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
  a = h implies a |^ i = h |^ i
proof
  assume
A1: a = h;
  now
    per cases;
    suppose
A2:   i >= 0;
      hence a |^ i = a |^ |.i.| by ABSVALUE:def 1
        .= h |^ |.i.| by A1,Th1
        .= h |^ i by A2,ABSVALUE:def 1;
    end;
    suppose
A3:   i < 0;
A4:   a |^ |.i.| = h |^ |.i.| by A1,Th1;
      thus a |^ i = (a |^ |.i.|)" by A3,GROUP_1:30
        .= (h |^ |.i.|)" by A4,GROUP_2:48
        .= h |^ i by A3,GROUP_1:30;
    end;
  end;
  hence thesis;
end;
