reserve A for set,
  C for non empty set,
  B for Subset of A,
  x for Element of A,
  f,g for Function of A,C;
reserve B for Element of Fin A;
reserve L for non empty LattStr,
  a,b,c for Element of L;

theorem Th11:
  the L_join of L is commutative associative & the L_meet of L is
commutative associative & the L_join of L absorbs the L_meet of L & the L_meet
  of L absorbs the L_join of L implies L is Lattice-like
proof
  assume that
A1: the L_join of L is commutative and
A2: the L_join of L is associative and
A3: the L_meet of L is commutative and
A4: the L_meet of L is associative and
A5: the L_join of L absorbs the L_meet of L and
A6: the L_meet of L absorbs the L_join of L;
  thus a"\/"b = b"\/"a by A1;
  thus a"\/"(b"\/"c) = (a"\/"b)"\/"c by A2;
  thus (a"/\"b)"\/"b = b
  proof
    thus (a"/\"b)"\/"b = b "\/" (a "/\" b) by A1
      .= b "\/" (b "/\" a) by A3
      .= b by A5;
  end;
  thus a"/\"b = b"/\"a by A3;
  thus a"/\"(b"/\"c) = (a"/\"b)"/\"c by A4;
  let a,b;
  thus thesis by A6;
end;
