
theorem ThSPEM2:
  for L being Z_Lattice holds
  ScProductEM(L) = (ScProductDM(L)) || (rng MorphsZQ(L))
  proof
    let L be Z_Lattice;
    P1: [:the carrier of EMbedding(L), the carrier of EMbedding(L):]
    = [:the carrier of EMbedding(L), rng MorphsZQ(L):] by ZMODUL08:def 3
    .= [:rng MorphsZQ(L), rng MorphsZQ(L):] by ZMODUL08:def 3;
    P2: [:the carrier of DivisibleMod(L), the carrier of DivisibleMod(L):]
    = [:the carrier of DivisibleMod(L), Class EQRZM(L):] by ZMODUL08:def 4
    .= [:Class EQRZM(L), Class EQRZM(L):] by ZMODUL08:def 4;
    A0: EMbedding(L) is Submodule of DivisibleMod(L) by ZMODUL08:24;
    then the carrier of EMbedding(L) c= the carrier of DivisibleMod(L)
    by VECTSP_4:def 2;
    then rng MorphsZQ(L) c= the carrier of DivisibleMod(L) by ZMODUL08:def 3;
    then rng MorphsZQ(L) c= Class EQRZM(L) by ZMODUL08:def 4;
    then [:rng MorphsZQ(L), rng MorphsZQ(L):]
    c= [:Class EQRZM(L), Class EQRZM(L):] by ZFMISC_1:96;
    then reconsider scd = (ScProductDM(L)) || (rng MorphsZQ(L)) as Function of
    [:rng MorphsZQ(L), rng MorphsZQ(L):], the carrier of F_Real
    by P2,FUNCT_2:32;
    for x being object st x in [:rng MorphsZQ(L), rng MorphsZQ(L):]
    holds (ScProductEM(L)).x = scd.x
    proof
      let x be object such that
      B1: x in [:rng MorphsZQ(L), rng MorphsZQ(L):];
      consider x1, x2 be object such that
      B2: x1 in rng MorphsZQ(L) & x2 in rng MorphsZQ(L) & x = [x1, x2]
      by B1,ZFMISC_1:def 2;
      reconsider x1, x2 as Vector of EMbedding(L) by B2,ZMODUL08:def 3;
      set a = 1.(INT.Ring);
      x1 in EMbedding(L);
      then x1 in DivisibleMod(L) by A0,ZMODUL01:24;
      then reconsider xx1 = x1 as Vector of DivisibleMod(L);
      x2 in EMbedding(L);
      then x2 in DivisibleMod(L) by A0,ZMODUL01:24;
      then reconsider xx2 = x2 as Vector of DivisibleMod(L);
      B3: x1 = a * xx1;
      B4: x2 = a * xx2;
      thus (ScProductEM(L)).x
      = (1.F_Real) * (1.F_Real)" * (ScProductEM(L)).(x1,x2) by B2
      .= (ScProductDM(L)).(xx1, xx2) by B3,B4,defScProductDM
      .= scd.x by B2,FUNCT_1:49,ZFMISC_1:87;
    end;
    hence thesis by P1,FUNCT_2:12;
  end;
