reserve L for D_Lattice;
reserve a, b, c for Element of L;

theorem
  (a"\/"b)"/\"(b"\/"c)"/\"(c"\/"a) = (a"/\"b)"\/"(b"/\"c)"\/"(c"/\"a)
proof
  thus (a"\/"b)"/\"(b"\/"c)"/\"(c"\/"a) = (((a"\/"b)"/\"(b"\/"c))"/\"c)"\/"(((
  a"\/"b)"/\"(b"\/"c))"/\"a) by Def11
    .= ((a"\/"b)"/\"((b"\/"c)"/\"c))"\/"(((a"\/"b)"/\"(b"\/"c))"/\"a) by Def7
    .= ((a"\/"b)"/\"c)"\/"(a"/\"((a"\/"b)"/\"(b"\/"c))) by Def9
    .= ((a"\/"b)"/\"c)"\/"((a"/\"(a"\/"b))"/\"(b"\/"c)) by Def7
    .= (c"/\"(a"\/"b))"\/"(a"/\"(b"\/"c)) by Def9
    .= ((c"/\"a)"\/"(c"/\"b))"\/"(a"/\"(b"\/"c)) by Def11
    .= ((a"/\"b)"\/"(c"/\"a))"\/"((c"/\"a)"\/"(b"/\"c)) by Def11
    .= (a"/\"b)"\/"((c"/\"a)"\/"((c"/\"a)"\/"(b"/\"c))) by Def5
    .= (a"/\"b)"\/"(((c"/\"a)"\/"(c"/\"a))"\/"(b"/\"c)) by Def5
    .= (a"/\"b)"\/"(b"/\"c)"\/"(c"/\"a) by Def5;
end;
