
theorem ThrEMLat2:
  for L being Z_Lattice, r being Element of F_Rat,
  v, u being Vector of EMLat(r, L), x, y being Vector of EMLat(L)
  st v = x & u = y holds <; v, u ;> = <; x, y ;>
  proof
    let L be Z_Lattice, r be Element of F_Rat,
    v, u be Vector of EMLat(r, L), x, y be Vector of EMLat(L) such that
    A1: v = x & u = y;
    v in the carrier of EMLat(L)& u in the carrier of EMLat(L) by A1;
    then A2: v in rng MorphsZQ(L) & u in rng MorphsZQ(L) by defEMLat;
    v in the carrier of EMLat(r, L) & u in the carrier of EMLat(r, L);
    then v in (r * rng MorphsZQ(L)) & u in (r * rng MorphsZQ(L))
    by defrEMLat;
    then A3: v is Vector of EMbedding(r, L) & u is Vector of EMbedding(r, L)
    by ZMODUL08:def 6;
    thus <; v, u ;> = ((ScProductDM(L)) || (r * rng MorphsZQ(L))).(v, u)
    by defrEMLat
    .= ((ScProductDM(L)) || the carrier of EMbedding(r, L)).(v, u)
    by ZMODUL08:def 6
    .= (ScProductDM(L)).(v, u) by A3,ThSPrEM1
    .= ((ScProductDM(L)) || rng MorphsZQ(L)).(v, u)
    by A2,FUNCT_1:49,ZFMISC_1:87
    .= (ScProductEM(L)).(x, y) by A1,ThSPEM2
    .= <; x, y ;> by defEMLat;
  end;
