
theorem Th1:
  for L being Lattice holds (for a,b,c being Element of L holds a
  "/\"(b"\/"c) = (a"/\"b)"\/"(a"/\"c)) iff for a,b,c being Element of L holds a
  "\/"(b"/\"c) = (a"\/"b)"/\"(a"\/"c)
proof
  let L be Lattice;
  hereby
    assume
A1: for a,b,c be Element of L holds a"/\"(b"\/"c)=(a"/\"b)"\/"(a"/\"c);
    let a,b,c be Element of L;
    thus a"\/"(b"/\"c)=(a"\/"(c"/\"a))"\/"(c"/\"b) by Def8
      .=a"\/"((c"/\"a)"\/"(c"/\"b)) by Def5
      .=a"\/"((a"\/"b)"/\"c) by A1
      .=((a"\/"b)"/\"a)"\/"((a"\/"b)"/\"c) by Def9
      .=(a"\/"b)"/\"(a"\/"c) by A1;
  end;
  assume
A2: for a,b,c be Element of L holds a"\/"(b"/\"c)=(a"\/"b)"/\"(a"\/"c);
  let a,b,c be Element of L;
  thus a"/\"(b"\/"c)=(a"/\"(c"\/"a))"/\"(c"\/"b) by Def9
    .=a"/\"((c"\/"a)"/\"(c"\/"b)) by Def7
    .=a"/\"((a"/\"b)"\/"c) by A2
    .=((a"/\"b)"\/"a)"/\"((a"/\"b)"\/"c) by Def8
    .=(a"/\"b)"\/"(a"/\"c) by A2;
end;
