
theorem LmEMDetX8:
  for L being Z_Lattice, b being FinSequence of L
  holds b is OrdBasis of L
  iff (MorphsZQ(L))*b is OrdBasis of EMbedding(L)
  proof
    let L be Z_Lattice, b be FinSequence of L;
    E1: ex T being linear-transformation of L, EMbedding(L)
    st T is bijective & T = MorphsZQ(L) &
    (for v being Vector of L holds T.v = Class(EQRZM(L),[v,1]))
    by ZMODUL08:21;
    E4: (MorphsZQ(L))*b is FinSequence of EMbedding(L) by E1,FINSEQ_2:32;
    P3: rng ((MorphsZQ(L))*b) =(MorphsZQ(L)).: (rng b) by RELAT_1:127;
    hereby
      assume b is OrdBasis of L;
      then P1: b is one-to-one & rng b is Basis of L by ZMATRLIN:def 5;
      then (MorphsZQ(L)).: (rng b) is Basis of EMbedding(L) by LmEMDetX6;
      hence (MorphsZQ(L))*b is   OrdBasis of EMbedding(L)
      by E1,E4,P1,P3,ZMATRLIN:def 5;
    end;
    assume (MorphsZQ(L))*b is OrdBasis of EMbedding(L);
    then P1: (MorphsZQ(L))*b is one-to-one
    & rng ((MorphsZQ(L))*b) is Basis of EMbedding(L)
    by ZMATRLIN:def 5;
    dom (MorphsZQ(L)) = the carrier of L by FUNCT_2:def 1;
    then rng b c= dom (MorphsZQ(L));
    then P2: b is one-to-one by P1,FUNCT_1:25;
    rng b is Basis of L by LmEMDetX6,P1,P3 ;
    hence b is OrdBasis of L by ZMATRLIN:def 5,P2;
  end;
