
theorem ZL2ThSc1X:
  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 <; v, b/.n;> = <; w, b/.n ;>
  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 <; v,b/.n ;> = <; w,b/.n;>;
    for n being Nat st n in dom b holds <; b/.n,v ;> = <; b/.n,w;>
    proof
      let n be Nat;
      assume A2: n in dom b;
      thus <; b/.n,v ;> = <; v,b/.n ;> by ZMODLAT1:def 3
      .= <; w,b/.n ;> by A1,A2
      .= <; b/.n,w ;> by ZMODLAT1:def 3;
    end;
    hence thesis by ZL2ThSc1;
  end;
