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

theorem LMEQR003:
  for V be Z_Module,
  z be Element of V,
  i, n be Element of INT.Ring
  st i <> 0.INT.Ring & n <> 0.INT.Ring & V is Mult-cancelable holds
  Class(EQRZM(V),[z,i]) = Class(EQRZM(V),[n*z,n*i])
  proof
    let V be Z_Module, z be Element of V,
    i, n be Element of INT.Ring;
    assume AS: i <> 0.INT.Ring & n <> 0.INT.Ring & V is Mult-cancelable; then
    B61: not i in {0} by TARSKI:def 1;
    i in INT \ {0} by XBOOLE_0:def 5,B61; then
    X1: [z,i] in [:the carrier of V,(INT \{0}):] by ZFMISC_1:87;
    (n*i)*z = (i*n)*z .= i*(n*z) by VECTSP_1:def 16;
    then [[z,i], [n*z,n*i]] in EQRZM(V) by AS,LMEQR001;
    hence Class(EQRZM(V),[z,i])= Class(EQRZM(V),[n*z,n*i]) by X1,EQREL_1:35;
  end;
