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;
reserve L for Lattice;
reserve a,b,c,u,v for Element of L;

theorem Th22:
  the L_join of L is_distributive_wrt the L_meet of L implies L is distributive
proof
  assume
A1: the L_join of L is_distributive_wrt the L_meet of L;
  then
A2: for a,b,c holds a "\/" (b "/\" c) = (a "\/" b) "/\" (a "\/" c)by BINOP_1:12
;
  let a,b,c;
  thus a"/\"(b"\/"c) = a"/\"(c"\/"a)"/\"(c"\/"b) by LATTICES:def 9
    .= a"/\"((c"\/"a)"/\"(c"\/"b)) by LATTICES:def 7
    .= a"/\"((a"/\"b)"\/"c) by A2
    .= ((a"/\"b)"\/"a)"/\"((a"/\"b)"\/"c) by LATTICES:def 8
    .= (a"/\"b)"\/"(a"/\"c) by A1,BINOP_1:12;
end;
