reserve x,y,y1,y2 for set;
reserve G for Group;
reserve a,b,c,d,g,h for Element of G;
reserve A,B,C,D for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve n for Nat;
reserve i for Integer;

theorem
  (a |^ i) |^ b = (a |^ b) |^ i
proof
  per cases;
  suppose
    i >= 0;
    then i = |.i.| by ABSVALUE:def 1;
    hence thesis by Lm4;
  end;
  suppose
A1: i < 0;
    hence a |^ i |^ b = (a |^ |.i.|)" |^ b by GROUP_1:30
      .= (a |^ |.i.| |^ b)" by Th26
      .= ((a |^ b) |^ |.i.|)" by Lm4
      .= (a |^ b) |^ i by A1,GROUP_1:30;
  end;
end;
