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

theorem LMEQRZM1:
  for V be Z_Module,
  v be Element of
  ModuleStr (# Class EQRZM(V), addCoset(V), zeroCoset(V), lmultCoset(V) #)
  st V is Mult-cancelable holds
  ex i be Element of INT.Ring, z be Element of V
  st i <> 0.INT.Ring & v = Class(EQRZM(V),[z,i])
  proof
    let V be Z_Module, v be Element of
    ModuleStr (# Class EQRZM(V), addCoset(V), zeroCoset(V), lmultCoset(V) #);
    assume V is Mult-cancelable;
    v in Class EQRZM(V);
    then consider A1 be object such that
    A1: A1 in [:the carrier of V,(INT \{0}):] & v=Class(EQRZM(V),A1)
    by EQREL_1:def 3;
    consider z, i be object such that
    A2: z in the carrier of V & i in INT \{0} & A1 = [z,i]
    by A1,ZFMISC_1:def 2;
    reconsider z as Element of V by A2;
    A31: i in INT & not i in {0} by XBOOLE_0:def 5,A2;
    reconsider i as Integer by A2;
    reconsider i as Element of INT.Ring by A2;
    take i,z;
    thus i <> 0.INT.Ring & v = Class(EQRZM(V),[z,i])
      by A1,A2,A31,TARSKI:def 1;
  end;
