reserve i1 for Element of INT;
reserve j1,j2,j3 for Integer;
reserve p,s,k,n for Nat;
reserve x,y,xp,yp for set;
reserve G for Group;
reserve a,b for Element of G;
reserve F for FinSequence of G;
reserve I for FinSequence of INT;

theorem Th10:
  for G being finite Group, a being Element of G holds
  (a|^n)" = a|^(card G - (n mod card G))
proof
  let G be finite Group;
  let a be Element of G;
  set q = card G;
  set p9 = n mod q;
  n mod q <=q by NAT_D:1;
  then reconsider q9=q -( n mod q) as Element of NAT by INT_1:5;
  a|^n*(a|^(q-(n mod q)))=a|^(n+q9) by GROUP_1:33
    .= a|^((q*(n div q)+(n mod q))+q9) by NAT_D:2
    .= a|^(q*(n div q)+q +(-p9+p9))
    .= a|^(q*(n div q)+q)*(a|^(-p9+p9)) by GROUP_1:33
    .= a|^(q*(n div q)+q)*(a|^(-p9)*(a|^p9)) by GROUP_1:33
    .= a|^(q*(n div q)+q)*((a|^p9)"*(a|^p9)) by GROUP_1:36
    .= a|^(q*(n div q)+q)*(1_G) by GROUP_1:def 5
    .= a|^(q*(n div q+1)) by GROUP_1:def 4
    .= a|^q|^(n div q+1) by GROUP_1:35
    .= (1_G)|^(n div q+1) by Th9
    .= 1_G by GROUP_1:31;
  hence thesis by GROUP_1:12;
end;
