
theorem Th14:
  for i,j being Integer holds i gcd j = i gcd |.j.|
proof
  let i,j be Integer;
  set k = i gcd |.j.|;
  k divides |.j.| by INT_2:def 2;
  then consider x being Integer such that
A1: k*x = |.j.| by INT_1:def 3;
  |.j.| divides j by Th13;
  then consider y being Integer such that
A2: |.j.| * y = j by INT_1:def 3;
A3: for m being Integer st m divides i & m divides j holds m divides k
  proof
    j divides |.j.| by Th13;
    then consider y being Integer such that
A4: j * y = |.j.| by INT_1:def 3;
    let m be Integer;
    assume that
A5: m divides i and
A6: m divides j;
    consider x being Integer such that
A7: m * x = j by A6,INT_1:def 3;
    m * (x * y) = |.j.| by A7,A4;
    then m divides |.j.| by INT_1:def 3;
    hence thesis by A5,INT_2:def 2;
  end;
  k * (x * y) = j by A1,A2;
  then
A8: k divides j by INT_1:def 3;
  k divides i by INT_2:def 2;
  hence thesis by A8,A3,INT_2:def 2;
end;
