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;

theorem Th5:
  a * A * A = a * A & a * (A * A) = a * A &
  A * A * a = A * a & A * (A * a) = A * a
proof
  thus a * A * A = a * (A * A) by GROUP_4:45
    .= a * A by GROUP_2:76;
  hence a * (A * A) = a * A by GROUP_4:45;
  thus A * A * a = A * a by GROUP_2:76;
  hence thesis by GROUP_4:46;
end;
