reserve i,j,k,n,m,l,s,t for Nat;
reserve a,b for Real;
reserve F for real-valued FinSequence;
reserve z for Complex;
reserve x,y for Complex;
reserve r,s,t for natural Number;

theorem
  m divides l implies m lcm (n gcd l) divides (m lcm n) gcd l
proof
  set M = m lcm n;
  set K = M gcd l;
  set N = n gcd l;
A1: m divides M by NAT_D:def 4;
A2: N divides n by NAT_D:def 5;
A3: N divides l by NAT_D:def 5;
  n divides M by NAT_D:def 4;
  then N divides M by A2,NAT_D:4;
  then
A4: N divides K by A3,NAT_D:def 5;
  assume m divides l;
  then m divides K by A1,NAT_D:def 5;
  hence thesis by A4,NAT_D:def 4;
end;
