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

theorem Th2:
  for T be _Theta, lambda, epsilon1,epsilon2 be Real st
     lambda = T * epsilon1 & |.epsilon1.| <= |.epsilon2.|
   ex T1 be _Theta st lambda = T1 * epsilon2
proof
  let T be _Theta, lambda, epsilon1,epsilon2 be Real such that
A1:  lambda = T * epsilon1 & |.epsilon1.| <= |.epsilon2.|;
  per cases;
  suppose
A2:   epsilon2=0;
    take T1=0;
    epsilon1=0 by A2,A1;
    hence thesis by A1;
  end;
  suppose
A3:   epsilon2 > 0;
    then
A4:   |.epsilon2.| = epsilon2 by ABSVALUE:def 1;
    per cases;
    suppose
A5:     epsilon1 >= 0;
      then |.epsilon1.| = epsilon1 by ABSVALUE:def 1;
      then reconsider T1 = epsilon1 / epsilon2 as _Theta
        by Def1,XREAL_1:183,A5,A1,A4;
      take TT=T*T1;
      thus TT * epsilon2 = T * (epsilon1 / epsilon2 * epsilon2)
        .= lambda by A1,XCMPLX_1:87,A3;
    end;
    suppose
A6:     epsilon1 < 0;
      then
A7:    -epsilon1 >0;
      |.epsilon1.| = - epsilon1 by A6,ABSVALUE:def 1;
      then
A8:     (- epsilon1) / epsilon2 is theta by XREAL_1:183,A7,A1,A4;
      -(- epsilon1) / epsilon2 = --((epsilon1) / epsilon2) by XCMPLX_1:187;
      then reconsider T1 = (epsilon1) / epsilon2 as _Theta by A8;
      take TT=T*T1;
      thus TT * epsilon2 = T * ((epsilon1) / epsilon2 * epsilon2)
        .= lambda by A1,XCMPLX_1:87,A3;
    end;
  end;
  suppose
A9:   epsilon2 < 0;
    then
A10:  |.epsilon2.| = - epsilon2 by ABSVALUE:def 1;
    per cases;
    suppose
A11:    epsilon1 >= 0;
      then |.epsilon1.| = epsilon1 by ABSVALUE:def 1;
      then
A12:   (epsilon1) / -epsilon2 is theta by XREAL_1:183,A11,A1,A10;
      -(epsilon1) / (-epsilon2) = --((epsilon1) / epsilon2) by XCMPLX_1:188;
      then reconsider T1 = (epsilon1) / epsilon2 as _Theta by A12;
      take TT=T*T1;
      thus TT * epsilon2 = T * (epsilon1 / epsilon2 * epsilon2)
        .= lambda by A1,XCMPLX_1:87,A9;
    end;
    suppose
A13:    epsilon1 < 0;
      then
A14:    -epsilon1 >0;
      |.epsilon1.| = - epsilon1 by A13,ABSVALUE:def 1;
      then (- epsilon1) / (- epsilon2) is theta by XREAL_1:183,A14,A1,A10;
      then reconsider T1 = (epsilon1) / epsilon2 as _Theta by XCMPLX_1:191;
      take TT=T*T1;
      thus TT * epsilon2 = T * ((epsilon1) / epsilon2 * epsilon2)
        .= lambda by XCMPLX_1:87,A9,A1;
    end;
  end;
end;
