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 Th62:
  for H being strict Subgroup of G holds H |^ a |^ a" = H & H |^ a " |^ a = H
proof
  let H be strict Subgroup of G;
  thus H |^ a |^ a" = H |^ (a * a") by Th60
    .= H |^ 1_G by GROUP_1:def 5
    .= H by Th61;
  thus H |^ a" |^ a = H |^ (a" * a) by Th60
    .= H |^ 1_G by GROUP_1:def 5
    .= H by Th61;
end;
