reserve a,b,c,h for Integer;
reserve k,m,n for Nat;
reserve i,j,z for Integer;
reserve p for Prime;

theorem
  for i,j,k being Integer st i <> 0 holds
  i divides k & i*j,k are_coprime implies i = 1 or i = -1
  proof
    let i,j,k be Integer such that
A1: i <> 0 and
A2: i divides k;
A3: |.i*j.| = |.i.|*|.j.| by COMPLEX1:65;
    assume i*j,k are_coprime;
    then |.i*j.|,|.k.| are_coprime by INT_2:34;
    then
A4: |.i.| = 1 by A1,A2,A3,INT_2:16,GROUP_22:2;
    |.i.| = i or |.i.| = -i by COMPLEX1:71;
    hence thesis by A4;
  end;
