
theorem Th70:
  for x be Element of F_Complex for n be Nat ex f be
  Function of COMPLEX,COMPLEX st f = FPower(x,n) & f is_continuous_on COMPLEX
proof
  let x be Element of F_Complex;
  defpred P[Nat] means
ex f be Function of COMPLEX,COMPLEX st f =
  FPower(x,$1) & f is_continuous_on COMPLEX;
A1: the carrier of F_Complex = COMPLEX by COMPLFLD:def 1;
A2: for n be Nat st P[n] holds P[n+1]
  proof
    let n be Nat;
    reconsider g=FPower(x,n+1) as Function of COMPLEX,COMPLEX by A1;
    given f be Function of COMPLEX,COMPLEX such that
A3: f = FPower(x,n) & f is_continuous_on COMPLEX;
    take g;
    thus g = FPower(x,n+1);
     ex f1 be Function of COMPLEX,COMPLEX st
     f1 = FPower(x,n) & FPower(x,n+1) = f1(#)id(COMPLEX) by Th69;
    hence thesis by A3,Th62,CFCONT_1:43;
  end;
A4: P[0]
  proof
    reconsider f=FPower(x,0) as Function of COMPLEX,COMPLEX by A1;
    take f;
    thus f = FPower(x,0);
    f = (the carrier of F_Complex) --> x by Th66;
    hence thesis by A1,Th63;
  end;
  thus for n be Nat holds P[n] from NAT_1:sch 2(A4,A2);
end;
