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 |^(-n) =(x``)|^(-n)
proof
  defpred P[Nat] means x |^(-$1) =(x``)|^(-$1);
A1: now
    let n;
    assume P[n];
    set m=-(n+1);
    x |^ m = BCI-power(X).(x`,|.m.|) by Def2
      .= BCI-power(X).((x``)`,|.m.|) by BCIALG_1:8
      .= (x``) |^(-(n+1)) by Def2;
    hence P[n+1];
  end;
  x |^(0) =0.X by Def1
    .=(x``)|^(0) by Def1;
  then
A2: P[0];
  for n holds P[n] from NAT_1:sch 2(A2,A1);
  hence thesis;
end;
