
theorem ThDM1:
  for V being torsion-free Z_Module, v being Vector of DivisibleMod(V) holds
  ex a being Element of INT.Ring st a <> 0 & a * v in EMbedding(V)
  proof
    let V be torsion-free Z_Module, v be Vector of DivisibleMod(V);
    A1: v in the carrier of DivisibleMod(V);
    reconsider vv = v as Element of Class EQRZM(V) by defDivisibleMod;
    AX1: v in Class EQRZM(V) by A1,defDivisibleMod;
    v is Element of ModuleStr (# Class EQRZM(V), addCoset(V),
    zeroCoset(V), lmultCoset(V) #) by defDivisibleMod;
    then consider a be Element of INT.Ring, z be Vector of V such that
    A2: a <> 0 & v = Class(EQRZM(V), [z, a]) by ZMODUL04:5;
    reconsider aq = a as Element of F_Rat by NUMBERS:14;
    A3: aq = a / 1 & 1 <> 0;
    AX4: v in Class EQRZM(V) by A1,defDivisibleMod;
    a * v = ((lmultCoset(V)) | [:INT, Class EQRZM(V):]).(a, v)
    by defDivisibleMod
    .= (lmultCoset(V)).(a, v) by AX4,FUNCT_1:49,ZFMISC_1:87
    .= Class(EQRZM(V), [a*z, 1.INT.Ring*a]) by AX1,A2,A3,ZMODUL04:def 4
    .= Class(EQRZM(V), [z, 1.INT.Ring]) by A2,ZMODUL04:4;
    hence thesis by A2,ThEM1;
  end;
