reserve A for non trivial Nat,
        B,C,n,m,k for Nat,
        e for Nat;

theorem Th9:
  for T be _Theta, alpha, epsilon be Real st
      alpha = (1+T*epsilon)|^n & 0<= epsilon <= 1 / (2*n) holds
  ex T1 be _Theta st alpha = 1+T1*2*n*epsilon
proof
  defpred P[Nat] means
    for T be _Theta, alpha, epsilon be Real st
      alpha = (1+T*epsilon)|^$1 & 0<= epsilon <= 1 / (2*$1) holds
    ex T1 be _Theta st alpha = 1+T1*2*$1*epsilon;
A1:P[0]
  proof
    let T be _Theta, alpha, epsilon be Real such that
    A2: alpha = (1+T*epsilon)|^0 & 0<= epsilon <= 1 / (2*0);
    take T;
    thus thesis by A2,NEWTON:4;
  end;
A3:P[m] implies P[m+1]
  proof
    assume
A4:   P[m];set m1=m+1;
    let T be _Theta, alpha, epsilon be Real such that
A5:   alpha = (1+T*epsilon)|^m1 & 0<= epsilon <= 1 / (2*m1);
    per cases;
    suppose
A6:     m=0;
      1<=m1 by NAT_1:11;
      then 1 <= 2*1 <= 2*m1 by XREAL_1:66;
      then 1<= 2*m1 by XXREAL_0:2;
      then reconsider I=1/(2*m1) as _Theta by Def1,XREAL_1:183;
      alpha = 1+T*(I*(2*m1))*epsilon by A5,A6;
      then alpha = 1+(T*I)*2*m1*epsilon;
      hence thesis;
    end;
    suppose
A7:     m>0;
      2*m*1 <= 2*m1*1 by XREAL_1:64,NAT_1:11;
      then 1/(2*m1) <= 1/(2*m) by A7,XREAL_1:102;
      then
A8:     epsilon <= 1/(2*m) by A5,XXREAL_0:2;
      then consider T1 be _Theta such that
A9:     (1+T*epsilon)|^m = 1+T1*2*m*epsilon by A5,A4;
A10:    (1+T*epsilon)|^m1 = (1+T1*(2*m*epsilon)) * (1+T*epsilon)
        by A9,NEWTON:6;
      epsilon*(2*m) <= 1/(2*m) *(2*m) by A8,XREAL_1:64;
      then 0 <= 2*m*epsilon <=1 by A5,XCMPLX_1:87,A7;
      then consider T2 be _Theta such that
A11:    (1+T*epsilon)|^m1 =
      1 + T2 * (2*m*epsilon +2 * epsilon ) by A5,A10,Th3;
      take T2;
      thus thesis by A5,A11;
    end;
  end;
  P[m] from NAT_1:sch 2(A1,A3);
  hence thesis;
end;
