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 is finite-period & x` is finite-period implies ord x = ord(x`)
proof
  assume that
A1: x is finite-period and
A2: x` is finite-period;
  set m = ord(x`);
  (x`)|^m in BCK-part(X) by A2,Def5;
  then ex zz being Element of X st zz = (x`)|^m & 0.X <= zz;
  then ((x`)|^m)` = 0.X;
  then (x``)|^m = 0.X by Th18;
  then ((x``)|^m)` = 0.X by BCIALG_1:def 5;
  then (x|^m)` = 0.X by Th21;
  then 0.X <= x|^m;
  then
A3: x|^m in BCK-part(X);
  set n = ord x;
  m<>0 by A2,Def5;
  then
A4: n<=m by A1,A3,Def5;
  x|^n in BCK-part(X) by A1,Def5;
  then ex zz being Element of X st zz = x|^n & 0.X <= zz;
  then (x|^n)` = 0.X;
  then (x`)|^n = 0.X by Th18;
  then ((x`)|^n)` = 0.X by BCIALG_1:def 5;
  then 0.X <= (x`)|^n;
  then
A5: (x`)|^n in BCK-part(X);
  n<>0 by A1,Def5;
  then m<=n by A2,A5,Def5;
  hence thesis by A4,XXREAL_0:1;
end;
