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 Th13:
  a|^i in AtomSet(X)
proof
  defpred P[Integer] means a |^ $1 in AtomSet(X);
  0.X in AtomSet(X);
  then
A1: P[0] by Def1;
  per cases;
  suppose
A2: i>=0;
A3: for i2 be Integer st i2>=0 holds P[i2] implies P[i2 + 1]
    proof
      let i2 be Integer;
      assume i2>=0;
      then reconsider j=i2 as Element of NAT by INT_1:3;
      (a|^(j+1))``=(a\(a |^ j )`)`` by Def1;
      then (a|^(j+1))``=(a`\(a |^ j )``)` by BCIALG_1:9;
      then (a|^(j+1))``=a``\(a |^ j)``` by BCIALG_1:9;
      then
A4:   (a|^(j+1))``=a\(a |^ j)``` by BCIALG_1:29;
      assume P[i2];
      then (a|^(j+1))``=a\(a |^ j)` by A4,BCIALG_1:29;
      then (a|^(j+1))``=a |^ (j+1) by Def1;
      hence thesis by BCIALG_1:29;
    end;
    for i st i>=0 holds P[i] from INT_1:sch 2(A1,A3);
    hence thesis by A2;
  end;
  suppose
A5: i<=0;
A6: for i2 be Integer st i2<=0 holds P[i2] implies P[i2 - 1]
    proof
      let i2 be Integer;
      assume
A7:   i2<=0;
      assume
A8:   P[i2];
      per cases by A7;
      suppose
A9:     i2=0;
        (a`)``=a` by BCIALG_1:8;
        then a` in AtomSet(X) by BCIALG_1:29;
        hence thesis by A9,Th5;
      end;
      suppose
A10:    i2<0;
        set j=i2;
        reconsider m=-j as Element of NAT by A10,INT_1:3;
        a|^(j-1)=(BCI-power(X).(a`,|.j-1.|)) by A10,Def2;
        then a|^(j-1)=(BCI-power(X).(a`,-(j-1))) by A10,ABSVALUE:def 1;
        then a|^(j-1)=a`|^(m+1);
        then a|^(j-1)=a`\(a` |^ m )` by Def1;
        then a|^(j-1)=a`\(a |^ --j )` by Th10;
        then (a|^(j-1))``=(a``\(a |^ j )``)` by BCIALG_1:9;
        then (a|^(j-1))``=(a\(a |^ j )``)` by BCIALG_1:29;
        then (a|^(j-1))``=(a\(a |^ j ))` by A8,BCIALG_1:29;
        then (a|^(j-1))``=a`\(a |^ -m )` by BCIALG_1:9;
        then (a|^(j-1))``=a`\(a` |^ m )` by Th10;
        then (a|^(j-1))``=a` |^ (m+1) by Def1;
        then (a|^(j-1))``=(BCI-power(X).(a`,-(j-1)));
        then (a|^(j-1))``=(BCI-power(X).(a`,|.j-1.|)) by A10,ABSVALUE:def 1;
        then (a|^(j-1))``=a|^(j-1) by A10,Def2;
        hence thesis by BCIALG_1:29;
      end;
    end;
    for i st i<=0 holds P[i] from INT_1:sch 3(A1,A6);
    hence thesis by A5;
  end;
end;
