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 Th20:
  the L_join of L is_distributive_wrt the L_join of L
proof
  now
    let a,b,c;
    thus (the L_join of L).(a,(the L_join of L).(b,c)) = a "\/" (b "\/" c)
      .= a "\/" b "\/" c by LATTICES:def 5
      .= a "\/" a "\/" b "\/" c
      .= (a "\/" b) "\/" a "\/" c by LATTICES:def 5
      .= (a "\/" b) "\/" (a "\/" c) by LATTICES:def 5
      .= (the L_join of L).((the L_join of L).(a,b),(the L_join of L).(a,c));
  end;
  hence thesis by BINOP_1:12;
end;
