
theorem LmEMDetX5:
  for L, E being free Z_Module, I being Subset of L, J being Subset of E
  st the ModuleStr of L = the ModuleStr of E & I = J
  holds ( I is Basis of L iff J is Basis of E)
  proof
    let L,E be free Z_Module, I be Subset of L, J be Subset of E;
    assume that
    A1: the ModuleStr of L = the ModuleStr of E and
    A2: I = J;
    hereby
      assume P1: I is Basis of L;
      then I is linearly-independent &
      Lin I = ModuleStr(# the carrier of L, the addF of L,
      the ZeroF of L, the lmult of L #) by VECTSP_7:def 3;
      then P2: J is linearly-independent by A1,A2,LmEMDetX51;
      Lin J = Lin I by A1,A2,LmEMDetX52
      .= ModuleStr(# the carrier of E, the addF of E,
      the ZeroF of E, the lmult of E #) by A1,P1,VECTSP_7:def 3;
      hence J is Basis of E by P2,VECTSP_7:def 3;
    end;
    assume P1: J is Basis of E;
    then J is linearly-independent &
    Lin J = ModuleStr(# the carrier of E, the addF of E,
    the ZeroF of E, the lmult of E #) by VECTSP_7:def 3;
    then P2: I is linearly-independent by A1,A2,LmEMDetX51;
    Lin I = Lin J by A1,A2,LmEMDetX52
    .= ModuleStr(# the carrier of L, the addF of L,
    the ZeroF of L, the lmult of L #) by A1,P1,VECTSP_7:def 3;
    hence I is Basis of L by P2,VECTSP_7:def 3;
  end;
