reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;

theorem LMEQR001:
  for V be Z_Module,
  z1, z2 be Element of V, i1, i2 be Element of INT.Ring
  st V is Mult-cancelable holds
  [[z1,i1],[z2,i2]] in EQRZM(V) iff
  i1 <> 0 & i2 <> 0 & i2*z1 = i1*z2
  proof
    let V be Z_Module,
    z1, z2 be Element of V,i1,i2 be Element of INT.Ring;
    assume AS: V is Mult-cancelable;
    hereby
      assume [[z1,i1],[z2,i2]] in EQRZM(V);
      then consider zz1, zz2 be Element of V,ii1,ii2 be Element of INT.Ring
        such that
      P2: [z1,i1] = [zz1,ii1] & [z2,i2] = [zz2,ii2]
      & ii1 <> 0 & ii2 <> 0 & ii2*zz1 = ii1*zz2 by AS,defEQRZM;
      z1 = zz1 & i1 = ii1 & z2 = zz2 & i2 = ii2 by P2,XTUPLE_0:1;
      hence i1 <> 0 & i2 <> 0 & i2*z1 = i1*z2 by P2;
    end;
    assume A2: i1 <> 0 & i2 <> 0 & i2*z1 = i1*z2; then
    A21: not i1 in {0} & not i2 in {0} by TARSKI:def 1;
    i1 in INT \{0} & i2 in INT \{0} by XBOOLE_0:def 5,A21;
    then [z1,i1] in [:the carrier of V,(INT \{0}):] &
    [z2,i2] in [:the carrier of V,(INT \{0}):] by ZFMISC_1:87;
    hence [[z1,i1],[z2,i2]] in EQRZM(V) by A2,AS,defEQRZM;
  end;
