
theorem
  for L being Lattice holds (for a,b,c being Element of L holds (a"/\"b)
  "\/"(b"/\"c)"\/"(c"/\"a) = (a"\/"b)"/\"(b"\/"c)"/\"(c"\/"a)) implies L is
  distributive
proof
  let L be Lattice;
  assume
A1: for a,b,c being Element of L holds (a"/\"b)"\/"(b"/\"c)"\/"(c"/\"a)
  = (a"\/"b)"/\"(b"\/"c)"/\"(c"\/"a);
A2: now
    let a,b,c be Element of L;
    assume
A3: c [= a;
    thus a"/\"(b"\/"c) = (b"\/"c)"/\"(a"/\"(a"\/"b)) by Def9
      .= (a"\/"b)"/\"((b"\/"c)"/\"a) by Def7
      .= (a"\/"b)"/\"((b"\/"c)"/\"(c"\/"a)) by A3
      .= (a"\/"b)"/\"(b"\/"c)"/\"(c"\/"a) by Def7
      .= (a"/\"b)"\/"(b"/\"c)"\/"(c"/\"a) by A1
      .= (a"/\"b)"\/"((b"/\"c)"\/"(c"/\"a)) by Def5
      .= (a"/\"b)"\/"((b"/\"c)"\/"c) by A3,Th2
      .= (a"/\"b)"\/"c by Def8;
  end;
  let a,b,c be Element of L;
A4: ((a"/\"b)"\/"(c"/\"a))"\/"a = (a"/\"b)"\/"((c"/\"a)"\/"a) by Def5
    .= (a"/\"b)"\/"a by Def8
    .= a by Def8;
  thus a"/\"(b"\/"c) = (a"/\"(c"\/"a))"/\"(b"\/"c) by Def9
    .= a"/\"((c"\/"a)"/\"(b"\/"c)) by Def7
    .= (a"/\"(a"\/"b))"/\"((c"\/"a)"/\"(b"\/"c)) by Def9
    .= a"/\"((a"\/"b)"/\"((b"\/"c)"/\"(c"\/"a))) by Def7
    .= a"/\"((a"\/"b)"/\"(b"\/"c)"/\"(c"\/"a)) by Def7
    .= a"/\"((b"/\"c)"\/"(a"/\"b)"\/"(c"/\"a)) by A1
    .= a"/\"((b"/\"c)"\/"((a"/\"b)"\/"(c"/\"a))) by Def5
    .= (a"/\"(b"/\"c))"\/"((a"/\"b)"\/"(c"/\"a)) by A2,A4,Def3
    .= (a"/\"b"/\"c)"\/"((a"/\"b)"\/"(c"/\"a)) by Def7
    .= ((a"/\"b"/\"c)"\/"(a"/\"b))"\/"(c"/\"a) by Def5
    .= (a"/\"b)"\/"(a"/\"c) by Def8;
end;
