
theorem ZL2ThSc1:
  for L being positive-definite Z_Lattice, b being OrdBasis of L,
  v, w being Vector of L
  st for n being Nat st n in dom b holds <; b/.n, v ;> = <; b/.n, w ;>
  holds v = w
  proof
    let L be positive-definite Z_Lattice, b be OrdBasis of L,
    v, w be Vector of L such that
    A1: for n being Nat st n in dom b holds <; b/.n, v ;> = <; b/.n, w ;>;
    reconsider I = rng b as Basis of L by ZMATRLIN:def 5;
    for u being Vector of L st u in I holds <; u, v ;> = <; u, w ;>
    proof
      let u be Vector of L such that
      B1: u in I;
      consider i be Nat such that
      B2: i in dom b & b.i = u by B1,FINSEQ_2:10;
      b/.i = u by B2,PARTFUN1:def 6;
      hence thesis by A1,B2;
    end;
    then <; v - w, v ;> = <; v - w, w ;> by ZL2LmSc1;
    then <; v - w, v ;> - <; v - w, w ;> = 0.INT.Ring;
    then ||. v - w .|| = 0.INT.Ring by ZMODLAT1:11;
    then v - w = 0.L by ZMODLAT1:16;
    hence thesis by RLVECT_1:21;
  end;
