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 Th24:
  for G being strict Group for a being Element of G st G = gr{a}
  for H being strict Subgroup of G st H <> (1).G holds
  ex k being Nat st 0 < k & a |^k in H
proof
  let G be strict Group;
  let a be Element of G such that
A1: G = gr{a};
  let H be strict Subgroup of G;
  assume H <> (1).G;
  then
A2: H <> (1).H by GROUP_2:63;

  consider c being Element of H such that
A3: c <> 1_H by A2,Th23;
A4: c in H;
  then c in G by GROUP_2:40;
  then reconsider c as Element of G;
A5: c in gr{a} by A1;
  ex j being Integer st c = a |^j & j <> 0
  proof
    assume
A6: not ex j being Integer st c = a |^j & j <> 0;
    consider i being Integer such that
A7: c = a |^ i by A5,GR_CY_1:5;
A8: a |^i <> 1_G by A3,A7,GROUP_2:44;
    i = 0 by A6,A7;
    hence contradiction by A8,GROUP_1:25;
  end;
  then consider j being Integer such that
A9: c = a |^ j and
A10: j <> 0;
  consider n being Nat such that
A11: j = n or j = -n by INT_1:2;
  per cases by A11;
  suppose
    j = n;
    hence thesis by A4,A9,A10;
  end;
  suppose
A12: j = - n;
    then
A13: 0 < n by A10;
    (a |^ n)" in H by A4,A9,A12,GROUP_1:36;
    then (a |^ n)"" in H by GROUP_2:51;
    hence thesis by A13;
  end;
end;
