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 Th19:
  for d be BiFunction of A,L for q be Element of [:A,A,the carrier
  of L,the carrier of L:] holds d c= new_bi_fun(d,q)
proof
  let d be BiFunction of A,L;
  let q be Element of [:A,A,the carrier of L,the carrier of L:];
  set g = new_bi_fun(d,q);
A1: A c= new_set A by XBOOLE_1:7;
A2: for z being object st z in dom d holds d.z = g.z
  proof
    let z be object;
    assume
A3: z in dom d;
    then consider x,y being object such that
A4: [x,y] = z by RELAT_1:def 1;
    reconsider x9 = x, y9 = y as Element of A by A3,A4,ZFMISC_1:87;
    d.[x,y] = d.(x9,y9) .= g.(x9,y9) by Def10
      .= g.[x,y];
    hence thesis by A4;
  end;
  dom d = [:A,A:] & dom g = [:new_set A,new_set A:] by FUNCT_2:def 1;
  then dom d c= dom g by A1,ZFMISC_1:96;
  hence thesis by A2,GRFUNC_1:2;
end;
