
theorem
  for n,k,m being Nat holds
    n divides k * m implies
      ex a,b being Nat st a divides k & b divides m & n = a * b
  proof
    let n,k,m be Nat;
    assume
A1: n divides k*m;
    take a = k gcd n;
A2: a divides k & a divides n by INT_2:def 2;
    per cases;
    suppose
A3:   a = 0;
      take m;
      thus thesis by A3;
    end;
    suppose
A3:   a <> 0;
      consider b being Nat such that
A4:   n = a * b by A2,NAT_D:def 3;
      take b;
      A7: n divides m * n;
      a * m = |.m.| * a .= (m*k) gcd (m*n) by INT_6:16;
      then n divides a * m by A1,A7,INT_2:def 2;
      hence thesis by A3,A4,INT_4:7,INT_2:def 2;
    end;
  end;
