
theorem
  for L being positive-definite Z_Lattice, I being Basis of L,
  v, w being Vector of L holds
  (for u being Vector of L st u in I holds <; v,u ;> = <; w,u ;>)
  implies
  (for u being Vector of L holds <; v,u ;> = <; w,u;>)
  proof
    let L be positive-definite Z_Lattice, I be Basis of L,
    v, w be Vector of L;
    assume AS:
    for u being Vector of L st u in I holds <; v,u ;> = <; w,u ;>;
    P1: for u being Vector of L st u in I holds <; u,v ;> = <; u,w ;>
    proof
      let u be Vector of L;
      assume P0: u in I;
      thus <; u,v ;> = <; v,u ;> by ZMODLAT1:def 3
      .= <; w,u ;> by AS,P0
      .= <; u,w ;> by ZMODLAT1:def 3;
    end;
    thus for u being Vector of L holds <; v,u ;> = <; w,u ;>
    proof
      let u be Vector of L;
      thus <; v,u ;> = <; u,v ;> by ZMODLAT1:def 3
      .= <; u,w ;> by P1,ZL2LmSc1
      .= <; w,u ;> by ZMODLAT1:def 3;
     end;
end;
