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 Th10:
  a` |^ n = a |^ -n
proof
  per cases;
  suppose
A1: n=0;
    hence a` |^ n = 0.X by Def1
      .= a|^ -n by A1,Th3;
  end;
  suppose
A2: n>0;
    set m=-n;
    -(-n)>0 by A2;
    then
A3: m<0;
    then a|^m = BCI-power(X).(a`,|.m.|) by Def2
      .=BCI-power(X).(a`,-(-n)) by A3,ABSVALUE:def 1;
    hence thesis;
  end;
end;
