
theorem Th15:
  for i,j being Integer st i,j are_coprime holds i lcm j = |.i * j.|
proof
  let i,j be Integer;
  assume
A1: i gcd j = 1;
  per cases;
  suppose
A2: i = 0 or j = 0;
    hence i lcm j = 0 by INT_2:4
      .= |.i*j.| by A2,ABSVALUE:2;
  end;
  suppose
A3: i <> 0 & j <> 0;
A4: for m being Integer st i divides m & j divides m holds |.i*j.| divides m
    proof
      j divides i lcm j by INT_2:def 1;
      then consider kj being Integer such that
A5:   j * kj = i lcm j by INT_1:def 3;
      i divides i lcm j by INT_2:def 1;
      then consider ki being Integer such that
A6:   i * ki = i lcm j by INT_1:def 3;
A7:   j divides i * j by INT_2:2;
      i divides i * j by INT_2:2;
      then i lcm j divides i * j by A7,INT_2:def 1;
      then consider kij being Integer such that
A8:   (i lcm j) * kij = i * j by INT_1:def 3;
      i * j = j * (kj * kij) by A5,A8;
      then i = kj * kij by A3,XCMPLX_1:5;
      then
A9:   kij divides i by INT_1:def 3;
      i * j = i * (ki * kij) by A6,A8;
      then j = ki * kij by A3,XCMPLX_1:5;
      then kij divides j by INT_1:def 3;
      then
A10:  kij divides 1 by A1,A9,INT_2:def 2;
      let m be Integer;
      assume that
A11:  i divides m and
A12:  j divides m;
A13:  i lcm j divides m by A11,A12,INT_2:def 1;
      per cases by A10,INT_2:13;
      suppose
        kij = 1;
        hence thesis by A8,A13,ABSVALUE:def 1;
      end;
      suppose
A14:    kij = -1;
        -(i * j) <> 0 by A3,XCMPLX_1:6;
        then -(-(i * j)) < 0 by A8,A14;
        hence thesis by A8,A13,A14,ABSVALUE:def 1;
      end;
    end;
    j divides |.j.| by Th13;
    then j divides |.i.| * |.j.| by INT_2:2;
    then
A15: j divides |.i * j.| by COMPLEX1:65;
    i divides |.i.| by Th13;
    then i divides |.i.| * |.j.| by INT_2:2;
    then i divides |.i * j.| by COMPLEX1:65;
    hence thesis by A15,A4,INT_2:def 1;
  end;
end;
