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) implies x|^ (-n) = 0.X
proof
  defpred P[Nat] means x |^ (-$1) = 0.X;
  assume x in BCK-part(X);
  then
A1: ex y being Element of X st y=x & 0.X<=y;
A2: now
    let n;
    assume
A3: P[n];
    per cases;
    suppose -n=0;
      then x |^(-(n+1)) =x` by Th5
        .=0.X by A1;
      hence P[n+1];
    end;
    suppose
A4:   -n<0;
      then BCI-power(X).(x`,|.-n.|) = 0.X by A3,Def2;
      then BCI-power(X).(x`,--n) = 0.X by A4,ABSVALUE:def 1;
      then x`\(x` |^ n )` =(x\0.X)` by BCIALG_1:9
        .=x` by BCIALG_1:2;
      then x`\(x` |^ n )`= 0.X by A1;
      then 0.X = x`|^ (n+1) by Th2
        .=BCI-power(X).(x`,-(-(n+1)))
        .=BCI-power(X).(x`,|.-(n+1).|) by ABSVALUE:def 1
        .=x |^ (-(n+1)) by Def2;
      hence P[n+1];
    end;
  end;
  x |^ (-0) = x |^ 0
  .= 0.X by Def1; then
A5: P[0];
  for n holds P[n] from NAT_1:sch 2(A5,A2);
  hence thesis;
end;
