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 in BCK-part(X) & n>=1 implies x|^n = x
proof
  assume that
A1: x in BCK-part(X) and
A2: n>=1;
  defpred P[Nat] means x |^ $1 = x;
A3: ex y being Element of X st y=x & 0.X<=y by A1;
A4: now
    let n;
    assume n>=1;
    assume P[n];
    then x |^ (n+1) =x\x` by Th2
      .=x\0.X by A3
      .= x by BCIALG_1:2;
    hence P[n+1];
  end;
A5: P[1] by Th4;
  for n st n>=1 holds P[n] from NAT_1:sch 8(A5,A4);
  hence thesis by A2;
end;
