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 lcm (n lcm k) = (m lcm n) lcm k
proof
  set M = n lcm k;
  set K = m lcm M;
  set N = m lcm n;
  set L = N lcm k;
A1: m divides K by NAT_D:def 4;
A2: M divides K by NAT_D:def 4;
  n divides M by NAT_D:def 4;
  then n divides K by A2,NAT_D:4;
  then
A3: N divides K by A1,NAT_D:def 4;
  k divides M by NAT_D:def 4;
  then k divides K by A2,NAT_D:4;
  then
A4: L divides K by A3,NAT_D:def 4;
A5: N divides L by NAT_D:def 4;
A6: k divides L by NAT_D:def 4;
  n divides N by NAT_D:def 4;
  then n divides L by A5,NAT_D:4;
  then
A7: M divides L by A6,NAT_D:def 4;
  m divides N by NAT_D:def 4;
  then m divides L by A5,NAT_D:4;
  then K divides L by A7,NAT_D:def 4;
  hence thesis by A4,NAT_D:5;
end;
