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 Th28:
  x is finite-period & a is finite-period & x in BranchV(a)
  implies ord x = ord a
proof
  assume that
A1: x is finite-period and
A2: a is finite-period and
A3: x in BranchV(a);
  set na = ord a;
  na<>0 by A2,Def5;
  then
A4: na>=1 by NAT_1:14;
  per cases by A4,XXREAL_0:1;
  suppose
A5: na=1;
    then a|^1 in BCK-part(X) by A2,Def5;
    then a in BCK-part(X) by Th4;
    then ex aa being Element of X st a = aa & 0.X <= aa;
    then
A6: a` = 0.X;
    a in AtomSet(X);
    then ex ab being Element of X st a = ab & ab is atom;
    then a = 0.X by A6;
    then
A7: x|^1 in BCK-part(X) by A3,Th4;
    for m being Element of NAT st x|^m in BCK-part(X) & m <> 0 holds 1 <=
    m by NAT_1:14;
    hence thesis by A1,A5,A7,Def5;
  end;
  suppose
A8: na>1;
    0.X in AtomSet(X);
    then
A9: x` = a` by A3,BCIALG_1:37;
A10: for m being Element of NAT st x|^m in BCK-part(X) & m <> 0 holds na <= m
    proof
      let m be Element of NAT;
      assume that
A11:  x|^m in BCK-part(X) and
A12:  m <> 0;
      ex xx being Element of X st x|^m = xx & 0.X <= xx by A11;
      then (x|^m)` = 0.X;
      then a`|^m = 0.X by A9,Th18;
      then (a|^m)` = 0.X by Th17;
      then 0.X <= a|^m;
      then a|^m in BCK-part(X);
      hence thesis by A2,A12,Def5;
    end;
    a|^na in BCK-part(X) by A2,Def5;
    then ex aa being Element of X st a|^na = aa & 0.X <= aa;
    then (a|^na)` = 0.X;
    then x`|^na = 0.X by A9,Th17;
    then (x|^na)` = 0.X by Th18;
    then 0.X <= x|^na;
    then x|^na in BCK-part(X);
    hence thesis by A1,A8,A10,Def5;
  end;
end;
