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

theorem Th3:
  for T1,T2 be _Theta, lambda, epsilon1,epsilon2 be Real st
     lambda = (1+ T1 * epsilon1) * (1+ T2 * epsilon2) & 0 <= epsilon1 <=1 &
        0<= epsilon2
  holds ex T be _Theta st lambda = 1+ T * (epsilon1 + 2*epsilon2)
proof
  let T1,T2 be _Theta, lambda, epsilon1,epsilon2 be Real such that
A1: lambda = (1+ T1 * epsilon1) * (1+ T2 * epsilon2) &
    0 <= epsilon1 <=1 & 0<= epsilon2;
  reconsider epsilon1 as _Theta by A1,Def1;
  set ep=epsilon1 + 2*epsilon2;
  per cases;
  suppose epsilon1 + 2*epsilon2=0;
    then
A2:   epsilon1=0 & 2*epsilon2=0 by A1;
    take T =0;
    thus thesis by A2,A1;
  end;
  suppose
A3:    ep<>0;
    -1<= T1<=1 & -1<= T2<=1 by Def1;
    then (-1)*epsilon1 <= T1 * epsilon1  <= 1* epsilon1 &
    (-1)*epsilon2 <= T2 * epsilon2  <= 1* epsilon2 by A1,XREAL_1:64;
    then
A4:  (-1)*epsilon1 + (-1)*epsilon2 <=
     (T1 * epsilon1) + (T2 * epsilon2) <= epsilon1+epsilon2 by XREAL_1:7;
    -1 <= T1*T2*epsilon1 <= 1 by Def1;
    then (-1)*epsilon2 <= T1*T2*epsilon1*epsilon2 <= 1*epsilon2
      by A1,XREAL_1:64;
    then (-1)*epsilon1 + (-1)*epsilon2 + (-1)*epsilon2 <=
      (T1 * epsilon1) + (T2 * epsilon2) + T1*T2*epsilon1*epsilon2
      <= epsilon1 + epsilon2 +epsilon2 by A4,XREAL_1:7;
    then (-1) * ep <= (T1 * epsilon1) + (T2 * epsilon2) +
      T1*T2*epsilon1*epsilon2 <= 1* ep;
    then -1 <= ((T1 * epsilon1) + (T2 * epsilon2) +
      T1*T2*epsilon1*epsilon2)/ep <= 1 by A3,A1,XREAL_1:77,79;
    then reconsider T =((T1 * epsilon1) + (T2 * epsilon2) +
    T1*T2*epsilon1*epsilon2)/ep as _Theta by Def1;
    take T;
    T * ep = (T1 * epsilon1) + (T2 * epsilon2) +
    T1*T2*epsilon1*epsilon2 by A3,XCMPLX_1:87;
    hence thesis by A1;
  end;
end;
