reserve a,b,c for Integer;
reserve i,j,k,l for Nat;
reserve n for Nat;
reserve a,b,c,d,a1,b1,a2,b2,k,l for Integer;
reserve p,p1,q,l for Nat;

theorem
  a lcm b = |.a.| lcm |.b.|
proof
A1: |.b.| = b or |.b.| = -b by ABSVALUE:1;
A2: |.a.| = a or |.a.| = -a by ABSVALUE:1;
A3: now
    let m be Integer;
    assume |.a.| divides m & |.b.| divides m;
    then a divides m & b divides m by A2,A1,Th10;
    hence a lcm b divides m by Def1;
  end;
  b divides a lcm b by Def1;
  then
A4: |.b.| divides a lcm b by A1,Th10;
  a divides a lcm b by Def1;
  then |.a.| divides a lcm b by A2,Th10;
  hence thesis by A4,A3,Def1;
end;
