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 Th7:
  0.X |^ n = 0.X
proof
  defpred P[Nat] means (0.X) |^ $1 = 0.X;
A1: now
    let n;
    assume P[n];
    then (0.X) |^ (n + 1) = ((0.X)`)` by Th2
      .=(0.X)` by BCIALG_1:def 5
      .= 0.X by BCIALG_1:def 5;
    hence P[n+1];
  end;
A2: P[0] by Def1;
  for n holds P[n] from NAT_1:sch 2(A2,A1);
  hence thesis;
end;
