reserve X for BCI-algebra;
reserve n for Nat;
reserve x,y for Element of X;
reserve a,b for Element of AtomSet(X);
reserve m,n for Nat;
reserve i,j for Integer;

theorem
  a\b is finite-period & x is finite-period & y is finite-period & a is
finite-period & b is finite-period & x in BranchV(a) & y in BranchV(b) implies
  ord(a\b) divides (ord x lcm ord y)
proof
  assume that
A1: a\b is finite-period and
A2: x is finite-period and
A3: y is finite-period and
A4: a is finite-period and
A5: b is finite-period and
A6: x in BranchV(a) and
A7: y in BranchV(b);
  ord y divides (ord x lcm ord y) by NAT_D:def 4;
  then consider yx being Nat such that
A8: ord x lcm ord y =(ord y)* yx by NAT_D:def 3;
  reconsider na = ord a as Element of NAT;
  reconsider xly = ord x lcm ord y as Element of NAT;
  ord x divides (ord x lcm ord y) by NAT_D:def 4;
  then consider xy being Nat such that
A9: ord x lcm ord y =(ord x)* xy by NAT_D:def 3;
  (a\b)|^xly = a|^((ord x)* xy)\b|^((ord y)* yx) by A9,A8,Th15
    .=(a|^(ord x))|^xy\b|^((ord y)* yx) by Th23
    .=(a|^(ord x))|^xy\b|^(ord y)|^ yx by Th23
    .=(a|^na)|^xy\b|^(ord y)|^ yx by A2,A4,A6,Th28
    .=(0.X)|^xy\b|^(ord y)|^ yx by A4,Th26
    .=(0.X)|^xy\b|^(ord b)|^ yx by A3,A5,A7,Th28
    .=(0.X)|^xy\(0.X)|^ yx by A5,Th26
    .=((0.X)|^ yx)` by Th7
    .=(0.X)` by Th7
    .=0.X by BCIALG_1:def 5;
  then ((a\b)|^xly)` = 0.X by BCIALG_1:def 5;
  then 0.X <=(a\b)|^xly;
  then (a\b)|^xly in BCK-part(X);
  hence thesis by A1,Th29;
end;
