
theorem Th10:
  for A being non empty set for L being lower-bounded LATTICE 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_fun2(d,q) is zeroed
proof
  let A be non empty set;
  let L be lower-bounded LATTICE;
  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_fun2(d,q);
  for u being Element of new_set2 A holds f.(u,u) = Bottom L
  proof
    let u be Element of new_set2 A;
A2: u in A or u in {{A},{{A}}} by XBOOLE_0:def 3;
    per cases by A2,TARSKI:def 2;
    suppose
      u in A;
      then reconsider u9 = u as Element of A;
      thus f.(u,u) = d.(u9,u9) by Def4
        .= Bottom L by A1;
    end;
    suppose
      u = {A} or u = {{A}};
      hence thesis by Def4;
    end;
  end;
  hence thesis;
end;
