reserve n for Element of NAT;
reserve i for Integer;
reserve G,H,I for Group;
reserve A,B for Subgroup of G;
reserve N for normal Subgroup of G;
reserve a,a1,a2,a3,b,b1 for Element of G;
reserve c,d for Element of H;
reserve f for Function of the carrier of G, the carrier of H;
reserve x,y,y1,y2,z for set;
reserve A1,A2 for Subset of G;
reserve N for normal Subgroup of G;
reserve S,T1,T2 for Element of G./.N;
reserve g,h for Homomorphism of G,H;
reserve h1 for Homomorphism of H,I;

theorem
  g.(a |^ i) = (g.a) |^ i
proof
  per cases;
  suppose
A1: i >= 0;
    hence g.(a |^ i) = g.(a |^ |.i.|) by ABSVALUE:def 1
      .= (g.a) |^ |.i.| by Th36
      .= (g.a) |^ i by A1,ABSVALUE:def 1;
  end;
  suppose
A2: i < 0;
    hence g.(a |^ i) = g.(a |^ |.i.|)" by GROUP_1:30
      .= (g.(a |^ |.i.|))" by Th32
      .= ((g.a) |^ |.i.|)" by Th36
      .= (g.a) |^ i by A2,GROUP_1:30;
  end;
end;
