
theorem LmDE22:
  for L being Z_Lattice,
  v being Vector of DivisibleMod(L), I being Basis of EMbedding(L)
  st for u being Vector of DivisibleMod(L) st u in I
  holds (ScProductDM(L)).(v, u) = 0 holds
  for u being Vector of DivisibleMod(L) holds (ScProductDM(L)).(v, u) = 0
  proof
    let L be Z_Lattice, v be Vector of DivisibleMod(L),
    I be Basis of EMbedding(L) such that
    A1: for u being Vector of DivisibleMod(L) st u in I
    holds (ScProductDM(L)).(v, u) = 0;
    defpred P[Nat] means
    for I being finite Subset of EMbedding(L) st card I = $1
    & I is linearly-independent
    & for u being Vector of DivisibleMod(L) st u in I
    holds (ScProductDM(L)).(v, u) = 0
    holds for w being Vector of DivisibleMod(L) st w in Lin(I) holds
    (ScProductDM(L)).(v, w) = 0;
    P1: P[0]
    proof
      let I be finite Subset of EMbedding(L) such that
      B1: card I = 0 & I is linearly-independent &
      for u being Vector of DivisibleMod(L) st u in I
      holds (ScProductDM(L)).(v, u) = 0;
      I = {}(the carrier of EMbedding(L)) by B1;
      then B2: Lin(I) = (0).EMbedding(L) by ZMODUL02:67;
      thus for w being Vector of DivisibleMod(L) st w in Lin(I)
      holds (ScProductDM(L)).(v, w) = 0
      proof
        let w be Vector of DivisibleMod(L) such that
        C1: w in Lin(I);
        w = 0.EMbedding(L) by B2,C1,ZMODUL02:66
        .= zeroCoset(L) by ZMODUL08:def 3
        .= 0.DivisibleMod(L) by ZMODUL08:def 4;
        hence thesis by ThScDM6;
      end;
    end;
    P2: for n being Nat st P[n] holds P[n+1]
    proof
      let n be Nat such that
      B1: P[n];
      let I be finite Subset of EMbedding(L) such that
      B2: card I = n+1 & I is linearly-independent &
      for u being Vector of DivisibleMod(L) st u in I
      holds (ScProductDM(L)).(v, u) = 0;
      I is non empty by B2;
      then consider u be object such that
      B3: u in I;
      reconsider u as Vector of EMbedding(L) by B3;
      set Iu = I \ {u};
      {u} is Subset of I by B3,SUBSET_1:41;
      then B4: card(Iu) = n+1 - card({u}) by B2,CARD_2:44
      .= n+1 - 1 by CARD_1:30
      .= n;
      reconsider Iu as finite Subset of EMbedding(L);
      I = Iu \/ {u} by B3,XBOOLE_1:45,ZFMISC_1:31;
      then B5: Lin(I) = Lin(Iu) + Lin{u} by ZMODUL02:72;
      B7: Iu c= I by XBOOLE_1:36;
      B6: Iu is linearly-independent by B2,XBOOLE_1:36,ZMODUL02:56;
      B8: for w being Vector of DivisibleMod(L) st w in Iu
      holds (ScProductDM(L)).(v, w) = 0 by B2,B7;
      thus for w being Vector of DivisibleMod(L) st w in Lin(I)
      holds (ScProductDM(L)).(v, w) = 0
      proof
        let w be Vector of DivisibleMod(L) such that
        C1: w in Lin(I);
        consider w1, w2 be Vector of EMbedding(L) such that
        C2: w1 in Lin(Iu) & w2 in Lin{u} & w = w1 + w2 by B5,C1,ZMODUL01:92;
        CX: EMbedding(L) is Submodule of DivisibleMod(L) by ZMODUL08:24;
        then C9: w1 is Vector of DivisibleMod(L) by ZMODUL01:25;
        reconsider ww1 = w1 as Vector of DivisibleMod(L) by CX,ZMODUL01:25;
        consider i be Element of INT.Ring such that
        C4: w2 = i*u by C2,ZMODUL06:19;
        u is Element of DivisibleMod(L) by CX,ZMODUL01:25;
        then C6: (ScProductDM(L)).(v, u) = 0 by B2,B3;
        reconsider uu = u as Element of DivisibleMod(L) by CX,ZMODUL01:25;
        i*uu in DivisibleMod(L);
        then reconsider ww2 = w2 as Vector of DivisibleMod(L)
        by C4,CX,ZMODUL01:29;
        C8: (ScProductDM(L)).(v, i*u) = (ScProductDM(L)).(v, i*uu)
        by CX,ZMODUL01:29
        .= (ScProductDM(L)).(i*uu, v) by ThSPDM1
        .= i* (ScProductDM(L)).(uu, v) by ThSPDM1
        .= i* (ScProductDM(L)).(v, u) by ThSPDM1;
        C10: w = ww1 + ww2 by C2,CX,ZMODUL01:28;
        (ScProductDM(L)).(v, w) = (ScProductDM(L)).(w, v) by ThSPDM1
        .= (ScProductDM(L)).(ww1, v) + (ScProductDM(L)).(ww2, v) by C10,ThSPDM1
        .= (ScProductDM(L)).(v, w1) + (ScProductDM(L)).(ww2, v) by ThSPDM1
        .= (ScProductDM(L)).(v, w1) + (ScProductDM(L)).(v, w2) by ThSPDM1;
        hence thesis by B1,B4,B6,B8,C2,C4,C6,C8,C9;
      end;
    end;
    P3: for n being Nat holds P[n] from NAT_1:sch 2(P1,P2);
    P4: card I is Nat;
    P5: I is linearly-independent & EMbedding(L) = Lin(I) by VECTSP_7:def 3;
    thus for w being Vector of DivisibleMod(L) holds
    (ScProductDM(L)).(v, w) = 0
    proof
      let w be Vector of DivisibleMod(L);
      consider a be Element of INT.Ring such that
      B1: a <> 0.INT.Ring & a * w in EMbedding(L) by ZMODUL08:29;
      (ScProductDM(L)).(v, a*w) = (ScProductDM(L)).(a*w, v) by ThSPDM1
      .= a * (ScProductDM(L)).(w, v) by ThSPDM1
      .= a * (ScProductDM(L)).(v, w) by ThSPDM1;
      hence thesis by A1,B1,P3,P4,P5;
    end;
  end;
