reserve X for non empty set;
reserve e,e1,e2,e19,e29 for Equivalence_Relation of X,
  x,y,x9,y9 for set;
reserve A for non empty set,
  L for lower-bounded LATTICE;

theorem Th16:
  for d being BiFunction of A,L st d is zeroed for q being Element
  of [:A,A,the carrier of L,the carrier of L:] holds new_bi_fun(d,q) is zeroed
proof
  let d be BiFunction of A,L;
  assume
A1: d is zeroed;
  let q be Element of [:A,A,the carrier of L,the carrier of L:];
  set f = new_bi_fun(d,q);
  for u being Element of new_set A holds f.(u,u) = Bottom L
  proof
    let u be Element of new_set A;
A2: u in A or u in {{A},{{A}},{{{A}}}} by XBOOLE_0:def 3;
    per cases by A2,ENUMSET1:def 1;
    suppose
      u in A;
      then reconsider u9 = u as Element of A;
      thus f.(u,u) = d.(u9,u9) by Def10
        .= Bottom L by A1;
    end;
    suppose
      u = {A} or u = {{A}} or u = {{{A}}};
      hence thesis by Def10;
    end;
  end;
  hence thesis;
end;
