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 Th25:
  x is finite-period implies x`` is finite-period
proof
  reconsider b = x`` as Element of AtomSet(X) by BCIALG_1:34;
  assume x is finite-period;
  then consider p being Element of NAT such that
A1: p<>0 and
A2: x|^p in BCK-part(X);
  ex y being Element of X st y = x|^p & 0.X <= y by A2;
  then (x|^p)` = 0.X;
  then (b|^p)` = 0.X by Th21;
  then 0.X <= b|^p;
  then b|^p in BCK-part(X);
  hence thesis by A1;
end;
