reserve G for strict Group,
  a,b,x,y,z for Element of G,
  H,K for strict Subgroup of G,
  p for Element of NAT,
  A for Subset of G;
reserve G for Group;
reserve H, B, A for Subgroup of G,
  D for Subgroup of A;

theorem Th22:
  for a being Element of G st a in H holds
  for j being Integer holds a |^ j in H
proof
  let a be Element of G;
  assume
A1: a in H;
  let j be Integer;
  consider k being Nat such that
A2: j = k or j = - k by INT_1:2;
  per cases by A2;
  suppose
A3: j =k;
    defpred P[Nat] means a |^ $1 in H;
    a |^0 = 1_G by GROUP_1:25;
    then
A4: P[ 0 ] by GROUP_2:46;
A5: for n being Nat st P[n] holds P[n + 1]
    proof
      let n be Nat;
      assume P[n];
      then (a |^ n) * a in H by A1,GROUP_2:50;
      hence thesis by GROUP_1:34;
    end;
    for n being Nat holds P[n] from NAT_1:sch 2(A4,A5);
    hence thesis by A3;
  end;
  suppose
A6: j = - k;
    defpred PP[Nat] means a |^ (- $1) in H;
    a |^0 = 1_G by GROUP_1:25;
    then
A7: PP[ 0 ] by GROUP_2:46;
A8: for n being Nat st PP[n] holds PP[n +1 ]
    proof
      let n be Nat such that
A9:   PP[n];
      a" in H by A1,GROUP_2:51;
      then a |^(-n) * a" in H by A9,GROUP_2:50;
      then a |^(-n) * a |^ (-1) in H by GROUP_1:32;
      then a |^(- n + (- 1)) in H by GROUP_1:33;
      hence thesis;
    end;
    for n being Nat holds PP[n] from NAT_1:sch 2(A7,A8);
    hence thesis by A6;
  end;
end;
