
theorem
  for L being INTegral Z_Lattice, r being non zero Element of F_Rat,
  a being Rational, v, u being Vector of EMLat(r, L)
  st r = a holds
  a" * a" * <; v, u ;> in INT
  proof
    let L be INTegral Z_Lattice, r be non zero Element of F_Rat,
    a be Rational, v, u be Vector of EMLat(r, L) such that
    A1: r = a;
    consider m0, n0 be Integer such that
    A2: n0 > 0 & a = m0/n0 by RAT_1:2;
    reconsider n=n0,m=m0 as Element of INT.Ring by INT_1:def 2;
    consider vv be Vector of EMLat(L) such that
    A3: n*v = m*vv by A1,A2,ThrEMLat1;
    consider uu be Vector of EMLat(L) such that
    A4: n*u = m*uu by A1,A2,ThrEMLat1;
    r <> 0.F_Rat;
    then A5: m <> 0 by A1,A2;
    A6: n * n * <; v, u ;> = n * (n * <; v, u ;>)
    .= n * <; v, n*u ;> by ZMODLAT1:9
    .= <; n*v, n*u ;> by ZMODLAT1:def 3
    .= <; m*vv, m*uu ;> by A3,A4,ThrEMLat2
    .= m * <; vv, m*uu ;> by ZMODLAT1:def 3
    .= m * (m * <; vv, uu ;>) by ZMODLAT1:9
    .= m * m * <; vv, uu ;>;
    A7: (1/m0) * (1/m0) * (n0 * n0 * <; v, u ;>)
    = (n0/m0) * (n0/m0) * <; v, u ;>
    .= a" * (n0/m0) * <; v, u ;> by A2,XCMPLX_1:213
    .= a" * a" * <; v, u ;> by A2,XCMPLX_1:213;
    (1/m0) * (1/m0) * (m0 * m0 * <; vv, uu ;>)
    = m0 * (1/m0) * (m0 * (1/m0) * <; vv, uu ;>)
    .= 1 * (m0 * (1/m0) * <; vv, uu ;>) by A5,XCMPLX_1:106
    .= 1.F_Real  * <; vv, uu ;> by A5,XCMPLX_1:106
    .= <; vv, uu ;>;
    hence thesis by A6,A7,ZMODLAT1:def 5;
  end;
