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
  x\y is finite-period & x in BranchV(a) & y in BranchV(a) implies ord(x \y)=1
proof
  assume that
A1: x\y is finite-period and
A2: x in BranchV(a) & y in BranchV(a);
A3: for m being Element of NAT st (x\y)|^m in BCK-part(X) & m <> 0 holds 1
  <= m by NAT_1:14;
  x\y in BCK-part(X) by A2,BCIALG_1:40;
  then (x\y)|^1 in BCK-part(X) by Th4;
  hence thesis by A1,A3,Def5;
end;
