
theorem ThSPEM3:
  for L being Z_Lattice, v1, v2 being Vector of DivisibleMod(L),
  u1, u2 being Vector of EMbedding(L) st v1 = u1 & v2 = u2 holds
  (ScProductEM(L)).(u1, u2) = (ScProductDM(L)).(v1, v2)
  proof
    let L be Z_Lattice, v1, v2 be Vector of DivisibleMod(L),
    u1, u2 be Vector of EMbedding(L) such that
    A1: v1 = u1 & v2 = u2;
    A2: u1 = 1.INT.Ring  * v1 & u2 = 1.INT.Ring * v2 by A1;
    thus (ScProductDM(L)).(v1, v2)
    = (1.F_Real)" * (1.F_Real)" * (ScProductEM(L)).(u1, u2)
    by A2,defScProductDM
    .= (ScProductEM(L)).(u1, u2);
  end;
