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

theorem Th4:
  for T1,T2 be _Theta,epsilon1,epsilon2 be Real st
    T1*epsilon1 <= epsilon2 & epsilon2 <= T2 * epsilon1
  holds ex T be _Theta st epsilon2 = T * epsilon1
proof
  let T1,T2 be _Theta,epsilon1,epsilon2 be Real such that
A1:  T1*epsilon1 <= epsilon2 & epsilon2 <= T2 * epsilon1;
  per cases;
  suppose
A2: epsilon1=0;
    then epsilon2=0 & 0 = 0*0 by A1;
    hence thesis by A2;
  end;
  suppose
A3: epsilon1>0; then
    -1<= T1 <= epsilon2/epsilon1 <= T2 <=1 by A1,XREAL_1:77,79,Def1;
    then -1 <= epsilon2/epsilon1 <=1 by XXREAL_0:2;
    then reconsider T=epsilon2/epsilon1 as _Theta by Def1;
    T*epsilon1 = epsilon2 by A3, XCMPLX_1:87;
    hence thesis;
  end;
  suppose
A4:   epsilon1<0;
    1>=T1 >= epsilon2/epsilon1 >= T2 >=-1 by A1,A4,XREAL_1:78,80,Def1;
    then -1 <= epsilon2/epsilon1 <=1 by XXREAL_0:2;
    then reconsider T=epsilon2/epsilon1 as _Theta by Def1;
    T*epsilon1 = epsilon2 by A4, XCMPLX_1:87;
    hence thesis;
  end;
end;
