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

theorem Th7:
  for T1 be _Theta, epsilon be Real st 0<= epsilon <= 1/2
    ex T2 be _Theta st
          1/ (1 + T1*epsilon) = 1 + T2*2* epsilon
proof
  let T1 be _Theta, epsilon be Real such that
A1:  0<= epsilon <= 1/2;
A2: 1 - epsilon >=1- 1/2 by A1, XREAL_1:10;
  -1<= T1 <= 1 by Def1;
  then (-1) *epsilon <= T1*epsilon by A1,XREAL_1:64;
  then
A3: 1 + (-1) *epsilon <= 1+ T1*epsilon by XREAL_1:6;
  then 1/(1+ T1*epsilon) <= 1 / (1- epsilon) by A2,XREAL_1:118;
  then 1/(1+ T1*epsilon)*epsilon <= 1 / (1- epsilon)*epsilon by A1,XREAL_1:64;
  then 1/(1+ T1*epsilon)*epsilon <= (1*epsilon) / (1- epsilon) by XCMPLX_1:74;
  then
A4: (1*epsilon) /(1+ T1*epsilon) <= (1*epsilon) / (1- epsilon) by XCMPLX_1:74;
  reconsider I=1 as _Theta by Def1;
A5: |. 2*epsilon.| = 2*epsilon by A1,ABSVALUE:def 1;
A6:  (epsilon / (1- epsilon)) <= epsilon/ (1/2)= 2*epsilon
    by A1,XREAL_1:118,A2;
  per cases;
  suppose T1<0;
    then
A7:    |.(-T1)* (epsilon / (1+T1 *epsilon)) .| =
    (-T1)*(epsilon / (1+T1 * epsilon)) by A1,A2,A3,ABSVALUE:def 1;
    -T1 <= 1 by Def1;
    then (-T1)*(epsilon /(1+ T1*epsilon))<= 1 *(epsilon /(1+ T1*epsilon))
      by A1,A2,A3, XREAL_1:64;
    then (-T1)*(epsilon /(1+ T1*epsilon))<= (epsilon) / (1- epsilon)
      by A4,XXREAL_0:2;
    then I* ((-T1)*(epsilon /(1+ T1*epsilon))) <= 2*epsilon by A6,XXREAL_0:2;
    then consider T be _Theta such that
A8:   (-T1)* (epsilon / (1+T1*epsilon)) = T * (2*epsilon) by Th2,A7,A5;
    1 - (T1*epsilon) / (1+T1*epsilon) =
    (1+ T1* epsilon)/(1+T1* epsilon) - (T1*epsilon) / (1+T1* epsilon)
    by A2,A3,XCMPLX_1:60
    .= (1+ T1* epsilon- T1*epsilon) / (1+T1* epsilon) by XCMPLX_1:120
    .= 1 / (1+T1* epsilon);
    then 1/ (1 + T1*epsilon) = 1 - T1*(epsilon / (1+T1*epsilon))
      by XCMPLX_1:74;
    then 1/ (1 + T1*epsilon) = 1 + T*2* epsilon by A8;
    hence thesis;
  end;
  suppose T1>=0; then
A9:   |.T1* (epsilon / (1+T1 *epsilon)) .| =
    (T1*(epsilon / (1+T1 * epsilon))) by A1,ABSVALUE:def 1;
    T1 <= 1 by Def1;
    then T1*(epsilon /(1+ T1*epsilon))<= 1 *(epsilon /(1+ T1*epsilon))
      by A1,A2,A3, XREAL_1:64;
    then T1*(epsilon /(1+ T1*epsilon))<= (epsilon) / (1- epsilon)
      by A4,XXREAL_0:2;
    then I* (T1*(epsilon /(1+ T1*epsilon))) <= 2*epsilon by A6,XXREAL_0:2;
    then consider T be _Theta such that
A10:  T1* (epsilon / (1+T1*epsilon)) = T * (2*epsilon) by Th2,A9,A5;
    1 - (T1*epsilon) / (1+T1*epsilon) =
    (1+ T1* epsilon)/(1+T1* epsilon) - (T1*epsilon) / (1+T1* epsilon)
      by A2,A3,XCMPLX_1:60
      .= (1+ T1* epsilon- T1*epsilon) / (1+T1* epsilon) by XCMPLX_1:120
      .= 1 / (1+T1* epsilon);
    then 1/ (1 + T1*epsilon) = 1 - T * (2*epsilon) by A10,XCMPLX_1:74;
    then 1/ (1 + T1*epsilon) = 1 + (-T)*2* epsilon;
    hence thesis;
  end;
end;
