
theorem Th20:
  for i,j being Integer st i,j are_coprime holds i * j divides i lcm j
proof
  let i,j be Integer;
  assume i,j are_coprime;
  then |.i * j.| divides i lcm j by Th15;
  then consider z being Integer such that
A1: |.i*j.| * z = (i lcm j) by INT_1:def 3;
  per cases;
  suppose
    0 <= i*j;
    then z*(i*j) = (i lcm j) by A1,ABSVALUE:def 1;
    hence thesis by INT_1:def 3;
  end;
  suppose
A2: 0 > i*j;
    (-z)*(i*j) = z*(-(i*j)) .= (i lcm j) by A1,A2,ABSVALUE:def 1;
    hence thesis by INT_1:def 3;
  end;
end;
