
theorem Th17:
  for i,j being Integer holds (i * j) gcd i = |.i.|
proof
  let i,j be Integer;
A1: for m being Integer st m divides i*j & m divides i holds m divides |.i.|
  proof
    let m be Integer;
    assume that
    m divides i*j and
A2: m divides i;
    consider k being Integer such that
A3: m * k = i by A2,INT_1:def 3;
    i divides |.i.| by Th13;
    then consider l being Integer such that
A4: i * l = |.i.| by INT_1:def 3;
    m * (k * l) = |.i.| by A3,A4;
    hence thesis by INT_1:def 3;
  end;
A5: |.i.| divides i by Th13;
  then |.i.| divides i*j by INT_2:2;
  hence thesis by A5,A1,INT_2:def 2;
end;
