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 Th54:
  A |^ a |^ a" = A & A |^ a" |^ a = A
proof
  thus A |^ a |^ a" = A |^ (a * a") by Th47
    .= A |^ 1_G by GROUP_1:def 5
    .= A by Th52;
  thus A |^ a" |^ a = A |^ (a" * a) by Th47
    .= A |^ 1_G by GROUP_1:def 5
    .= A by Th52;
end;
