theorem
  [.a",b".] = [.a,b.] |^ (a * b)" & [.a",b".] = [.a,b.] |^ (b * a)"
proof
  thus [.a",b".] = [.b",a.] |^ a" by Th27
    .= ([.a,b.] |^ b") |^ a" by Th27
    .= [.a,b.] |^ (b" * a") by GROUP_3:24
    .= [.a,b.] |^ (a * b)" by GROUP_1:17;
  thus [.a",b".] = [.b,a".] |^ b" by Th28
    .= [.a,b.] |^ a" |^ b" by Th28
    .= [.a,b.] |^ (a" * b") by GROUP_3:24
    .= [.a,b.] |^ (b * a)" by GROUP_1:17;
end;
