reserve x,y,X,Y for set,
  k,l,n for Nat,
  i,i1,i2,i3,j for Integer,
  G for Group,
  a,b,c,d for Element of G,
  A,B,C for Subset of G,
  H,H1,H2, H3 for Subgroup of G,
  h for Element of H,
  F,F1,F2 for FinSequence of the carrier of G,
  I,I1,I2 for FinSequence of INT;

theorem Th4:
  a in H implies a |^ i in H
proof
  assume
A1: a in H;
  now
    per cases;
    suppose
      i >= 0;
      then a |^ i = a |^ |.i.| by ABSVALUE:def 1;
      hence thesis by A1,Th3;
    end;
    suppose
      i < 0;
      then
A2:   a |^ i = (a |^ |.i.|)" by GROUP_1:30;
      a |^ |.i.| in H by A1,Th3;
      hence thesis by A2,GROUP_2:51;
    end;
  end;
  hence thesis;
end;
