reserve a,b,c for Integer;
reserve i,j,k,l for Nat;

theorem Th4:
  a = 0 or b = 0 iff a lcm b = 0
proof
A1: b = 0 implies (a lcm b) = 0
  proof
    assume b = 0;
    then 0 divides (a lcm b) by Def1;
    hence thesis;
  end;
A2: (a lcm b) = 0 implies a = 0 or b = 0
  proof
A3: b divides b implies b divides b*a;
    assume
A4: a lcm b = 0;
    a divides a implies a divides a*b;
    then 0 divides a*b by A4,A3,Def1;
    hence thesis by XCMPLX_1:6;
  end;
  a = 0 implies a lcm b = 0
  proof
    assume a = 0;
    then 0 divides (a lcm b) by Def1;
    hence thesis;
  end;
  hence thesis by A1,A2;
end;
