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 :: P39
  x is positive Element of X iff x is nilpotent & ord x=1
proof
  thus x is positive Element of X implies x is nilpotent&ord x=1
  proof
    assume x is positive Element of X;
    then 0.X<=x by Def2;
    then
A1: x`=0.X;
    thus
A2: x is nilpotent
    proof
      set k=1;
      take k;
      thus thesis by A1,Th2;
    end;
    thus ord x=1
    proof
      set k=1;
      reconsider k as non zero Nat;
      (0.X,x) to_power k=0.X & for m being Nat st (0.X,x)
      to_power m=0.X &m <> 0 holds k<= m by A1,Th2,NAT_1:14;
      hence thesis by A2,Def8;
    end;
  end;
  assume x is nilpotent & ord x=1;
  then (0.X,x) to_power 1 = 0.X by Def8;
  then x`=0.X by Th2;
  then 0.X<=x;
  hence thesis by Def2;
end;
