reserve X for BCI-algebra;
reserve I for Ideal of X;
reserve a,x,y,z,u for Element of X;
reserve f,f9,g for sequence of  the carrier of X;
reserve j,i,k,n,m for Nat;

theorem
  x is nilpotent implies ord x = ord (x`)
proof
  assume
A1: x is nilpotent;
  then consider k being non zero Nat such that
A2: (0.X,x) to_power k = 0.X;
A3: x` is nilpotent
  proof
    take k;
    (0.X,x`) to_power k = ((0.X,x) to_power k)` by Th9
      .=0.X by A2,BCIALG_1:def 5;
    hence thesis;
  end;
  set k=ord x;
A4: now
    let m be Nat;
    assume that
A5: (0.X,x`)to_power m=0.X and
A6: m <> 0;
    ((0.X,x)to_power m)`=0.X by A5,Th9;
    then (((0.X,x)to_power m)`)`=0.X by BCIALG_1:def 5;
    then (0.X,x)to_power m=0.X by Th12;
    hence k<=m by A1,A6,Def8;
  end;
  (0.X,x`) to_power k = ((0.X,x) to_power k)` by Th9
    .=(0.X)` by A1,Def8
    .=0.X by BCIALG_1:def 5;
  hence thesis by A3,A4,Def8;
end;
