theorem
  [.a |^ b",b.] = [.b",a.]
proof
  thus [.a |^ b",b.] = (a" |^ b") * b" * (a |^ b") * b by GROUP_3:26
    .= (a" |^ b") * (b" * (b"" * a * b")) * b by GROUP_1:def 3
    .= (a" |^ b") * (b" * (b"" * (a * b"))) * b by GROUP_1:def 3
    .= (a" |^ b") * (a * b") * b by GROUP_3:1
    .= (a" |^ b") * ((a * b") * b) by GROUP_1:def 3
    .= [.b",a.] by GROUP_3:1;
end;
