theorem Th52:
  x in sproduct f implies x is PartFunc of dom f, union rng f
proof
  assume x in sproduct f;
  then consider g such that
A1: x = g and
A2: dom g c= dom f and
A3: for x being object st x in dom g holds g.x in f.x by Def9;
  rng g c= union rng f
  proof
    let y be object;
    assume y in rng g;
    then consider z being object such that
A4: z in dom g and
A5: y = g.z by FUNCT_1:def 3;
A6: g.z in f.z by A3,A4;
    f.z in rng f by A2,A4,FUNCT_1:def 3;
    hence thesis by A5,A6,TARSKI:def 4;
  end;
  hence thesis by A1,A2,RELSET_1:4;
end;
