Th42:
  for r being Element of F_Real holds
  FPower(r,n) is differentiable Function of REAL,REAL
  proof
    let r be Element of F;
    defpred P[Nat] means
    FPower(r,$1) is differentiable Function of REAL,REAL;
A1: P[0]
    proof
      FPower(r,0) = (the carrier of F) --> r by POLYNOM5:66;
      hence thesis;
    end;
A2: for n being Nat st P[n] holds P[n+1]
    proof
      let n be Nat;
      FPower(r,n+1) = FPower(r,n)(#)id(REAL) by Th41;
      hence thesis;
    end;
    for n being Nat holds P[n] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;
