
theorem ThND1:
  for V being non trivial free Z_Module, v being non zero Vector of V
  holds v is non divisible
  proof
    let V be non trivial free Z_Module, v be non zero Vector of V;
    reconsider i2=2 as Element of INT.Ring by INT_1:def 2;
    set I = the Basis of V;
    A1: I is linearly-independent & (Omega).V = Lin(I) by VECTSP_7:def 3;
    consider L be Linear_Combination of I, u be Vector of V such that
    A2: v = Sum(L) & u in I & L.u <> 0 by LmND1;
    assume v is divisible;
    then consider w be Vector of V such that
    A5: (i2*L.u)*w = v by A2;
    w in Lin(I) by A1;
    then consider Lw be Linear_Combination of I such that
    A6: w = Sum(Lw) by ZMODUL02:64;
    reconsider Luw = (i2*L.u)*Lw as Linear_Combination of I by ZMODUL02:31;
    A8: Sum(Luw) = Sum(L) by A2,A5,A6,ZMODUL02:53;
    Carrier(Luw) c= I & Carrier(L) c= I by VECTSP_6:def 4;
    then Luw = L by A8,VECTSP_7:def 3,ZMODUL03:3;
    then L.u = (i2*L.u)*Lw.u by VECTSP_6:def 9
    .= (i2*Lw.u)*L.u;
    then i2*Lw.u = 1 by A2,XCMPLX_1:7;
    hence contradiction by INT_1:9;
  end;
